Qualitative Observations on Production Line Behaviorweb.mit.edu/.../www/oldcell1/presentations/gershwin-qualitative.pdf · Author: Stanley B. Gershwin Created Date

Qualitative Observations onProduction Line Behavior —

Intuitive and Counter-Intuitive Explanations

Stanley B. Gershwin

Department of Mechanical EngineeringMassachusetts Institute of Technology

4th International Conference on Information Systems, Logistics and Supply ChainQuebec, Canada

August 26-29, 2012

Qualitative Observations on Production Line Behavior 1/54 Copyright c©2012 Stanley B. Gershwin. All rights reserved.

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


Introduction

Objectives:

I To describe interesting behaviors, properties, and phenomena offlow line models.

I To propose explanations.

I To provide intuition.


Kinds of Systems

Machine Buffer

M1 M2 M3 4 M5 M6MB1 B2 B3 B4 B5

Focus:

I Flow lines

I Randomness due to unreliable machines

I Finite buffers

I Propagation of failures through starvation and blockage


Kinds of Systems

M M M M MM

1 n n n nn

N N N NN1 2 3 4 5

5432

1 B B B B B1 2 2 3 3 4 4 5 5 6

Performance measures:

I Production rate P, a benefitI Production rate increases with buffer sizes.

I Average in-process inventory n̄i , a costI In-process inventory varies with buffer sizes.

I Other costs: machines and buffer space


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 ofup- and down-time.

I Notation:I p=1/MTTF is failure probability or failure rate.I r=1/MTTR is repair probability or repair rate.


Methodology

Models:







Methodology

Models:







Methodology

Models:







Methodology

Performance evaluation:

I Two-machine lines are evaluated analytically, exactly. We obtainthe steady-state probability distribution and use it to determine theaverage production rate and buffer level.

I Short lines can be evaluated numerically, exactly.

I Long lines are evaluated analytically, approximately.


Methodology






Methodology






Methodology

Mi

M (i)u

M (i)d

Line L(i)

Mi−2

Bi−2

Mi−1

Bi−1

Bi

Mi+1

Bi+1

Mi+2

Mi+3i+2

B

I Long lines are approximately evaluated by decomposition .


General Observations

I In discrete models, Ni can be treated as continuous because P(N) andn̄(N) change only a little when Ni changes by 1.

I Monotonicity and concavity have been proven for some 2-machine linemodels.

I P(N) appears to be monotonic and concave for longer lines.

M B M B M1 1 2 2 3

N N1 2

0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100

0.79 0.8 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9

P(N)

N1

N2

P(N)


Infinite Buffers

M8 8 M B9 9 M10BM B5 5 M B6 6 B7 7MM B3 3 B4 4MM B2 2M B1 1

0

2000

4000

6000

8000

10000

12000

14000

0 100000 200000 300000 400000 500000 600000 700000 800000 900000 1e+06

Buffer 1Buffer 2Buffer 3Buffer 4Buffer 5Buffer 6Buffer 7Buffer 8Buffer 9

Single bottleneck


Infinite Buffers

I The system is not in steady state.

I An infinite amount of inventory accumulates in the buffer upstreamof the bottleneck.

I A finite amount of inventory appears downstream of the bottleneck.


Infinite Buffers


0

2000

4000

6000

8000

10000

12000

0 100000 200000 300000 400000 500000 600000 700000 800000 900000 1e+06

Buffer 1Buffer 2Buffer 3Buffer 4Buffer 5Buffer 6Buffer 7Buffer 8Buffer 9

Two bottlenecks


Infinite Buffers

I The second bottleneck is the slowest machine upstream of thebottleneck. An infinite amount of inventory accumulates justupstream of it.

I A finite amount of inventory appears between the secondbottleneck and the machine upstream of the first bottleneck.

I Et cetera.


Infinite Buffers


0

2000

4000

6000

8000

10000

12000

0 100000 200000 300000 400000 500000 600000 700000 800000 900000 1e+06

Buffer 4Buffer 9

Two bottlenecks


Infinite Buffers

I Each slope is the difference between the production rate of theimmediate downstream bottleneck and the next upstreambottleneck.

I The largest accumulation of inventory is located at the largestdifference, not (necessarily) the first bottleneck.


Two-Machine LinesSimulations

0

2

4

6

8

10

0 200 400 600 800 1000

n(t

)

t

r1 = .1, p1 = .01, r2 = .1, p2 = .01,N = 10

Buffer size is the same as MTTR. Blockage and starvation are frequent.



0

20

40

60

80

100

0 2000 4000 6000 8000 10000

n(t

)

t

r1 = .1, p1 = .01, r2 = .1, p2 = .01,N = 100

Buffer size is 10 × MTTR. Blockage and starvation are infrequent.



0

20

40

60

80

100

0 2000 4000 6000 8000 10000

n(t

)

t

ri = .1, i = 1, 2, p1 = .02, p2 = .01,N = 100

First machine is a bottleneck. Blockage is infrequent. Starvation isfrequent.



0

20

40

60

80

100

0 2000 4000 6000 8000 10000

n(t

)

t

ri = .1, i = 1, 2, p1 = .01, p2 = .02,N = 100

Second machine is a bottleneck. Starvation is infrequent. Blockage isfrequent.


Two-Machine LinesAnalytic Numerical Results

Probabilities of the buffer being empty or full are much larger thanprobabilities of intermediate buffer levels.



r = .061

r = .121

r = .141

N

P

1r = .08

r = .11

0.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0 20 40 60 80 100 120 140 160 180 200

r = .061

r = .081

r = .11

r = .121

r = .141

N

n

0

20

40

60

80

100

120

140

160

180

200

0 20 40 60 80 100 120 140 160 180 200

As N →∞,I Production rate approaches an upper limit.

I If the first machine is a bottleneck, average inventory n̄ approaches anupper 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).



Exponential and Continuous Two−Machine Lines

µ

P

0 1 20

0.2

0.4

0.6

0.8

1

ExponentialContinuous

2 µ

n

0 1 20

5

10

15

20

ExponentialContinuous

2

I Two models. Both have exponentially distributed repair and failure times.

I Exponential: discrete material, exponentially distributed operating times(mean operation time=µ).

I Continuous: continuous material, constant flow through machines whenoperational (flow rate=µ).

I Parameters of the models are the same.Qualitative Observations on Production Line Behavior 27/54 Copyright c©2012 Stanley B. Gershwin. All rights reserved.


µ

P

0 1 20

0.2

0.4

0.6

0.8

1

ExponentialContinuousNo−variability limit

2 µ

n

0 1 20

5

10

15

20

Exponential

No−variability limitContinuous

2

I Third model, No-variability limit .

I Same model as continuous but perfectly reliable and µ adjusted so thatisolated production rates of corresponding machines are the same.



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.


Long Lines — Behavior of P(N) and n̄i (N)

0

5

10

15

20

0 10 20 30 40 50

Avera

ge B

uff

er

Level

Buffer Number

50 Machines; r=0.1; p=0.01; mu=1.0; N=20

Distribution of average inventory in a line with identical machines andbuffers.

P= .79; Total average inventory= 490



0

5

10

15

20

0 10 20 30 40 50

Avera

ge B

uff

er

Level

Buffer Number

50 Machines; r=0.1; p=0.01; mu=1.0; N=20

I “Lazy integral”

I Inventory distribution is anti-symmetric: n̄i = 20 − n̄50−i .

I More inventory at the upstream end because

I each buffer near the beginning of the line has few machinesupstream and many downstream.

I This is like a 2-machine line with a faster upstream machine and aslower 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 thesame number of machines upstream and downstream.


Long Lines — Behavior of P(N) and n̄i (N)Analytical vs simulation

4

6

8

10

12

14

16

0 5 10 15 20 25 30 35 40 45 50

Avera

ge b

uffer

leve

Buffer number

Analytic10,000 steps50,000 steps

200,000 steps

Time steps Decomp 10,000 50,000 200,000

Production rate 0.786 0.740 0.751 0.750


Long Lines — Behavior of P(N) and n̄i (N)Effects of a Bottleneck

0

5

10

15

20

0 10 20 30 40 50

Avera

ge B

uff

er

Level

Buffer Number

50 Machines; r=0.1; p=0.01 EXCEPT p(10)=.02; mu=1.0; N=20


Long Lines — Behavior of P(N) and n̄i (N)Effects of Very Large Buffers

0

5

10

15

20

0 10 20 30 40 50

Avera

ge B

uffer

Level

Buffer Number

50 Machines; r=0.1; p=0.01; mu=1.0; N=20.0 EXCEPT N(25)=2000.0

∗∗ A very large buffer breaks a line into two smaller lines.



0

5

10

15

20

0 10 20 30 40 50

Avera

ge B

uffer

Level

Buffer Number

25 Machines; r=0.1; p=0.01; mu=1.0; N=20.0

This line is the same as the first half of ∗∗.



0

5

10

15

20

0 10 20 30 40 50

Avera

ge B

uffer

Level

Buffer Number

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

Upstream same as ∗∗; downstream faster.


Long Lines — Behavior of P(N) and n̄i (N)Effect of one buffer’s size on the average inventory in others

0

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50

n1

n2

n3

n4

n5

n6

n7

N

Av

era

ge

Bu

ffe

r L

ev

el

6

6

6

I 8-machine line; we vary the size ofBuffer 6, N6

I Observation: as N6 increases,upstream inventories decrease,downstream inventory increase.

I Explanation: Any Buffer i divides the

line in two parts.

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.



I Consider a three-machine line.

I Break up the line at each buffer into a single machine and a two-machineone-buffer line and compare the production rates of each.

N1 N2

Md(1) Mu(2)

M1 B1 M2 B2 M3

N1 N2

M1 B1 M2 B2 M3

I For the split at B1, consider the average inventory in B1 as the size ofB2 varies from 0 to ∞.

I For the split at B2, consider the average inventory in B2 as the size ofB1 varies from 0 to ∞.



N1 N2

Md(1) Mu(2)

M1 B1 M2 B2 M3

N1 N2

M1 B1 M2 B2 M3

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 ∞.



There are five possible cases. Two cases are interesting.

Properties

Type 1P3 ≥ P{1,2}(∞) andP1 ≥ P{2,3}(∞)

Type 2P3 ≥ P{1,2}(∞) andP{2,3}(0) < P1 < P{2,3}(∞)

Type 3P3 ≥ P{1,2}(∞) andP1 ≤ P{2,3}(0)

Type 4P{1,2}(0) < P3 < P{1,2}(∞) andP1 ≥ P{2,3}(∞)

Type 5P3 ≤ P{1,2}(0) andP1 ≥ P{2,3}(∞)



N1 N2

Md(1) Mu(2)

M1 B1 M2 B2 M3

N1 N2

M1 B1 M2 B2 M3

Properties

Type 2P3 ≥ P{1,2}(∞) and

? P{2,3}(0) < P1 < P{2,3}(∞)

I If N2 increases, n̄1 decreases.I ? means that

I when N2 is small, Md(1) is a bottleneck.I when N2 is large, M1 is a bottleneck.



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}(∞) andP{2,3}(0) < P1 < P{2,3}(∞).

n̄1

N1 = 100

N1 = 500

N1 = 1000

N2

BM1 1 M (1)dfaster

BM1 1 M (1)dfaster

for small N2

for large N2

0 50 100 150 200 250N2

n̄1

0

200

400

600

800

1000



Consider profit J(N1,N2) = 1000P(N1,N2)− N1 − N2 − n̄1 − n̄2.

0 100 200 300 400 500 600 700 800 900 1000 0 50

100 150

200 250

300 350

400 450

500-1500

-1000

-500

0

500

1000J(N)

infeasiblefeasible

iso-pro�t 700iso-pro�t 606.8

iso-pro�t 500iso-pro�t 400iso-pro�t 300iso-pro�t 200iso-pro�t 100

iso-pro�t 0iso-pro�t -200iso-pro�t -400iso-pro�t -600iso-pro�t -800

Optimalboundary

N1

N2

J(N)

0 50 100 150 200 250 300 350 400 450 500 0 100

200 300

400 500

600 700

800 900

1000-2000

-1500

-1000

-500

0

500

1000J(N) infeasible

feasibleiso-pro�t 720iso-pro�t 630

iso-pro�t 546.3iso-pro�t 450iso-pro�t 300iso-pro�t 100

iso-pro�t -100iso-pro�t -400iso-pro�t -800

iso-pro�t -1200iso-pro�t -1600

Optimalboundary

N1

N2

J(N)

0 100

200 300

400 500

600 0 100

200 300

400 500

600-1500

-1000

-500

0

500

1000

1500J(N)

infeasiblefeasible

iso-pro�t 1240iso-pro�t 1188.3

iso-pro�t 1150iso-pro�t 1100iso-pro�t 1000

iso-pro�t 800iso-pro�t 600iso-pro�t 200

iso-pro�t -200iso-pro�t -600

Optimalboundary

N1

N2

J(N)

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.



Longer lines: Effect on n̄4 of varying N2 or N6.

P (1) = .8990

P (2) = .9122

P (3) = .9097

P(2),(3)(0) = .8614

Type 2

P (1) = .9092

P (2) = .9092

P (3) = .9014

P(2),(3)(0) = .8535

Type 4

B4 B6

M(1) M(2) M(3)B(1) B(2)

B4B2

M(1) M(2) M(3)B(1) B(2)

(a)

(b)



0

200

400

600

800

1000

0 20 40 60 80 0

200

400

600

800

1000

0 50 100 150 200

n̄4

N6 N2(a) (b)

n̄4

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 ina long line can be treated as one of the five types.

I n̄i (Nj) can be non-convex and non-concave.


Long Lines — Optimization

0

5

10

15

20

0 10 20 30 40 50

Opti

mal B

uff

er

Siz

e

Buffer Number

50 Machines; r=0.1; p=0.01; mu=1.0; Total Buffer Space = 980

Allocation of buffer space that maximizes production rate.P= .79; Total average inventory= 469



0

5

10

15

20

0 10 20 30 40 50

Op

tim

al B

uff

er

Siz

e

Buffer Number

50 Machines; r=0.1; p=0.01; mu=1.0; Total Buffer Space = 980

I Buffers are smaller at the ends of the line because no disruptionsare considered that originate outside of the line.

I Larger buffers in the rest of the line are all about the same sizebecause they are affected by local disruptions more than remotedisruptions.

I Similar to the “bowl phenomenon” of Hillier and Boling.



I BUT, there seems to be a paradox.

I The distribution of inventory in a line with the same machines hasmore inventory at the upstream end and less at the downstreamend.

I Doesn’t that mean there should be more space at the upstreamend and less at the downstream end?



��

��

0

5

10

15

20

0 10

I No! The level of buffer B1

rarely gets below 10.

I Thought experiment:Suppose the first buffer wasLast-In-First-Out. Then thebottom 10 parts would staythere for a very long time.

I Therefore we would getalmost the same productionrate if we replaced thebottom 10 parts with bricks.


Long Lines — OptimizationEffect of a bottleneck

5

10

15

20

25

30

35

0 10 20 30 40 50

Opti

mal B

uff

er

Siz

e

Buffer Number

50 Machines; r=0.1; p=0.01 EXCEPT p(10)=.02; mu=1.0; Total Buffer Space = 980



Problem: pick buffer sizes so that P ≥ .88 to minimize total bufferspace.

Three cases:

I Case 1 MTTF= 200 minutes and MTTR = 10.5 minutes for allmachines (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).


Long Lines — OptimizationEffect of a bottleneck

0

10

20

30

40

50

60

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Buffer

Bu

ffer S

ize

Case 1 −− All Machines Identical

Case 2 −− Machine 5 Bottleneck −− MTTF = 100

Case 3 −− Machine 5 Bottleneck −− MTTR = 21

Bottleneck


Other Phenomena

I Reversibility/equivalence

I Buffers as low-pass filters.


Conclusions

I Buffers absorb variability. Buffers are needed most whereI variability is greatest (largest MTTR); orI vulnerability to variability is greatest (a bottleneck).

I Two-machine lines can tell us a lot about the behavior of largersystems.


Documents

Qualitative Observations on Production Line Behaviorweb.mit.edu/.../www/oldcell1/presentations/gershwin-qualitative.pdf · Author: Stanley B. Gershwin Created Date