WINTER 2012IE 368. FACILITY DESIGN AND OPERATIONS MANAGEMENT 1 IE 368: FACILITY DESIGN AND...
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Transcript of 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
WINTER 2012IE 368. FACILITY DESIGN AND OPERATIONS MANAGEMENT
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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
WINTER 2012IE 368. FACILITY DESIGN AND OPERATIONS MANAGEMENT
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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
WINTER 2012IE 368. FACILITY DESIGN AND OPERATIONS MANAGEMENT
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Performance Evaluation (cont.)
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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Performance Evaluation (cont.)
Blocking in a production system
WS 1 WS 2 WS 3Jobs
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Performance Evaluation (cont.)
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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Performance Evaluation (cont.)
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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Probability/Statistics Concepts (cont.)
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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Probability/Statistics Concepts (cont.)
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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Probability/Statistics Concepts (cont.)
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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Probability/Statistics Concepts (cont.)
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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Probability/Statistics Concepts (cont.)
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
WINTER 2012IE 368. FACILITY DESIGN AND OPERATIONS MANAGEMENT
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Probability/Statistics Concepts (cont.)
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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Probability/Statistics Concepts (cont.)
Calculating and estimating CV(X) From data From a known or assumed distribution
Example 1 – Calculating CV(X) from data
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Probability/Statistics Concepts (cont.)
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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Performance Evaluation (cont.)
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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Performance Evaluation (cont.)
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)
WINTER 2012IE 368. FACILITY DESIGN AND OPERATIONS MANAGEMENT
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Performance Evaluation (cont.)
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.
WINTER 2012IE 368. FACILITY DESIGN AND OPERATIONS MANAGEMENT
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Performance Evaluation (cont.)
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
WINTER 2012IE 368. FACILITY DESIGN AND OPERATIONS MANAGEMENT
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Performance Evaluation (cont.)
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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Performance Evaluation (cont.)
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)
WINTER 2012IE 368. FACILITY DESIGN AND OPERATIONS MANAGEMENT
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Performance Evaluation (cont.)
Job arrival rate
TimeXX XX X X X
x5x4 x6 x7x3x2x1
Xi = Time between job arrivals (a random variable)
0
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Performance Evaluation (cont.)
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
WINTER 2012IE 368. FACILITY DESIGN AND OPERATIONS MANAGEMENT
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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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Performance Evaluation (cont.)
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
WINTER 2012IE 368. FACILITY DESIGN AND OPERATIONS MANAGEMENT
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Performance Evaluation (cont.)
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
WINTER 2012IE 368. FACILITY DESIGN AND OPERATIONS MANAGEMENT
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Queuing Models for Performance Evaluation
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
WINTER 2012IE 368. FACILITY DESIGN AND OPERATIONS MANAGEMENT
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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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Queuing Models for Performance Evaluation
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
WINTER 2012IE 368. FACILITY DESIGN AND OPERATIONS MANAGEMENT
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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
WINTER 2012IE 368. FACILITY DESIGN AND OPERATIONS MANAGEMENT
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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
22