Qualitative Observations on
Production Line Behavior —
Intuitive and Counter-Intuitive Explanations
Stanley B. Gershwin
Department of Mechanical Engineering
Massachusetts Institute of Technology
4th International Conference on Information Systems, Logistics and Supply Chain
Quebec, Canada
August 26-29, 2012
Qualitative Observations on Production Line Behavior
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Outline
I
Introduction
I
General Observations
I
Infinite Buffers
I
Two-Machine Lines
I
Long Lines — Behavior of P(N) and n̄i (N)
I
Long Lines — Optimization
I
Conclusions
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Introduction
Objectives:
I
To describe interesting behaviors, properties, and phenomena of
flow line models.
I
To propose explanations.
I
To provide intuition.
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Kinds of Systems
M1
B1
M2
Machine
B2
M3
B3
M4
B4
M5
B5
M6
Buffer
Focus:
I
Flow lines
I
Randomness due to unreliable machines
I
Finite buffers
I
Propagation of failures through starvation and blockage
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Kinds of Systems
N1
M1
B1
N2
M2
n1
B2
N3
B3
M3
n2
N4
M4
n3
B4
n4
N5
M5
B5
M6
n5
Performance measures:
I
Production rate P, a benefit
I
I
Average in-process inventory n̄i , a cost
I
I
Production rate increases with buffer sizes.
In-process inventory varies with buffer sizes.
Other costs: machines and buffer space
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Methodology
Models:
I
Lines are modeled as Markov processes.
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Methodology
Models:
I
Lines are modeled as Markov processes.
I
Material and time can be discrete or continuous.
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Methodology
Models:
I
Lines are modeled as Markov processes.
I
Material and time can be discrete or continuous.
I
Machines fail and are repaired at random times.
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Methodology
Models:
I
Lines are modeled as Markov processes.
I
Material and time can be discrete or continuous.
I
Machines fail and are repaired at random times.
I
Models used here have geometric or exponential distributions of
up- and down-time.
Notation:
I
I
I
p=1/MTTF is failure probability or failure rate.
r =1/MTTR is repair probability or repair rate.
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Methodology
Performance evaluation:
I
Two-machine lines are evaluated analytically, exactly. We obtain
the steady-state probability distribution and use it to determine the
average production rate and buffer level.
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Methodology
Performance evaluation:
I
Two-machine lines are evaluated analytically, exactly. We obtain
the steady-state probability distribution and use it to determine the
average production rate and buffer level.
I
Short lines can be evaluated numerically, exactly.
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Methodology
Performance evaluation:
I
Two-machine lines are evaluated analytically, exactly. We obtain
the steady-state probability distribution and use it to determine the
average production rate and buffer level.
I
Short lines can be evaluated numerically, exactly.
I
Long lines are evaluated analytically, approximately.
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Methodology
M i−2 Bi−2 M i−1 Bi−1 M i
Bi
M i+1 Bi+1 M i+2 Bi+2 M i+3
Line L(i)
M u (i)
I
M d (i)
Long lines are approximately evaluated by decomposition .
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General Observations
I
In discrete models, Ni can be treated as continuous because P(N) and
n̄(N) change only a little when Ni changes by 1.
I
Monotonicity and concavity have been proven for some 2-machine line
models.
I
P(N) appears to be monotonic and concave for longer lines.
N1
M1
B1
P(N)
N2
M2
B2
Qualitative Observations on Production Line Behavior
M3
0.9
0.89
0.88
0.87
0.86
0.85
0.84
0.83
0.82
0.81
0.8
0.79
0
10
20
30
40
N1
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50
60
70
80
90
1000
10
20
30
40
50
60
70
80
90
100
N2
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Infinite Buffers
M1
B1
M2
B2
M3
B3
M4
B4
M5
B5
M6
B6
M7
B7
M8
B8
M9
B9
M10
14000
Buffer 1
Buffer 2
Buffer 3
Buffer 4
Buffer 5
Buffer 6
Buffer 7
Buffer 8
Buffer 9
12000
10000
8000
6000
4000
2000
0
0
100000 200000 300000 400000 500000 600000 700000 800000 900000 1e+06
Single bottleneck
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Infinite Buffers
I
The system is not in steady state.
I
An infinite amount of inventory accumulates in the buffer upstream
of the bottleneck.
I
A finite amount of inventory appears downstream of the bottleneck.
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Infinite Buffers
M1
B1
M2
B2
M3
B3
M4
B4
M5
B5
M6
B6
M7
B7
M8
B8
M9
B9
M10
12000
Buffer 1
Buffer 2
Buffer 3
Buffer 4
Buffer 5
Buffer 6
Buffer 7
Buffer 8
Buffer 9
10000
8000
6000
4000
2000
0
0
100000 200000 300000 400000 500000 600000 700000 800000 900000 1e+06
Qualitative Observations on Production Line Behavior
Two bottlenecks
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Infinite Buffers
I
The second bottleneck is the slowest machine upstream of the
bottleneck. An infinite amount of inventory accumulates just
upstream of it.
I
A finite amount of inventory appears between the second
bottleneck and the machine upstream of the first bottleneck.
I
Et cetera.
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Infinite Buffers
M1
B1
M2
B2
M3
B3
M4
B4
M5
B5
M6
B6
M7
B7
M8
B8
M9
B9
M10
12000
10000
Buffer 4
Buffer 9
8000
6000
4000
2000
0
0
100000 200000 300000 400000 500000 600000 700000 800000 900000 1e+06
Qualitative Observations on Production Line Behavior
Two bottlenecks
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Infinite Buffers
I
Each slope is the difference between the production rate of the
immediate downstream bottleneck and the next upstream
bottleneck.
I
The largest accumulation of inventory is located at the largest
difference, not (necessarily) the first bottleneck.
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Two-Machine Lines
Simulations
10
8
n(t)
6
4
2
0
0
200
400
600
800
1000
t
r1 = .1, p1 = .01, r2 = .1, p2 = .01, N = 10
Buffer size is the same as MTTR. Blockage and starvation are frequent.
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Two-Machine Lines
Simulations
100
80
n(t)
60
40
20
0
0
2000
4000
6000
8000
10000
t
r1 = .1, p1 = .01, r2 = .1, p2 = .01, N = 100
Buffer size is 10 × MTTR. Blockage and starvation are infrequent.
Qualitative Observations on Production Line Behavior
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Two-Machine Lines
Simulations
100
80
n(t)
60
40
20
0
0
2000
4000
6000
8000
10000
t
ri = .1, i = 1, 2, p1 = .02, p2 = .01, N = 100
First machine is a bottleneck. Blockage is infrequent. Starvation is
frequent.
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Two-Machine Lines
Simulations
100
80
n(t)
60
40
20
0
0
2000
4000
6000
8000
10000
t
ri = .1, i = 1, 2, p1 = .01, p2 = .02, N = 100
Second machine is a bottleneck. Starvation is infrequent. Blockage is
frequent.
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Two-Machine Lines
Analytic Numerical Results
Probabilities of the buffer being empty or full are much larger than
probabilities of intermediate buffer levels.
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Two-Machine Lines
Analytic Numerical Results
200
0.92
180
0.9
160
0.88
r1
r1
r1
r1
r1
140
P
120
0.86
n
100
0.84
r1 =
r1 =
r1 =
r1 =
r1 =
0.82
0.8
80
.06
.08
.1
.12
.14
60
=
=
=
=
=
.14
.12
.1
.08
.06
40
20
0
0.78
0
20
40
60
80
100
120
140
160
180
200
0
20
40
60
80
100
120
140
160
N
180
200
N
As N → ∞,
I
Production rate approaches an upper limit.
I
If the first machine is a bottleneck, average inventory n̄ approaches an
upper limit.
I
If the second machine is a bottleneck, N − n̄ approaches an upper limit.
I
If the machines are identical, n̄ = N/2.
n̄ increases as the first machine becomes faster (i.e., more productive).
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Two-Machine Lines
Analytic Numerical Results
20
Exponential and Continuous Two−Machine Lines
1
Exponential
Continuous
15
0.8
0.6
10
P
0.4
n
5
0.2
Continuous
Exponential
0
0
1
2
0
µ2
0
1
µ2
2
I Two models. Both have exponentially distributed repair and failure times.
I Exponential: discrete material, exponentially distributed operating times
(mean operation time=µ).
continuous material, constant flow through machines when
operational (flow rate=µ).
I Parameters of the models are the same.
I Continuous:
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Two-Machine Lines
Analytic Numerical Results
20
1
Exponential
Continuous
No−variability limit
n
P
0.8
15
0.6
10
0.4
5
0.2
0
No−variability limit
Continuous
Exponential
0
0
1
µ2
2
0
1
µ2
2
I
Third model, No-variability limit .
I
Same model as continuous but perfectly reliable and µ adjusted so that
isolated production rates of corresponding machines are the same.
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Two-Machine Lines
Analytic Numerical Results
Observations:
I
Variability of exponential model > variability of continuous model
> variability of no-variability limit.
I
Production rate is greatest when variability is least.
I
Average inventory is closer to 0 or N when variability is least.
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Long Lines — Behavior of P(N) and n̄i (N)
50 Machines; r=0.1; p=0.01; mu=1.0; N=20
Average Buffer Level
20
15
10
5
0
0
10
20
30
40
50
Buffer Number
Distribution of average inventory in a line with identical machines and
buffers.
P= .79; Total average inventory= 490
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Long Lines — Behavior of P(N) and n̄i (N)
50 Machines; r=0.1; p=0.01; mu=1.0; N=20
Average Buffer Level
20
15
10
5
0
0
10
20
30
40
50
Buffer Number
I “Lazy integral”
I Inventory distribution is anti-symmetric: n̄i = 20 − n̄50−i .
I More inventory at the upstream end because
I
I
each buffer near the beginning of the line has few machines
upstream and many downstream.
This is like a 2-machine line with a faster upstream machine and a
slower downstream machine.
I Less inventory at the downstream end for a similar reason.
I The buffers in the middle are close to half full because they have close to the
same number of machines upstream and downstream.
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Long Lines — Behavior of P(N) and n̄i (N)
Analytical vs simulation
16
Analytic
10,000 steps
50,000 steps
200,000 steps
14
Average buffer leve
12
10
8
6
4
0
5
10
Time steps
Production rate
15
20
25
30
Buffer number
Decomp
0.786
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10,000
0.740
35
40
50,000
0.751
45
50
200,000
0.750
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Long Lines — Behavior of P(N) and n̄i (N)
Effects of a Bottleneck
50 Machines; r=0.1; p=0.01 EXCEPT p(10)=.02; mu=1.0; N=20
Average Buffer Level
20
15
10
5
0
0
10
Qualitative Observations on Production Line Behavior
20
30
40
50
Buffer Number
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Long Lines — Behavior of P(N) and n̄i (N)
Effects of Very Large Buffers
50 Machines; r=0.1; p=0.01; mu=1.0; N=20.0 EXCEPT N(25)=2000.0
20
Average Buffer Level
15
10
5
0
0
10
20
30
40
50
Buffer Number
∗∗ A very large buffer breaks a line into two smaller lines.
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Long Lines — Behavior of P(N) and n̄i (N)
25 Machines; r=0.1; p=0.01; mu=1.0; N=20.0
20
Average Buffer Level
15
10
5
0
0
10
20
30
40
50
Buffer Number
This line is the same as the first half of
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∗∗.
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Long Lines — Behavior of P(N) and n̄i (N)
50 Machines; upstream r=0.1; p=0.01; mu=1.0; N=20.0; N(25)=2000.0 downstream r=0.15; p=0.01; mu=1.0, N=50.0
20
Average Buffer Level
15
10
5
0
0
10
20
30
40
50
Buffer Number
Upstream same as
Qualitative Observations on Production Line Behavior
∗∗; downstream faster.
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Long Lines — Behavior of P(N) and n̄i (N)
Effect of one buffer’s size on the average inventory in others
I 8-machine line; we vary the size of
25
n1
n2
n3
n4
n5
n6
n7
20
Buffer 6, N6
Average Buffer Level
I Observation:
as N6 increases,
upstream inventories decrease,
downstream inventory increase.
15
I Explanation:
10
Any Buffer i divides the
line in two parts.
5
I
If Buffer i is upstream of Buffer 6,
the increase of N6 makes the
downstream part of the line faster.
I
If Buffer i is downstream of Buffer 6,
the increase of N6 makes the
upstream part of the line faster.
0
0
5
10
15
20
25
30
35
40
45
50
N6
Qualitative Observations on Production Line Behavior
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6
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Long Lines — Behavior of P(N) and n̄i (N)
Effect of one buffer’s size on the average inventory in others
I
Consider a three-machine line.
I
Break up the line at each buffer into a single machine and a two-machine
one-buffer line and compare the production rates of each.
N1
M1
B1
N2
M2
B2
N1
M3
M1
B1
N2
M2
B2
M3
M u (2)
M d (1)
I
For the split at B1 , consider the average inventory in B1 as the size of
B2 varies from 0 to ∞.
I
For the split at B2 , consider the average inventory in B2 as the size of
B1 varies from 0 to ∞.
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Long Lines — Behavior of P(N) and n̄i (N)
Effect of one buffer’s size on the average inventory in others
N1
M1
B1
N2
M2
B2
N1
M3
M1
B1
N2
M2
B2
M3
M u (2)
M d (1)
Definitions:
I
P1 is the isolated production rate of Machine 1.
I
P{2,3} (N2 ) is the production rate of the line formed by M2 , B2 , M3 .
I
P3 is the isolated production rate of Machine 3.
I
P{1,2} (N1 ) is the production rate of the line formed by M1 , B1 , M2 .
I
We consider n̄1 when N2 = 0 or ∞.
I
We consider n̄2 when N2 = 0 or ∞.
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Long Lines — Behavior of P(N) and n̄i (N)
Effect of one buffer’s size on the average inventory in others
There are five possible cases. Two cases are interesting.
Type 1
Type 2
Type 3
Type 4
Type 5
Properties
P3 ≥ P{1,2} (∞) and
P1 ≥ P{2,3} (∞)
P3 ≥ P{1,2} (∞) and
P{2,3} (0) < P1 < P{2,3} (∞)
P3 ≥ P{1,2} (∞) and
P1 ≤ P{2,3} (0)
P{1,2} (0) < P3 < P{1,2} (∞) and
P1 ≥ P{2,3} (∞)
P3 ≤ P{1,2} (0) and
P1 ≥ P{2,3} (∞)
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Long Lines — Behavior of P(N) and n̄i (N)
Effect of one buffer’s size on the average inventory in others
N1
M1
B1
N2
M2
B2
N1
M3
I
I
B1
M2
B2
M3
M u (2)
M d (1)
Type 2
M1
N2
Properties
P3 ≥ P{1,2} (∞) and
? P{2,3} (0) < P1 < P{2,3} (∞)
If N2 increases, n̄1 decreases.
? means that
I
I
when N2 is small, M d (1) is a bottleneck.
when N2 is large, M1 is a bottleneck.
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Long Lines — Behavior of P(N) and n̄i (N)
Effect of one buffer’s size on the average inventory in others
An example of n̄1 (N2 ) in Type 2
I
r1 = .8, p1 = .096, r2 = .1, p2 = .01, r3 = .1, and p3 = .01.
I
These parameters satisfy P3 ≥ P{1,2} (∞) and
P{2,3} (0) < P1 < P{2,3} (∞).
1000
faster
M1
600
400
d
M (1)
N1 = 100
N1 = 500
N1 = 1000
for large N 2
200
0
B1
for small N 2
faster
d
M1
B1
M (1)
n̄1
n̄1
800
0
50
100
Qualitative Observations on Production Line Behavior
N2
N2
150
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Long Lines — Behavior of P(N) and n̄i (N)
Effect of one buffer’s size on the average inventory in others
Consider profit J(N1 , N2 ) = 1000P(N1 , N2 ) − N1 − N2 − n̄1 − n̄2 .
J(N)
1000
500
0
-500
-1000
-1500
0
J(N)
1000
500
0
-500
-1000
-1500
-2000
0
50
100 150
200 250
300 350
400 450
N1
5000
100 200
300 400
500 600
700 800
50
N1
900 1000 0
infeasible
feasible
iso-profit 720
iso-profit 630
iso-profit 546.3
iso-profit 450
iso-profit 300
iso-profit 100
iso-profit -100
iso-profit -400
1000 iso-profit -800
iso-profit -1200
900
iso-profit -1600
800
700
Optimal
600
boundary
500
400
N2
300
200
100
infeasible
feasible
iso-profit 700
iso-profit 606.8
iso-profit 500
iso-profit 400
iso-profit 300
iso-profit 200
iso-profit 100
iso-profit 0
iso-profit -200
iso-profit -400
500 iso-profit -600
450
iso-profit
-800
400
Optimal
350
boundary
300
250
200
N2
150
100
infeasible
feasible
iso-profit 1240
iso-profit 1188.3
iso-profit 1150
iso-profit 1100
iso-profit 1000
iso-profit 800
iso-profit 600
iso-profit 200
iso-profit -200
iso-profit -600
Optimal
600 boundary
J(N)
1500
1000
500
0
-500
-1000
500
-1500
0
400
100
200
300
300
N1
200
400
500
100
N2
6000
In spite of the non-convexity of the profit, we ran 5000 randomly-generated
examples and observed that they all had a single local maximum.
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Long Lines — Behavior of P(N) and n̄i (N)
Effect of one buffer’s size on the average inventory in others
Longer lines: Effect on n̄4 of varying N2 or N6 .
B4
B6
P (1) = .8990
P (2) = .9122
P (3) = .9097
P(2),(3) (0) = .8614
M (1)
B2
B(1)
M (2)
B(2)
M (3)
(a)
Type 2
B4
P (1) = .9092
P (2) = .9092
P (3) = .9014
P(2),(3) (0) = .8535
M (1)
B(1)
M (2)
B(2)
Qualitative Observations on Production Line Behavior
M (3)
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(b)
Type 4
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Long Lines — Behavior of P(N) and n̄i (N)
Effect of one buffer’s size on the average inventory in others
800
600
600
n̄4
1000
800
n̄4
1000
400
200
0
0
400
200
20
40
60
N6
80
0
0
50
100
150
N2
(a)
200
(b)
I
In this example, n̄4 is affected differently as N2 and N6 vary.
I
The effect of each buffer’s size on each other buffer’s average inventory in
a long line can be treated as one of the five types.
I
n̄i (Nj ) can be non-convex and non-concave.
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Stanley B. Gershwin. All rights reserved.
Long Lines — Optimization
50 Machines; r=0.1; p=0.01; mu=1.0; Total Buffer Space = 980
Optimal Buffer Size
20
15
10
5
0
0
10
20
30
40
50
Buffer Number
Allocation of buffer space that maximizes production rate.
P= .79; Total average inventory= 469
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Stanley B. Gershwin. All rights reserved.
Long Lines — Optimization
50 Machines; r=0.1; p=0.01; mu=1.0; Total Buffer Space = 980
Optimal Buffer Size
20
15
10
5
0
0
10
20
30
40
50
Buffer Number
I
Buffers are smaller at the ends of the line because no disruptions
are considered that originate outside of the line.
I
Larger buffers in the rest of the line are all about the same size
because they are affected by local disruptions more than remote
disruptions.
I
Similar to the “bowl phenomenon” of Hillier and Boling.
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Stanley B. Gershwin. All rights reserved.
Long Lines — Optimization
I
BUT, there seems to be a paradox.
I
The distribution of inventory in a line with the same machines has
more inventory at the upstream end and less at the downstream
end.
I
Doesn’t that mean there should be more space at the upstream
end and less at the downstream end?
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Stanley B. Gershwin. All rights reserved.
Long Lines — Optimization
20
11
00
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
15
I
No! The level of buffer B1
rarely gets below 10.
I
Thought experiment:
Suppose the first buffer was
Last-In-First-Out. Then the
bottom 10 parts would stay
there for a very long time.
I
Therefore we would get
almost the same production
rate if we replaced the
bottom 10 parts with bricks.
10
5
0
0
10
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Stanley B. Gershwin. All rights reserved.
Long Lines — Optimization
Effect of a bottleneck
50 Machines; r=0.1; p=0.01 EXCEPT p(10)=.02; mu=1.0; Total Buffer Space = 980
35
Optimal Buffer Size
30
25
20
15
10
5
0
10
Qualitative Observations on Production Line Behavior
20
30
40
50
Buffer Number
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Stanley B. Gershwin. All rights reserved.
Long Lines — Optimization
Problem: pick buffer sizes so that P ≥ .88 to minimize total buffer
space.
Three cases:
I
Case 1 MTTF= 200 minutes and MTTR = 10.5 minutes for all
machines (P = .95 parts per minute).
I
Case 2 Like Case 1 except Machine 5. For Machine 5, MTTF =
100 and MTTR = 10.5 minutes (P = .905 parts per minute).
I
Case 3 Like Case 1 except Machine 5. For Machine 5, MTTF =
200 and MTTR = 21 minutes (P = .905 parts per minute).
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Stanley B. Gershwin. All rights reserved.
Long Lines — Optimization
Effect of a bottleneck
60
Bottleneck
Case 1 −− All Machines Identical
50
Case 2 −− Machine 5 Bottleneck −− MTTF = 100
Buffer Size
Case 3 −− Machine 5 Bottleneck −− MTTR = 21
40
30
20
10
0
1
2
3
4
5
6
7
Qualitative Observations on Production Line Behavior
8
9
10
11
12
13
14
15
16
17
18
19
Buffer
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Other Phenomena
I
Reversibility/equivalence
I
Buffers as low-pass filters.
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Conclusions
I
Buffers absorb variability. Buffers are needed most where
I
I
I
variability is greatest (largest MTTR); or
vulnerability to variability is greatest (a bottleneck).
Two-machine lines can tell us a lot about the behavior of larger
systems.
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Stanley B. Gershwin. All rights reserved.