Manufacturing Systems Modeling, Analysis and Design IME 452 IME 545 Chapter 3, Sections 3'1, 3'2

1
Manufacturing SystemsModeling, Analysis and
DesignIME 452 / IME 545Chapter 3, Sections
3.1, 3.2
Amy Thompson Instructor
2
Transfer Lines and General Serial Type Systems

• Transfer Line a paced, serial system subject to
  breakdowns
• Used for their high rates of throughput for
  single, static products
• Breakdown could be due to machine failure or
  transport failure of system or unavailable
  resources
• Usually have common material handling and control
  system, so usually capital intensive and
  availability and effectiveness are crucial design
  features

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Size of Line

• As the number of stations along a line increases,
  the probability of all stations functioning
  decreases. (Given that each station is not 100
  reliable.)
• Buffers, though expensive to install and
  maintain, insulate stations from failures
  elsewhere in the line.
• For system design, want to determine
  effectiveness of a line given buffer capacities,
  failures, and repair rates

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Station Failures

• Buffered Station Operational States
• Station Failure
• Total Line Failure
• Station Blocked
• Station Starved
• Operational (cycle) Dependent Failures (_at_80)
  occur only while the system is running, measured
  in number of cycles between failures
• Time Dependent Failures measured in time units
  between failures

EmptyStarved
BlockedFull
X
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Some Typical Statistical Distributions Used in
Reliability

• Exponential
• Weibull
• Binomial
• Lognormal

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Probabilistic System Reliability

• Serial Components
• Parallel Components
• A Simple Common Cause Model
• Hybrid System
• Bridge Circuit

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System Maintainability and Supportability

• Maintenance
• Breakdown, Preventative, Productive, Total
  Productive Maintenance
• Availability (a measure of uptime)

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Overall Equipment Effectiveness

• Process rate ( a measure of the ability to
  operate at a standard speed)
• Quality rate (a measure of the ability to produce
  to a standard product quality)
• Equipment effectiveness (an overall measure of
  the system effectiveness)

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Throughput and Productivity

• Throughput the average rate at which product
  comes off the line
• Effectiveness or Productivity a measure of
  performance of a line configuration defined
• E() operator is expected value
• Downtime period during which product doesnt
  leave the line

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Productivity Evaluation

• Methods
• Discrete Event Simulation
• Analysis
• What we will cover
• Configuration and Productivity
• Paced Serial lines w/o buffers
• Two-stage paced lines w/buffers
• Unpaced Serial lines

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Configuration, Reliability and Productivity
Availability of all the machines is R
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Calculating System Productivity

• mi Production rate at state i

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Configuration Throughput Distribution

• Parallel systems have significantly higher
  expected throughput than serial systems (given
  the same machine availability).

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Productivity of Pure Serial and Pure Parallel
Lines
Normalizing the production rate to one, and
assuming the availability of each machine in the
system is the same, the productivity is
NOTE
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Paced Serial Lines w/o Buffers

• Operation Dependent Failures
• Assumptions
• Geometric distribution of failures
• Mean cycles to failure (MCTF) is and
  is the failure rate
• The average time to repair any station is 1/b
• All uptime and downtime variables are independent
• Idle stations do not fail
• Failures occur at the end of a cycle, dont
  destroy product.
• At most one station can fail on any cycle

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Paced Serial Lines w/o Buffers

• Let
• representing the probability that no machine
  fails
• Then with some manipulation, an m station line
  behaves like a one station line with parameter ß

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Paced Serial Lines w/o Buffers

• Can approximate ?, and calculate productivity
• Time Dependent Failures exponential
  distributions, failure rate ?, repair rate b,
  availability
  productivity

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Paced Serial Line Example

• Op dependent
• ??.02.05.03.02.12
• Exact
• Error 0.86

Average Repair time 2 cycles
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Paced Serial Line Example
mean time to failure
50
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33 1/3
50

• Time dependent productivity

Average Repair Time 2 minutes
Difference from Operation Dependent 2.5
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Time-Dependent vs.Operation-Dependent Models

• Time not suspended for time-dependent failures,
  so time-dependent models have a lower
  effectiveness

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Assignment, Problem 1

• Operation Dependent Failure Model
• Create a 2 station series transfer line in Pro
  Model.
• Make both stations processing times equal to
  some arbitrary value of C (cycle time), and use
  no move times.
• Use one repair person as a resource to perform
  repairs.
• The first station fails every 10 cycles and the
  second station every 15 cycles. Repair time is 2
  cycles for both stations.
• Do not allow stations to fail on the same cycle
  (dont allow station 2 to fail while station 1
  has failed.) Assume a repair on station 1 and
  station 2 cant occur at the same time.
• Determine the line availability.
• Add a second repair person and allow simultaneous
  failures. What is the new line availability? How
  does adding a second repair person and allowing
  simultaneous failures effect the calculation of
  the expected availability?

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Assignment, Problem 2

• Time Dependent Failure Model
• Create a 2 station series transfer line in Pro
  Model.
• Make both stations processing times equal to
  some arbitrary value of C (cycle time), and use
  no move times.
• Use one repair person as a resource to perform
  repairs.
• The first station fails every 10 cycles according
  to time and the second station every 15 cycles
  according to time. The station can not fail again
  until it is repaired. (Time to failure measured
  since station was last repaired.) Repair time is
  2 cycles according to time for both stations.
• Assume a repair on station 1 and station 2 cant
  occur at the same time.
• Determine the line availability.
• How does this line availability compare with that
  of the operation-dependent model?

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Manufacturing SystemsModeling, Analysis and
DesignIME 452 / IME 545Chapter 3, Sections 3.3
Amy Thompson Instructor
24
Parallel-Serial Production Lines
No Crossover
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Parallel-Serial Production Lines
Crossover
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Parallel-Serial Production Line Example
Width m3 (parallel lines), Length n2
(operations)
Let mttf 270 minutes, mttr 30 minutes The
availability R
Crossover
No Crossover
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Parallel-Serial Production Line with Crossover
Example
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Productivity Comparison Parallel-Serial
LineMachine Reliability 0.9
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Productivity Ratio Comparison Parallel-Serial
LineMachine Reliability 0.9
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Parallel-Serial Observations

• Synergistic Improvements to Productivity
• Lower Variability to Throughput
• Additional Parallel Lines produce Diminishing
  Returns
• More Value at Lower Machine Availability
• Productivity/ Capital Cost Trade-off vs. Buffers

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Two-stage paced lines w/buffer with
operation-dependent failures

• Describe model with a Markov Chain
• States of the Markov Chain are (S1,S2, z) where
  S(i) is the station status for station i
• Book uses W for operational and R for needs
  repair. Other nomenclature uses A and B for
  operational machines, respectively, and A and B
  for machines needing repair, respectively.
• Whether buffers are full are empty effect whether
  machines become idle.

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The Buffer-Machine Interaction

• Basic Building Block
• Assumptions
• View System at Start of Cycle
• Failure and Repairs occur at end of cycle
• Part is delivered downstream at end of cycle
  before failure or repair
• If a station starts a cycle under repair, it will
  not deliver a part downstream and its end.
• If station is operational at start, will send a
  part downstream, unless station is starved.
• Z is the buffer capacity, x is the buffer level
• Buffer full Station A Blocked
• Buffer Empty Station B Starved
• If Both Stations Failed or Operational, Buffer
  size unaffected
• Both machines never fail at the same time

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The Buffer-Machine Interaction

• State Diagram (Internal Conditions) See Table 3.1
  for transition states

B fails to repair
A repaired
A fails to repair
A,B Functional
B fails Buffer Full
A,B A Idle
A,B B Idle
A fails Buffer Empty
A,B Xx
B repaired
Buffer Empty
Buffer Empty
B fails to repair
A fails
A fails to repair
A,B Xx1
B fails
A repaired
A,B Xx-1
B repaired
A fails
AB fail to repair
A,B Xx
A repairs
B repairs
B fails
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Two-stage paced lines w/buffer with
operation-dependent failures

• Let S be the set of states of the system. Define
  the steady-state balance equations by applying
  the Chapman-Kolmogorov result
• P(s) is the probability of being in state s and
  p(u,v) is the transition probability for ending
  in state v given that we began the cycle in state
  u
• We group the rows of Table 3.1 with similar
  resultant states to obtain steady-state equations.

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Two-stage paced lines w/buffer with
operation-dependent failures

• Example state AB0 or (WW0)
• 3 possible ways to have reached this state, what
  are they?
• RW1
• WR0
• WW0
• All equations are shown on p. 75

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Two-stage paced lines w/buffer with
operation-dependent failures

• Using steady-state probability theory and the
  fact that states are mutually exclusive and
  exhaustive, these sets of equations can be
  reduced to the following Productivity
  equation(eq. 3-9)
• Closed Form Solution for non-simultaneous station
  failures (see eq. 3-10) by Buzacott.
• Limit theory on increasing buffer size as a
  buffer capacity is increased, asymptotic
  effectiveness approaches the capacity of the
  least effective station.

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Deterministic Failure Repair Time and Buffers

• Time spent waiting for repair is when buffers get
  filled or emptied
• Rule of Thumb
• Buffers should be large enough to accommodate at
  least the average repair time
• See Figure 3.4 on page 79.

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Longer Line with a Single Buffer

• System Reduction
• Rule 1 A set of stations can be aggregated into
  a single, virtual station, using equation 3.4,
  assuming all stations have a common repair rate,
  b, and all stations must stop if any individual
  station fails (no buffers between stations).
• Rule 2 Median Buffer Location if only one
  buffer is placed, place buffer in middle of line
• This is w.r.t. the failure rates up and
  downstream of the buffer
• Rule 3 Reversibility If direction is
  reversed, production rate stays the same. (Dont
  have to investigate a reverse design, gives same
  effectiveness.)

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Single Buffer Location
40
Manufacturing SystemsModeling, Analysis and
DesignIME 452 / IME 545Chapter 3, Sections 3.3
Amy Thompson Instructor
41
Time vs. Operation Dependent Errors

• Time failures occur independent of the number of
  cycles since the last failure.
• Operation dependent errors only occur when the
  station is running.
• Most failures (80) are operation dependent
  errors.

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Compare Time and Operation Dependent Models

• Specify in Downtime/Locations in Pro Model
• Specify frequency between downtimes in terms of
  cycle time (c)
• Specify to use the operator for 2 cycles in terms
  of c
• Operation Dependent Model in Pro Model
• Use Usage function in Downtime/Locations
• Time Dependent Model in Pro Model
• Use Clock function in Downtime/Locations

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Operation Dependent Model 2-Machine Example

• Operation Dependent
• Total Throughput Time for 1000 parts 126,600
  using usage function
• Resource used 166 times
• Time Dependent
• Total Throughput Time for 1000 parts 149,800
  using clock function
• Resource used 248 times

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Buffer Placement Example

• Average Repair Duration 16 cycles
• Total Failure Rate
• Median Rule
• Place Buffer after 2nd Station
• How Big? Rule of Thumb Accommodate average
  repair time say 16 pieces

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Buffer Placement Example
16
.003
.004
.005
.003
.002

• Productivity of Line

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Assignment

• Redo this problem, placing the buffer after
  station 3. Does availability improve?

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Manufacturing SystemsModeling, Analysis and
DesignIME 452 / IME 545Chapter 3, Sections 3.5
Amy Thompson Instructor
48
Unpaced Serial LineVariation in Production Rate

• Even though average workload may be balanced,
  variation in processing time can occur.
• Throughput with no buffers or breakdowns

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Unpaced Serial LineVariation in Production Rate

• If CV1.0, then if production rate is 0.1, actual
  throughput is 0.10.5 0.05 units completed per
  minute, 3 per hour
• If CV0.1, then if production rate is 0.1, actual
  throughput is 0.10.9 0.09 units completed per
  minute, 5.4 per hour

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Unpaced Serial LineVariation in Production Rate

• What effect does adding stations to this type of
  line have?
• What effect does lowering coefficient of
  variation have on this type of line?
• What happens when you lower the average time it
  takes to produce a part, without lowering the
  standard deviation of time for producing the
  part?

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Unpaced Serial LinesVariation in Production Rate

• If you place buffers of the same size between
  successive stations, assume the first station is
  never starved, and the first buffer tends to be
  full.
• Since the last workstation is never blocked, the
  last buffer will tend to be empty.
• In general, buffer utilization decreases from the
  front to the rear of the line, with the middle
  buffer being about ½ full on the average.
• Throughput is dependent upon the ratio of the
  buffer capacity (Z) to processing time
  coefficient of variation (cv).

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Unpaced Serial LinesVariation in Production Rate

• For Z/cv 10, 80 of capacity lost because of
  variability of processing times is recovered.
• For Z/cv 20, 90 of capacity lost because of
  variability of processing times is recovered. (2
  times larger buffer gives only 10 increase in
  capacity)
• The capacity lost in the unbuffered line that is
  recovered by adding buffers is only marginally
  dependent on line length.

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Unpaced Serial LinesVariation in Production Rate

• The capacity lost in the unbuffered line that is
  recovered by adding buffers is only marginally
  dependent on line length.
• Variation in processing time with equal buffers
  and no breakdowns

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Unpaced Serial Lines Variation in Production
Rate

• Blumenfeld Equation (approximation) (eq. 3-18)

T is mean service time at each station
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Buffers for Unpaced Lines

• For lines with identical workstations, the best
  allocation of buffers is to have many buffers of
  nearly equal size between stations vs. one buffer
  in the middle of the line.
• The largest buffers should be in the middle of
  the line, but the difference in size should be no
  larger than one slot.
• Buffer size should be symmetrical around line.
• As stations become non-identical, the less
  reliable stations should have larger input and
  output buffers.
• Buffers are less useful in unbalanced lines.
• If breakdowns or high processing time variability
  occurs at a workstation other than the
  bottleneck, input and output buffers at the
  bottleneck are more important.