1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding the impact of machine failures Understanding the role of buffers Able to correctly dimension buffer capacities Textbook : J. Li and S.M. Meerkov, Production Systems Engineering
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1
Chapter 8
Performance analysis and design of Bernoulli lines
Learning objectives :Understanding the mathematical models of production lines
Understanding the impact of machine failures
Understanding the role of buffers
Able to correctly dimension buffer capacities
Textbook :J. Li and S.M. Meerkov, Production Systems Engineering
• Machines are subject to Time Dependent Failures (TDF).
M1 B1 M2 B2 M3 B3 M4
Justifications:
•For most practical cases, the difference of performance measures with TDF and ODF models is within 1% - 3% (especially when buffers are not too small).
•The error resulting from the selection of failure model is small with respect to usual errors in identification of reliability parameters (rarelly known with accuracy better than 5% - 10%.
• Each machine is characterized by a Bernoulli reliability model.
• At the beginning of each time slot,
─ the status of each machine Mi - UP or DOWN - is determined by a chance experiment.
─ It is UP with proba pi and DOWN with proba 1-pi, independent of its status in all previous time slots and independent of the status of remaining system.
M1 B1 M2 B2 M3 B3 M4
Justification:
•It is practical for describing assembly operations where the downtime is typically very short and comparable with the cycle time of the machine.
─ an UP machine is blocked if its downstream buffer is full at the end of previous time slot and the downstream machine cannot produce.
─ It is starved if its upstream buffer is empty at the end of the previous time slot.
• At the end of a time slot, an UP machine that is neither blocked nor starved removes one part from its upstream buffer and adds one part in its downstream buffer.
• The first machine is never starved; the last machine is never blocked.
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Transformation of Transformation of a failure-prone line into a Bernoulli linea failure-prone line into a Bernoulli line
M1 B1 M2 B2 M3 B3 M4
• A failure-prone line with parameters :
i = 1/Ui, i, i, hi
• Bernoulli Line transformation
= min{ii}
pi = ei/i, with ei = 1/(1+i/i)
Ni = min{hiii+1, hii+1i} + 1Justifications:•From numerical results with real data, the error between the two models is quite small (less than 4%) for the case Ni ≥ 2 and is up to 7% - 8% for the case Ni < 2.
•The theory and results work for fractional buffer sizes as well.
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Transformation of Transformation of a failure-prone line into a Bernoulli linea failure-prone line into a Bernoulli line
M1 B1 M2 B2 M3 B3 M4
Why Ni = Ni = min{hiii+1, hii+1i} +1:
•A Bernoulli buffer can prevent starvation of the downstream machine and the blockage of upstream machine for a number of time slots at most equal to Ni
•hii+1 = largest time during which the downstream machine is protected from failure of upstream machine
•hiii+1= fraction of average downtime of the upstream machine that can be accommodated by the buffer.
•hii+1i= fraction of average downtime of the downstream machine that can be accommodated by the buffer.
•Fractional buffer sizes are allowed in this chapter
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Transformation of Transformation of a failure-prone line into a Bernoulli linea failure-prone line into a Bernoulli line
4.For 3/, the optimal buffer assignment is of the "inverted bowl" pattern. However, the difference with respect to "equal capacity" assignment is not significant.
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Continuous Improvement of Bernoulli Lines
42
Two improvability concepts
Constrained improvability :
Can a production system be improved by redistributing its limited buffer capacity and workforce resources?
Unconstrained Improbability :
Identify the bottleneck resource (buffer capacity or machine capability) such that its improvement best improves the system?
43
Constrained Improvability
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Resource constraints
Buffer capacity constraint (BC):
Workforce constraints (WF):
1
1
*M
ii
N N
1
*M
ii
p p
Production rates of the machines depend on workforce assignment
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Definitions
Definition: A Bernoulli line is
•improvable wrt BC if there exists a buffer assignment N'i such that iN'i = N* and
Theorem: A Bernoulli line is unimprovable wrt WF iff
where are the steady states of the recursive aggregation procedure.
Corollary. Under condition (WF1),
which implies
1
2 1i i
i fi i
N NWIP
N p
1, 1,..., 1 ( 1)f bi ip p i M WF
1,f bi ip p
1,
2 2i i
i
N NWIP i
Half buffer capacity usage
47
Improvability with respec to WF
WF-improvability indicator:
A Bernoulli line is practically unimprovable wrt workforce if each buffer is, on the average, close to half full.
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WF unimprovable allocation
Unimprovable allocation
Theorem. If i Ni-1 ≤ M/2, then the series x(n) defined below
converges to PR* where
1 1 1: *
* max ,... , ,...,i i
i
M Mp p p
PR PR p p N N
2
11
1
1 * , 0 (0,1)1
M Mi
M
i i
N x nx n p x
N
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WF unimprovable allocation
Theorem. The sequence p*i such that ip*i = p*, which renders the line unimprovable wrt WF, is given by
* 11
1
* 1
1
* 1
1
1*
*
1 1*, 2,..., 1
* *
1*
*
i ii
i i
MM
M
Np PR
N PR
N Np PR i M
N PR N PR
Np PR
N PR
Corollary. If all buffers are of equal capacity, i.e. Ni = N, then
which is a "flat" inverted bowl allocation.
Example : M = 5, Ni = 2, p* = 0.95. Compare with equal capacity.
* * * * *1 2 3 1...M Mp p p p p
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WF continuous improvement
WF continuous improvement procedure:
•Determine WIPi, for all i
•Determine the buffer with the largest |WIPi - Ni/2|. Assume this is buffer k
•If WIPk - Nk/2 > 0, re-allocate a sufficient small amount of work, pk, from Mk to Mk+1; If WIPk - Nk/2 <0, re-allocate pk+1 from Mk+1 to Mk.
•Return to step 1)
Example (home work): Continuous improvement of a 4 machine line with Ni = 5, p* = 0.94 and = 0.01. Initially, p = (0.9675, 0.9225, 0.8780, 0.8372)
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Improvability wrt WF and BC simultaneously
Theorem: A Bernoulli line is unimprovable wrt WF and BC simultaneously iff
Corollary. Under condition (WF&BC1),
and, moreover
where N is the capacity of each buffer, i.e. equal capacity buffers.
1, 2,..., 1iWIP WIP i M
1 , 2,..., 1 ( & 1)f bi i Mp p p p i M WF BC
1
1,
2 1i
N NWIP i
N p
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Unimprovabe allocation wrt WF and BC
Unimprovable allocation
Theorem. Let N* be a multiple of M-1. Then the series p*i and N*i, which render the line unimprovable wrt WF and BC, are given by
1 1 1: *
: *
** max ,... , ,...,i ii
i ii
M MN N N
p p p
PR PR p p N N
* *
** *1 *
2**
*
*
1
1**
**
1**, 2,..., 1
**
i opt
optM
opt
opti
opt
NN N
M
Np p PR
N PR
Np PR i M
N PR
PR** can be determined as PR* with N*i.
flat inverted bowl WF dist.
uniform BC dist.
Improvabiblity wrt BC
Theorem: A Bernoulli line is unimprovable wrt BC iff the quantity
is maximized over all sequences N'i such that iN'i = N*.
Condition of little practical importance.
1,...,min min , ( 1)
f bi i
i b fi Mi i
p pp BC
p p
Improvabiblity wrt BC
Numerical Fact.
The production rate ensured by the buffer capacity allocation defined by (BC1) is almost always the same as the production rate defined by the allocation that minimizes
over all sequences N'i such that iN'i = N*.
Implication:
A line is practically unimprovable wrt BC if the occupancy of each buffer Bi-1 is as close to the availability of buffer Bi as possible.
12,..., 1max ( 2)i i i
i MWIP N WIP BC
MiBi-1 Bi
Improvabiblity wrt BC
BC continuous improvement procedure:
•Determine WIPi, for all i
•Determine the buffer with the largest |WIPi - (Ni+1 - WIPi+1)|. Assume this is buffer k
•If WIPk - (Nk+1 - WIPk+1) > 0, transfer a unit of capacity from Bk to Bk+1; If WIPk - (Nk+1 - WIPk+1) < 0, re-allocate a unit from Bk+1 to Bk.
•Return to step 1)
Example (home work): Continuous improvement of a 11 machine line with pi = 0.8, i = 6, and p6 = 0.6. N* = 24. Determine the unimprovable buffer allocation (PR = 0.5843).
Unconstrained Improvability
Bottleneck machine
Definition:
•A machine Mi is the bottleneck machine (BN-m) of a Bernoulli line if
,i j
PR PRj i
p p
Problems with this definition:
1/ Gradient information cannot be measured on shopfloor2/ No analytical methods for evaluation of the gradients
Remark: gradient estimation is possible with sample path approaches (to be addressed).
Bottleneck machine
The best machine is the bottleneck
The worst machine is not the bottleneck
• Machine with the smalllest pi is not always the BN-m
• Machine with the largest WIP in front is not always the BN-m
•A buffer Bi is the bottleneck buffer (BN-b) of a Bernoulli line if
1 1
1 1
,..., , ,..., 1,...,
,..., , ,..., 1,..., ,
M i M
M j M
PR p p N N N
PR p p N N N j i
Buffer with the smallest Ni is not necessarily the BN-b
0.8 3 0.85 3 0.85 2 0.9
PR(Ni+1) 0.769 0.766 0.763
Bottlenecks in 2-machine lines
Theorem: For a 2-machine Bernoulli line,
if and only if
BL1 < ST2 (respectively, BL1 > ST2).
Remarks :
•The theorem reformulates partial derivatives in terms of "measurable" and "calculable" probabilities.
•It offers the possibility to identify BN-m without knowing the parameters of the system.
•It offers a simple graphic way of representing the BN-m.
1 2 2 1
(respectively, )PR PR PR PR
p p p p
Bottlenecks in 2-machine lines
STi 0 0.0215
BLi 0.1215 0
0.9 2 0.8
• Arrow in the direction of the inequality of the two probability
• Arrow pointing to the BN-m
Bottlenecks in long lines
Arrow Assignment Rule:
If BLi > STi+1, assign the arrow pointing from Mi to Mi+1.If BLi < STi+1, assign the arrow pointing from Mi+1 to Mi.
Bottlenecks in long lines
Bottleneck indicator: •If there is a single machine with no outgoing arrows, it is the BN-m
•If there are multiple machines with no outgoing arrows, the one with the largest severity is the Primary BN-m (PBN-m), where the severity of each BN-m is defined by
Si = |STi+1 - BLi| + |BLi-1-STi|, i = 2, ..., M-1
S1 = |ST2 - BL1|
SM = |BLM-1-STM|
•The BN-b is the buffer immediately upstream the BN-m (or PBN-m) if it is more often starved than blocked, or immediately downstream the BN-m ( or PBN-m) if it is more often blocked than starved.
Remark : It was shown numerically that Bottleneck Indicator correctly identifies the BN in most cases.
BN-m
Bottlenecks in long lines
Single Bottleneck
Multiple Bottlenecks
0.9 6 0.7 6 0.8 1 0.7 1 0.75 4 0.6 6 0.7 2 0.85
STi 0 0 0 0.09 0.23 0.1 0.2 0.36
BLi 0.4 0.2 0.3 0.14 0.03 0 0.01 0
BN-bBN-m
0.9 2 0.5 2 0.9 2 0.9 2 0.9 2 0.9 2 0.6 2 0.9
STi 0 0 0.39 0.37 0.33 027 0.11 0.41
BLi 0.41 0.01 0.03 0.05 0.1 0.17 0 0
BN-bPBN-m
Potency of buffering
Motivation: •When the worst machine is not the BN of the system, the buffer capacity is often incorrectly set.•Need to assess the buffering quality
Definition : The buffering of a production system is
•weakly potent if the BN-m is the worst machine; otherwise it is not potent
•potent if it is weakly potent and its production rate is sufficiently close to the BN-m efficiency (i.e. within 5%)
•strongly potent if it is potent and the system has the smallest possible total buffer capacity.