factory physics post

Modeling and Analysis of High Volume Manufacturing

Systems

Discrete vs. Continuous Flow and Repetitive Manufacturing Systems

(Figures borrowed from Heizer and Render)

Operation Process Chart Example for discrete part manufacturing(borrowed from Francis et. al.)

A typical (logical) Organization of the Production Activity in

Repetitive Manufacturing

RawMaterial& Comp.Inventory

FinishedItem

Inventory

S1,2S1,1 S1,n

S2,1 S2,2 S2,m

Assembly Line 1: Product Family 1

Assembly Line 2: Product Family 2

Fabrication (or Backend Operations)

Dept. 1 Dept. 2 Dept. k

S1,i

S2,i

Dept. j

Major Layout Types(borrowed from Francis et. al.)

Advantages and Limitations of the various layout types (borrowed from Francis et. al.)

Advantages and Limitations of the various layout types (cont. - borrowed from Francis et. al.)

The product-process matrix

Jumbledflow (jobShop)

Disconnectedline flow(batch)

Connectedline flow(assemblyLine)

Continuousflow (chemicalplants)

Processtype

Production volume & mix

Low volume,low standardi-zation

Multiple products,low volume

Few major products,high volume

High volume, highstandardization,commodities

Commercialprinter

HeavyEquipment

Autoassembly

Sugarrefinery

Void

Void

A conceptual characterization of the considered environments

• Flow line: A sequence of workstations supporting the production of a single part type.

• Each workstation consists of one or more identical servers executing one particular stage of the entire production process.

• The part processing time at each workstation follows some general distribution which must be defined in such a way that accounts for the various detractors affecting the station operations; these detractors will include machine downtime, lack of consumables, operator unavailability, experienced set-up times, preventive maintenance, etc.

• Finished parts could constitute end items or raw material for some other downstream process.

• The operation of the line workstations can be – synchronized through an interconnecting conveyor system, or– decoupled through the installation of some buffering capacity between

them.

Line Performance Measures

• Production rate or throughput, i.e., the number of parts produced per unit time

• Line capacity, i.e., the maximum sustainable production rate

• Line (expected) cycle time, i.e., the average time that is spend by any part into the line (this quantity includes both, processing and waiting time).

• Average Work-In-Porcess (WIP) accumulated at different stations

• Expected utilization of the station servers.

Remark: The above performance measures essentially provide a link between the directly quantifiable and manageable aspects and attributes of the considered operational environments and the primary strategic concerns of the company, especially those of responsiveness and cost efficiency.

Production Authorization Mechanisms• The issue here is to what extent part production is triggered from actual orders or from

forecasted demand.• In a “produce-to-stock” scheme, a certain amount of end-item inventory is maintained in

an effort to serve the experienced demand with zero lead time.• “Produce-to-stock” operation is most appropriate for highly commoditized and

standardized items.• A key performance measure for “produce-to-stock” production systems is the fill rate,

i.e. the percentage of the experienced demand that is actually met from stock. A desired fill rate is attained by maintaining an appropriate safety stock level, that is computed from the statistics of the experienced demand and cycle times.

• In a “produce-to-order” scheme, end items are produced in response to particular orders. • “Produce-to-order” operation is more appropriate for (highly) customized items.• A key performance measure for “produce-to-order” production is the attained service

level, i.e. the percentage of orders that are served within the quoted lead time. A target service level is attained through informed lead time quotation and some appropriate “capacity buffer”.

• Currently, many systems are a hybrid scheme consisting of “produce-to-stock” and “produce-to-order” components. In particular, mass customization is supported by an “assemble-to-order” scheme where end-items are assembled to order from a number of sub-assemblies that are produced to stock.

Shop-Floor / Line Control Mechanisms• Mechanisms that control the part release and advancement through the line.• They are broadly distinguished into “push” and “pull” mechanisms.• A “push” mechanism releases material into the line according to a target

production rate, and material is advanced to downstream stations as early as possible.

• Typical instantiations of “push” systems are: the asynchronous transfer line and the synchronous transfer line.

• A “pull” system controls the part release and advancement in the line taking into consideration the status of the various workstations in the line.

• Typical instantiations of “pull” systems are: the KANBAN and the CONWIP (controlled) production lines.

• In general, “pull” systems reduce congestion, since they take into consideration the actual shop-floor status in their decision making, but the same mechanisms will also make them more inert in case of shifts in the production level.

• Both mechanisms are effectively implementable in a “produce-to-stock” or “produce-to-order” context.

Asynchronous Transfer LinesW1 W2 W3

TH THTH THB1 B2 B3M1 M2 M3

Some important issues:• What is the maximum throughput that is sustainable through this line?• What is the expected cycle time through the line?• What is the expected WIP at the different stations of the line?• What is the expected utilization of the different machines?• How does the adopted batch size affect the performance of the line?• How do different detractors, like machine breakdowns, setups, and maintenance, affect the performance of the line?

Synchronous Transfer LinesThe key issue: Assembly Line Balancing (ALB)Given• a set of tasks to be supported by the line stations• each possessing a nominal processing time,• a number of precedence constraints among these tasks,• and a target throughput,determine• a partitioning of these tasks to a number of stations that observes the aforementioned specifications, while it minimizes the resulting number of stations (and therefore, the resulting labor cost).

KANBAN-based production linesStation 1 Station 2 Station 3

Some important issues:• What is the throughput attainable by a certain selection of KANBAN levels?• What is the resulting cycle time?• How do we select the KANBAN levels that will attain a desired production rate?• How do we introduce the various operational detractors into the model?

CONWIP-based production lines

Station 1 Station 2 Station 3 FGI

Some important issues:• Same as those for the KANBAN model, plus • How can we compare the performance of such a system to that of an asynchronous transfer line and/or a KANBAN-based system?

Module Objectives

• Provide an analytical characterization of the operation of HVM systems and their performance, based on queueing-theoretic models

• Derive qualitative insights and quantitative results on the attributes and factors that shape the behavior and performance of these systems.

• Demonstrate the application of these results to the design and control of the considered class of systems

• Focus primarily on flow lines, since they are the main layout used in the context of high-volume, repetitive manufacturing.

• Also, flow line dynamics are easier to trace and analyze, and therefore, more enlightening in terms of qualitative and quantitative insights.

• However, many of the derived insights and results are extensible to more complex environments either directly or through some appropriate decomposition.

Plan for the remaining part of the module

• Modeling and Performance Analysis of Asynchronous Transfer Lines through a Series of G/G/m queues

• Modeling the impact of operational detractors• Employing the above results in line diagnostics• Design of Asynchronous Transfer Lines• Design of Synchronous Transfer Lines (cf. the ALB problem in the module

on Sequencing, Dispatching and Scheduling of HVM systems)• Modeling and Performance Analysis of CONWIP-based production lines

through Closed Queueing Networks• An integrating framework for bounding and shaping the performance of a

production line• Analyzing the impact of batching on the system performance and designing

optimized batching policies• Understanding the relative advantages and disadvantages of the various

“push” and “pull”-based production systems

The G/G/1 model

Modeling Assumptions: • Part release rate = Target throughput rate = TH • Infinite Buffering Capacity• m identical servers• Server mean processing time = te

• St. deviation of processing time = e

• Coefficient of variation (CV) of processing time: ce = e / te

• Coefficient of variation of inter-arrival times = ca

M1

BTH TH

M2

Mm

Performance measures for the G/G/1 station

• Server utilization:

• Expected cycle time in the buffer: (Kingman’s approx.)

• Expected cycle time in the station:

• Average WIP in the buffer: (by Little’s law)

• Average WIP in the station:

• Squared CV of the inter-departure times:

M1

BTH TH

M2

Mm

€

u = (TH ⋅ te ) m

€

CTq ≅ca

2 + ce2

2

u 2(m +1)−1

m(1− u)te

€

CT = CTq + te

€

WIPq = TH ⋅CTq

€

WIP = TH ⋅CT = WIPq + m ⋅u

€

cd2 ≅1+ (1− u2)(ca

2 −1) +u2

m(ce

2 −1)

Some Important Remarks• The entire analysis of the previous slide is an approximation, since it derives from the

provided approximation for CTq.

• The provided formula for CTq is exact for the M/M/m and M/G/1 stations.

• The station operation will be stable only if u = TH te / m < 1.0, i.e., only if the effective workload released into the station per unit of time is less than the available processing time of the station. Otherwise WIPq, CTq and CT will grow infinitely large.

• Notice that the requirement u < 1.0 is also suggested by the provided expression for CTq since only then CTq takes finite positive values.

• The expression for CTq comprises three factors, each depending respectively upon (i) the variability of the processing and inter-arrival times, as expressed by the corresponding SCV’s, (ii) the server utilization, and (iii) the mean processing time. An increase of any of these factors results in an increase of CTq, and also of CT, WIPq and WIP.

• Assuming a non-zero mean processing time, CTq=0.0 only if ca2= ce

2=0.0; these are essentially the conditions of a paced / synchronous production line.

• The SCV of the inter-departure times can be approximated as a linear function of the SCVs of the inter-arrival and processing times. Furthermore, for u1.0, cd

2 depends primarily on ce

2, while for u0, cd2 depends primarily on ca

2.

Analyzing an entire Production Line

TH

Key observations:• A target production rate TH is achievable only if each station satisfies the stability requirement u < 1.0.

• For a stable system, the average production rate of every station will be equal to TH.

• For every pair of stations, the inter-departure times of the first constitute the inter-arrival times of the second.

• Then, the entire line can be evaluated on a station by station basis, working from the first station to the last, and using the equations for the basic G/G/1 model.

Modeling the impact of operational detractors• Effective processing time = time that the part occupies the server• Effective processing time = Actual processing time +

any additional non-processing time• Actual or otherwise natural processing time typically presents fairly low

variability ( SCV < 1.0). • Non-processing time is due to detractors like machine breakdowns, setups,

operator unavailability, lack of consumables, etc.• Detractors are distinguished to preemptive and non-preemptive. Each of these

categories requires a different analytical treatment.• Preemptive detractors are outages that take place during the actual processing

of the part. Typical examples are machine breakdowns, lack of consumables, operator unavailability, etc.

• Non-preemptive detractors are activities that may take place between the processing of two consecutive parts. Typical examples are setups, preventive maintenance, operator breaks, etc.

• We want to determine the mean, variance and SCV of the effective processing time from the corresponding attributes of the natural processing time and some additional attributes characterizing the behavior of the various detractors.

Modeling the impact of preemptive detractors• X = random variable modeling the natural processing time, following a general

distribution.

• to = E[X]; o2=Var[X]; co=o / to .

• T = random variable modeling the effective processing time = where

• Ui = random variable modeling the duration of the i-th outage, following a general distribution, and

• N = random variable modeling the number of outages during a the processing of a single part.

• mr=E[Ui]; r2=Var[Ui]; cr = r / mr

• Time between outages is exponentially distributed with mean mf.

• Availability A = mf / (mf+mr) = percentage of time the system is up.

• Then,

te = E[T] = to / A or equivalently re = 1/te = A (1/to) = A ro

€

X + U ii=1

N

∑

€

e2 = Var[T] = (σ o

2 A2) + to((mr2 + σ r

2) m f )

€

ce2 = σ e

2 / te2 = co

2 + (1+ cr2)A(1− A)(mr / to)

Breakdown Example

• Data: Injection molding machine has:

– 15 second stroke (to = 15 sec)

– 1 second standard deviation (o = 1 sec)

– 8 hour mean time to failure (mf = 28800 sec)

– 1 hour repair time (mr = 3600 sec)

• Natural variabilityco = 1/15 = 0.067 (which is very low)

Example Continued

• Effective variability:

41.4715

3600)888.01)(888.0(2)067.0()1(2

875.16888.0/15/

888.018

8

222 =−+=−+=

===

=+

=+

=

o

roe

oe

rf

f

tm

AAcc

Att

mmm

A

Which is very high!

Example Continued

• Suppose through a preventive maintenance program, we can reduce mf to 8 min and mr to 1 min

79.015

60)888.01)(888.0(2)067.0()1(2

875.16888.0/15/

888.018

8

222 =−+=−+=

===

=+

=+

=

o

roe

oe

rf

f

tm

AAcc

Att

mmm

A

Which is low!

(the same as before)

Modeling the impact of non-preemptive detractors

• X = random variable modeling the natural processing time, following a general distribution.

• to = E[X]; o2=Var[X]; co=o / to .

• NS = average number of parts processed between two consecutive setups• It is also assumed that the number of parts between two consecutive setups follows a

geometric distribution, which when combined with the previous bullet, it implies that probability for a setup after any given job = 1/ NS.

• Z = random variable modeling the duration of a setup• tS = E[Z]; S

2 = Var[Z]• S = random variable modeling the setup time experienced by any given job =

• T = random variable modeling the effective processing time = X+S• Then,

E[S] = tS / NS ; Var[S] = (S2 / NS) + tS

2((NS-1) / NS2);

te = E[T] = to+tS / NS ; ;

€

=Z with probability 1/Ns

0 with probability 1-1/N s

⎧ ⎨ ⎩

€

e2 = Var[T] = σ o

2 + σ S2 NS( ) + tS

2 (NS −1) NS2

( )

€

ce2 = σ e

2 / te2

Setup Example

• Data:

– Fast, inflexible machine: (2 hr setup every 10 jobs)

– Slower, flexible machine: (no setups)

0

0

0

0

0

0

1 hour

0.25 hours

10 jobs/setup

2 hours

21 1.2 hours

10

1 10.83 jobs/hour

1.2

1.2 hours

0.6 hours

1 10.83 jobs/hour

1.2

s

s

se

s

ee

e

t

N

t

tt t

N

rt

t

rt

=⎧⎪ =⎪⎪ =⎪

=⎪⎨⎪ = + = + =⎪⎪⎪ = = =⎪⎩

=⎧⎪ =⎪⎨⎪ = = =⎪⎩

No difference!

Setup Example (cont.)• Compare mean and variance

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

–Slower, flexible machine – no setups

• Conclusion:

( ) ( )

( )

00 0

0

2222 2 2 2

0 2 2

22

22

0.25 hours 0.25 0.25

0.251 10 10.25 2 0.4475

10 10

0.44750.3107638

1.2

s

s se s

s s

ee

e

c ct

Nó t

N N

ct

σσ

σσ

σ

= = = =

− −= + + = + + =

= = =

( )( )

222 2 0

0 220

0.60.25

1.2ec c

t

σ= = = =

Flexibility can reduce variability.

Setup Example (cont.)• New Machine: Consider a third machine same as previous machine with

setups, but with shorter, more frequent setups

• Analysis:

• Conclusion:( ) ( )

( )

2222 2 2 2

0 2 2

22

22

5 jobs/setup

1 hour

1 10.83 jobs/hr

1.2

0.251 5 10.25 1 0.235

5 5

0.2350.163194

1.2

s

s

ee

s se s

s s

ee

e

N

t

rt

Nó t

N N

ct

σσ

σ

=

=

= = =

− −= + + = + + =

= = =

Shorter, more frequent setups induce less variability.

Example: Employing the presented results for line diagnostics

M1 B M2

to1 =19 min

co12=0.25

mf1=48 hrsmr1=8 hrs MTTR ~ expon.

to2 =22 min

co22=1.0

mf2=3.3 hrsmr2=10 min MTTR ~ expon.

Ca2=1.0

20parts

Desired throughput is TH = 2.4 jobs / hr but practical experience has shown that it is not attainable by this line. We need to understand why this is not possible.

Diagnostics example continued:Capacity analysis based on mean values

M1 B M2

to1 =19 min

co12=0.25

mf1=48 hrsmr1=8 hrs MTTR ~ expon.

to2 =22 min

co22=1.0

mf2=3.3 hrsmr2=10 min MTTR ~ expon.

Ca2=1.0

20parts

€

A1 = m f 1 /(m f 1 + mr1) = 48 /(48 + 8) = 0.857

te1 = to1 / A1 =19 /0.857 = 22.17min

A2 = m f 2 /(m f 2 + mr2) = 3.3/(3.3 +10 /60) = 0.952

te 2 = to2 / A2 = 22 /0.952 = 23.11min ⇒ Bottleneck machine

Bottleneck utilization : u2 = TH ⋅ te2 = 2.4 ⋅(23.11/60) = 0.9244 <<1.0 (!)

Diagnostics example continued:An analysis based on the G/G/1 model

€

ce12 = co1

2 + (1+ cr12 )A1(1− A1)mr1 / to1 = 0.25 + (1+1) ⋅0.857 ⋅(1− 0.857) ⋅8 ⋅60 /19 = 6.442

u1 = TH ⋅ te1 = 2.4 ⋅22.17 /60 = 0.8868

CTq1 =ca1

2 + ce12

2

u1

1− u1

te1 =1+ 6.442

2

0.8868

1− 0.886822.17 = 646.256min

WIPq1 = TH ⋅CTq1 = 2.4 ⋅646.256 /60 = 25.85

ca 22 = cd1

2 = u12ce1

2 + (1− u12)ca1

2 = 0.88682 ⋅6.442 + (1− 0.88682) ⋅1= 5.28

ce22 = c2 + (1+ cr2

2 )A2(1− A2)mr2 / to2 =1+ (1+1) ⋅0.952 ⋅(1− 0.952) ⋅10 /22 =1.04

u2 = TH ⋅ te2 = 2.4 ⋅23.11/60 = 0.9244

CTq 2 =ca 2

2 + ce22

2

u2

1− u2

te2 =5.28 +1.04

2

0.9244

1− 0.924423.11 = 892.94min

WIPq 2 = TH ⋅CTq 2 = 2.4 ⋅892.94 /60 = 35.72 >> 20 (!)

i.e., the long outages of M1, combined with the inadequate capacity of the interconnecting buffer, starve the bottleneck!

Example: Designing an asynchronous prod. line• Design of a new 4-station assembly line for circuit board assembly.• The four consecutive stations and the currently considered “technology options”

for them (each option defines the processing rate in pieces per hour, the CV of the processing time, and the cost per unit in thousands of dollars).

Station Option 1 Option 2 Option 31 42, 2.0, 50 42, 1.0, 85 10, 2.0, 110.52 42, 2.0, 50 42, 1.0, 853 25, 1.0, 100 25, 0.7, 1204 50, 0.75, 20 6, 0.75, 24

• The above data correspond to the effective processing times.• Each station can employ only one technology option.• The maximum production rate to be supported by the line is 1000 panels / day.• The desired average cycle time through the line is one day.• One day is equivalent to one 8-hour shift.• Workpieces will go through the line in totes of 50 panels each, which will be released into the line at a constant rate determined by the target production rate.• Design task: Identify a line configuration that meets the above requirements while minimizing the equipment cost.• Also, estimate the expected WIP at every station, when the line is operated at maximum production rate.

A baseline design:Meeting the desired prod. rate with a low cost

1000 42,2.0,50 42,2.0,50 25,1.0,100 50,0.75,2050 42,1.0,85 42,1.0,85 25, 0.7,120 6,0.75,248 10,2.0,110.5

(1000 / 50) / 8 2.5

Station 1 Station 2 Station 3 Station 41/te 42 42 25 50Ce 2 2 1 0.75P 50 50 100 20m 3 3 6 3

te 0.0238 0.0238 0.04 0.02tb=B*te 1.1905 1.1905 2 1

Cb^2=Ce^2/B 0.08 0.08 0.02 0.0113u=TH*tb/m 0.9921 0.9921 0.8333 0.8333

Ca^2 0 0.4615 0.4687 0.5598Cd^2 = 1+(1-u^2)(Ca^2-1)+(u^2/sqrt(m))*(Cb^2-1) 0.4615 0.4687 0.5598 0.4691

CT = [(Ca^2+Cb^2)/2]*[u^(sqrt(2*(m+1))-1)/(m*(1-u))]*tb+tb 3.17 14.6 2.3 1.41CT1+CT2+CT3+CT4 21.48

WIPq 4.9 33.5 0.8 1

m*P 150 150 600 60960

Reducing the line cycle time by adding capacity to Station 2

1000 42,2.0,50 42,2.0,50 25,1.0,100 50,0.75,2050 42,1.0,85 42,1.0,85 25, 0.7,120 6,0.75,248 10,2.0,110.5

(1000 / 50) / 8 2.5


te 0.0238 0.0238 0.04 0.02tb=B*te 1.1905 1.1905 2 1

Cb^2=Ce^2/B 0.08 0.08 0.02 0.0113u=TH*tb/m 0.9921 0.7441 0.8333 0.8333



WIPq 4.9 0.4 0.8 1.1

m*P 150 200 600 601010

Adding capacity at Station 1, the new bottleneck

1000 42,2.0,50 42,2.0,50 25,1.0,100 50,0.75,2050 42,1.0,85 42,1.0,85 25, 0.7,120 6,0.75,248 10,2.0,110.5

(1000 / 50) / 8 2.5


te 0.0238 0.0238 0.04 0.02tb=B*te 1.1905 1.1905 2 1

Cb^2=Ce^2/B 0.08 0.08 0.02 0.0113u=TH*tb/m 0.7441 0.7441 0.8333 0.8333



WIPq 0.1 0.3 0.7 1

m*P 200 200 600 601060

An alternative option:Employ less variable machines at Station 1

1000 42,2.0,50 42,2.0,50 25,1.0,100 50,0.75,2050 42,1.0,85 42,1.0,85 25, 0.7,120 6,0.75,248 10,2.0,110.5

(1000 / 50) / 8 2.5


te 0.0238 0.0238 0.04 0.02tb=B*te 1.1905 1.1905 2 1

Cb^2=Ce^2/B 0.02 0.08 0.02 0.0113u=TH*tb/m 0.9921 0.7441 0.8333 0.8333



WIPq 1.2 0.4 0.8 1

m*P 255 200 600 601115

This option is dominated by the previous one since it presents a higher CT andalso a higher deployment cost. However, final selection(s) must be assessed and validated through simulation.

Analyzing CONWIP-based flowlines with single-machine stations as

Closed Queueing Networks (CQNs)Mean Value Analysis - the key underlying ideas: I. A CONWIP-based flow line with single-machine stations and its WIP level

set to W, can be modeled by a closed queueing network (CQN) with general processing distributions, W jobs in it, and the following structure:

II. In a CQN with W jobs and exponential processing times, the expected number of jobs observed at the various stations by a job arriving at some station Sj, is equal to the expected number of jobs observed at any random time at the same stations when the system is operated with W-1 jobs in it.

III. Assuming that this effect applies in an approximate sense for more general distributions of the processing times, we proceed to develop an algorithm that will compute the performance measures of interest iteratively, for various W levels, starting with W=0.

M1 M2 Mn

Notation

n = number of stations

te(j) = mean effective processing time at station j

ce2(j) = SCV for effective processing time at station j

TH(W) = the line throughput when operated with WIP level W

CT(W) = expected job cycle time through the line

CTj(W) = expected job cycle time at station j when the WIP level is W

WIPj(W) = expected WIP level at station j when the WIP level is W

uj(W) = utilization of the server at station j when the WIP level is W

Deriving the algorithm iterationCTj(W) = E[remaining processing time for the job at the server of Sj] +

(E[number of jobs at station Sj]-E[number of jobs in service])te(j) + te(j)

But• E[remaining processing time for the job at the server of Sj] =

Prob(Server of Sj busy)E[remaining process time | busy] =uj(W-1)E[remaining process time | busy]

(b) E[number of jobs at station Sj] WIPj(W-1)

• E[number of jobs in service] uj(W-1)

(d) uj(W-1) = TH(W-1) te(j)

€

u j (W −1)te ( j)(ce

j ( j) +1)

2 (Kleinrock)

Deriving the algorithm iteration (cont.)Combining the results of the previous slide:

But then,

Obviously, for W=0, CT(0) =TH(0) = WIPj(0) = 0Furthermore, application of the above formulae for W=1 gives:

€

CTj (W ) ≈te

2( j)

2(ce

2( j) −1)TH(W −1) + (WIPj (W −1) +1)te ( j)

€

CT(W ) = CTj (W )j=1

n

∑

TH(W ) =W

CT(W )

WIPj (W ) = TH(W ) ⋅CTj (W )

€

CT j (1) = te ( j)

CT(1) = te ( j) ≡ To

j=1

n

∑ (Raw Process Time)

TH(1) =1

CT(1)=

1

To

WIPj (1) =1

To

te ( j) = u j (1)

(from Little’s law)

Bounding the line throughput:Upper bounds

• For W, TH(W)rbminj{1/te(j)}, the bottleneck rate of the line.

• rb can also be achieved with finite WIP in a deterministic setting, i.e., in a line with ce(j) = 0, j, and synchronized with pace tb=1/rb.

• However, by Little’s law, a line with raw process time To, in order to produce at rate rb, will need a WIP level of Wo=rbTo; this WIP level is known as critical WIP.

• An interpretation of Wo is given by the following formula:

i.e., Wo is the level of WIP that we must maintain in the system in order to maintain the bottleneck utilization at 100%. Otherwise, the bottleneck will starve.

If W<Wo, then in a deterministic setting we can pace the jobs through the system in such a way that CT=To. Hence, the maximal line throughput will be

€

Wo = rbTo =1

tb

te ( j) = u j (rb ) ≤ nj=1

n

∑j=1

n

∑

€

TH(W ) =W

To

Example: Attaining the throughput upper bound with balanced, deterministically paced line

W=Wo=rbTo=5

W=6

W=4

t1 t2 t3 t4 t5= = = = = 1.0

TH=rb=1

CT=To=5

TH=rb=1

CT=To+tb=6

TH=W/CT =4/5

CT=To=5

Bounding the line throughput:Lower bounds (under global non-idleness)

• Clearly, 1/To is a lower bound to TH(W) under global non-idleness, since this is the rate of a line with only one job in it, and therefore, no parallelism.

• This bound is also achievable under any other finite WIP level W, by a non-idling policy that moves all W jobs as a single batch from station to station. Indeed, for that policy

€

CT(W ) = W ⋅To and TH(W ) =W

W ⋅To

=1

To

Example: Attaining the throughput lower bound through batching

T = 0

T = 6

T = 12

T = 18

t1 t2 t3 t4= = = = 2.0W=3

TH = W / (W To) = 3 / 24= 1 / 8

T = 24

The W-TH(W) space

rb

TH(W)

W1 Wo=rbTo

1/To

1/To

Ideal Operational Point

The W-CT(W) Space

To

Wo1

CT(W)

W

To 1/rb

Ideal Operational Point

The depicted curves are induced from those depicted in the W-TH(W) space, through Little’s law. In particular, upper bounds in the W-TH(W) space provide lower bounds in the W-CT(W) space, and vice versa, as follows:

€

Upper Bounds in W − TW (W ) :

W ≤ Wo : TH(W ) =W

To

⇒ CT(W ) =W

TH(W )= To

W > Wo : TH(W ) = rb ⇒ CT(W ) =W

rb

Lower Bounds in W − TH(W ) :

TH(W ) =1

To

⇒ CT(W ) =W

1 To

= To ⋅W

Practical System Performance• The “ideal” performance is attained in an optimized, deterministic setting. this performance is typically compromised by

• the variability that is inherent in the system operations;• the impact of the applied control policies (e.g., the batching policy that defines the lower bounds for the system throughput - Notice, however, that these policies might be justified from some broader considerations).

• A “benchmark” case: Maximizing the variability inherent in the system operation• Single-machine stations• Exponential processing times• Balanced line, i.e., te(j) = t, j

Key feature: All feasible states for this line are equiprobable. Mean Value Analysis (exact, since processing times are exponential):

€

CT j (W ) =W −1

nt + t = (1+

W −1

n)t (from symmetry)

CT(W ) = n(1+W −1

n)t = nt + (W −1)t = To +

W −1

rb

(notice the linear dependence on W )

TH(W ) =W

CT(W )=

W

To + (W −1) /rb

=W

Wo /rb + (W −1) /rb

=W

Wo + W −1rb

Remarks: As expected, TH(1) = rb/Wo =1/To and TH() = rb. A performance that is worse than that of the “benchmark” case is a strong indication of mismanagement / bad practice.

Effective Mechanisms for Improving the System Performance

The problem: Given a line operating at a desired throughput rate, TH, what are some possible mechanisms to reduce the expected cycle time through the line, CT (and through Little’s law, the line WIP, W) ?The key idea: We need to “pull” the curve describing the line performance in the W-TH(W) space to the left.

(i) Increase rb (by adding capacity or making more effective use of the existing capacity at the line bottleneck(s))

€

Notice that

• rb ↑ rb' ⇒ tb ↓ tb

' ⇒ To ↓ To' ⇒ 1/To ↑1/To

'

• Wo' = rb

' te ( j) =1+ rb' te ( j)

j≠b

∑j

∑ >

1+ rb te ( j)j≠b

∑ = Wo

⇒ W ' < W

TH(W)

WWo1

TH

W

1/To1/To

rb

rb’

1/To’

1/To’

Wo’W’

Effective Mechanisms for Improving the System Performance (cont.)

(ii) Add capacity to some non-bottleneck station(s) (this addition essentially enables the better catering to the bottleneck needs, but it can help only to a limited extent)

TH(W)

WWo1

TH

W

1/To1/To

rb=rb’

1/To’1/To’

Wo’W’

€

Notice that

• ∃j ≠ b : te ( j)↓ te' ( j)⇒ To ↓ To

' ⇒ 1/To ↑1/To'

• rb = rb'

• Wo' =1+ rb

' te' ( j) <1+

j≠b

∑ rb te ( j)j≠b

∑ = Wo

⇒ W ' < W

(iii) Reduce the inherent variability at the different stations; the corresponding reduction of the station CVs will “pull” the performance curve in the W-TH(W) space closer to the curve characterizing the upper bound.

(iv) Increase the line flexibility, which essentially enables the better utilization of the bottleneck capacity (and takes us back to item (i) above).

On the relative advantages of “push” and “pull”-based production systems

• The WIP cap employed by “pull” systems provides a feedback mechanism for reacting to operational disruptions and prevents congestion.

• The avoidance of congestion further facilitates more flexible and responsive decision making in reaction to arising contingencies.

• Another benefit from avoiding congestion is that the WIP necessary to elicit a certain throughput, TH, from a “pull” system is typically lower than the WIP accumulated in this system when it is operated in a “push” mode and with the feeding rate equal to TH.

• WIP as a control variable is more easily measurable than throughput.• The performance attained by a “pull” system is robust to the selection of the WIP

level.• However, this robustness also implies that the production rate of a “pull” system

cannot be easily controlled through WIP cap adjustments, which further implies that effective deployment of “pull” systems necessitates non-volatile demand profiles (and raises the issue of active demand management).

• CONWIP systems can achieve higher throughput than their KANBAN-based counterparts because they provide the aforementioned advantages of “pull”-based operation while eliminating the negative impact of (internal) blocking that is inherent in the KANBAN mechanism.

factory physics post

Documents

production activity

entire production process

line workstations

francis et

stock production systems

massem line

assembly line

processing time