winter 2012ie 368. facility design and operations management 1 ie 368: facility design and...

WINTER 2012IE 368. FACILITY DESIGN AND OPERATIONS MANAGEMENT

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IE 368: FACILITY DESIGN AND OPERATIONS MANAGEMENT

Lecture Notes #3

Production System DesignPart #2


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Performance Evaluation

So far: Identified the general type of production system flow For each workstation in the system we have calculated

the equipment fraction based on throughput, reliability, efficiency, etc.• Calculations are based on averages


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Performance Evaluation (cont.)

Will the design meet performance requirements? Throughput

• Quantity over time System responsiveness

• Time-In-System WIP Levels

• Inventory on the floor


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Throughput After computing the equipment fraction, can the

system meet long-run throughput requirements?• Over the short-run many things are possible

Can workstations block each other?• If so, it is not simple to determine if throughput

requirements can be met• If workstations do not block each other, throughput

requirements can be met


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Blocking in a production system

WS 1 WS 2 WS 3Jobs


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System responsiveness/WIP Unless all workstations work with complete

predictability and reliability, this is generally not known The lack of complete predictability and reliability

creates variability in system operations


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How do you evaluate performance? Computer simulation

• Needed for evaluating throughput when blocking occurs• Can be detailed and time consuming

Queuing (waiting line) models• Mathematical formulas• Applied at an early design stage• Can be used to evaluate TIS & WIP


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Queuing Models

Mathematical models of waiting line systems e.g., a workstation receiving jobs for processing

Their use requires some background in probability and statistics


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Probability/Statistics Concepts

Random variable A measurable quantity whose value is unpredictable

Equipment fractions were calculated assuming fixed average values

In reality many of the quantities in the equipment fraction equation vary e.g., the time per job, reliability, scrap, etc.


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Probability/Statistics Concepts (cont.)

Random variables are characterized by distribution functions Distribution functions describe the probability of

observing various outcomes for the random variable

Examples of distributions Normal Uniform Binomial …


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To utilize the queuing models, an understanding of the following concepts is needed Averages Variance Coefficient of variation

The concepts of the true and estimated values for these quantities are also needed


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Expected Value The true average of a random variable It is a measure of the central tendency of X Denoted E(X) for the random variable X E(X) is estimated using the sample average

E(X) is a constant and is a random variable

n

xx

n

ii

1

x


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Variance A measure of the true spread or predictability of a

random variable Denoted as Var(X) or V(X) Var(X) is estimated by the sample variance

Var(X) is a constant and s2 is a random variable

2

2 2 1

2 1 1

( )

1 1

n

in ni

i ii i

x

x x xn

sn n


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Example

125 130 135 140120115110x

Normal distribution E(X) = 125, Var(X) = 25

Exponential distribution E(X) = 1/2, Var(X) = 1/4


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The coefficient of variation (or CV) for a random variable X is a measure of relative variability Denoted CV(X)

CV(X) is dimensionless

( ) ( )( )

( ) ( )

ˆ( )

Var X StdDev XCV X

E X E X

sCV X

x


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For waiting line systems (e.g., a workstation receiving jobs) it is relative variability that affects performance instead of absolute variability

125 130 135 140120115110x

N(125,25) N(39.5,2.5)

39.5x


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Random Outages

Different types of downtimes have different impacts on variability

The most important distinction is between preemptive and non-preemptive downtimes Preemptive outages occur right in the middle of a process

Typically, these are outages for which there is no control as to when they happen (e.g., failures)

In contrast, non-preemptive outages require the tool to be idle before they can happen This means that we have some control as to exactly when

they occur. This is usually the case for planned maintenance activities or setup times

Schmidt, K., Rose, O. (2007). Queue time and x-factor characteristics for semiconductormanufacturing with small lot sizes. Proceedings of the 3rd Annual IEEE Conference on Automation Science and Engineering, 1069-1074.


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Calculating and estimating CV(X) From data From a known or assumed distribution

Example 1 – Calculating CV(X) from data


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Example 2 – Calculating CV(X) from known distribution X is assumed to follow a triangular distribution with

• Min (a) = 2• Mode (b) = 5• Max (c) = 10


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In-class Exercise

Compute the estimated CV for the following data: 1, 1, 2, 1, 1, 1, 1, 25, 1, 1

If X is normally distributed with E(X) =10, and Var(X) = 100


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Performance Evaluation

To evaluate the performance of a workstation (TIS and WIP), the concept of workstation utilization is needed

Utilization for a workstation is the ratio of: The average time between job departures (if each machine

in the workstation always has work), and average time between job arrivals

The average rate of job arrivals, and the average rate of job departures (if each machine in the workstation always has work)

To compute either one of these ratios the concept of effective process time of a job at a workstation is needed


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Effective Process Time The total time seen by a job at a workstation Effective process time is a random variable From the perspective of the output side of a

workstation, if a job is being processed at a workstation and is delayed, it does not matter if the delay is due to• Product type,• Machine down time,• Operator break time,• Human variability,…

Effective process time does NOT include idle time


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Effective Process Time

Examples Manual operations – Process time of each job will vary.

Effective process time is the average. Automated operations – Identical process times, interrupted

by down time. Combination – Varying process times interrupted by

equipment down times.

WSJobs

ObserverEff. Proc. Time= Avg. TimeBetween Jobs

Eff. Proc. Time= 1/(Avg. Rateof Jobs Seen)


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Effective Process Time Down time or other delays occurring during the

processing of a job are included as part of the effective process times

Idle time• Time when the WS is not working on a job, is NOT included

in the effective process time• e.g., a workstation is starved for jobs because its

upstream workstation is experiencing a long down time. This idle time is taken out of calculations of effective process time.


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Job

Departure Time from

WS

Effective Process

Time Notes

1 1 12 2 13 3 14 7.25 1 WS starved for 3.25 min5 8.25 16 9.25 17 15.5 6.25 WS down for 5.25 min8 16.5 19 17.5 1

10 18.5 1

Automated Workstation (WS) - 1 Machine1 Minute Processing per Job


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Job

Departure Time from

WS

Effective Process

Time Notes

1 1.1 1.12 2 0.93 2.8 0.84 6.1 1.2 WS starved for 2.1 min5 7 0.96 8.25 1.257 14.5 6.25 Operator unplanned break for 5.05 min8 15.6 1.19 16.4 0.8

10 17.45 1.05

Manual Workstation (WS) - 1 Person1 Minute Average Processing per Job


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Job arrival rate If each machine in the workstation always has work,

utilization for a workstation is the ratio of:• The average time between job departures (if each

machine in the workstation always has work), and average time between job arrivals

• The average rate of job arrivals, and the average rate of job departures (if each machine in the workstation always has work)

WSJobs

ObserverAvg Intearrival Time= Avg. TimeBetween Jobs

Arrival Rate= 1/(Avg. Intearrival Time)


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Job arrival rate

TimeXX XX X X X

x5x4 x6 x7x3x2x1

Xi = Time between job arrivals (a random variable)

0


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Workstation utilization

te = Average effective process time for a job at a workstation (on a single machine)

ta = Average inter-arrival (time between job arrivals)

time of jobs to the workstation. Note that ta is the inverse of the arrival rate of jobs to a workstation.

u = Utilization = a

e

tm

t

*

= The percent of time machines in a workstation are busy

m = The number of machines working in parallel at a workstation


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In-class Exercise – Estimate u

The following sample data was collected as part of a study of a single machine WS.

Arrival time of jobs Log of job completion times(to the nearest minute) (collected on a separate day)

8:00 AM 8:05 AM8:04 8:108:13 8:12 Start Idle Time8:21 8:15 End Idle Time8:22 8:198:23 8:228:25 8:238:26 8:29 Start Idle Time8:29 8:30 End Idle Time8:33 8:338:50 8:358:55 8:398:56 8:458:58 8:478:59 8:509:02 8:51

8:538:568:57


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To evaluate the performance of a workstation (TIS and WIP) using queuing models, the concept of workstation relative variability is needed The measure for relative variability is the coefficient of

variation For a random variable X, the CV of X is

( ) ( ) ˆ( ) ( )( ) ( )

Var X StdDev X sCV X CV X

E X E X x


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Variance of the effective process times at a workstation

Variance of the job interarrival times to a workstation Then

Coefficient of variation of the effective process times

Coefficient of variation of the job interarrival times

2e

2a

2e

ee

CVt

2a

aa

CVt


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Classification of CVs

Classification CV Examples

Low CV < 0.75 - Manual repetitive operations- Machines with short

frequent interruptions

Moderate 0.75 ≤CV ≤1.33

- Machines with setups- Machines with failures

(mostly shorter downtimes)

High CV > 1.33 - Processes with occasional long failures/downtimes


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The following sample data was collected as part of a study of a single machine WS.

Arrival time of jobs Log of job completion times(to the nearest minute) (collected on a separate day)

8:00 AM 8:05 AM8:04 8:108:13 8:12 Start Idle Time8:21 8:15 End Idle Time8:22 8:198:23 8:228:25 8:238:26 8:29 Start Idle Time8:29 8:30 End Idle Time8:33 8:338:50 8:358:55 8:398:56 8:458:58 8:478:59 8:509:02 8:51

8:538:568:57

In-class Exercise – Estimate the CVs


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Queuing Models for Performance Evaluation

So far, we have learned about (and calculated) seven different parameters to describe a production system made up of workstations What are these seven parameters?

In the next few slides, the basic theory (i.e., formulations and assumptions) to evaluate the performance of a production system made up of workstations is presented


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Will evaluate long-run average Throughput (with no blocking) Time-In-System WIP

Consider the simplest case A single machine workstation

WSJob Arrivals

Job Departures


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Evaluating Average Throughput

When a workstation (WS) is never blocked

Why?

WS throughput =1/ta , if utilization < 1

m/te , otherwise


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TIQ = Average time in queue Time in line before the start of processing

2 2

2 (1 )a e

e

CV CV uTIQ t

u

eeeea tTIQtt

u

uCVCVTIS

)1(2

22


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Example


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Example


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Example


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Evaluating Average WIP

Apply a result from queuing theory called Little’s Law

Where TP = ThroughputAssumes no limit on storage space

Q

WIP TP TIS

WIP TP TIQ


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Example


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In-class Exercise

Suppose option 2 in the prior example will be adopted but there is uncertainty in the exact throughput (job arrival rate)

Plot the average TIS as a function of throughput for the following throughput values (jobs per hour) 2.5, 2.6, 2.7, 2.8, 2.9, and 2.95


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In-class Exercise


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Examination of the Queuing Model

TIQ/TIS and WIP depend linearly on CVa2 and CVe

2

TIQ/TIS and WIP depend non-linearly on uAs CVa

2 and CVe2 get smaller it is possible to

Operate at a higher utilization (throughput) with the same TIQ/TIS and WIP

Have less TIQ/TIS and WIP for the same throughputThis is the fundamental idea behind many Just-In-Time

production systems

eea t

u

uCVCVTIQ

)1(2

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winter 2012ie 368. facility design and operations management 1 ie 368: facility design and...

Documents

facility design

system operations

production system design

performance evaluation

jobs slide

floor slide

early design stage

throughput requirements