1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding...

65
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

Transcript of 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding...

Page 1: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

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

Page 2: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

2

PlanPlan

• Definitions and justifications

• Two-machine Bernoulli lines

• Long Bernoulli Lines

• Continuous Improvement of Bernoulli Lines

• Constrained Improvability

• Unconstrained Improvability

Page 3: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

3

Definitions and justifications

Page 4: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

4

Bernoulli linesBernoulli linesDefinitionDefinition

• A Bernoulli line is a synchronuous line with all machines having identical cycle time.

• It is a slotted time model with time indexed t = 0, 1, 2, ...

M1 B1 M2 B2 M3 B3 M4

Justification: appropriate for high volume assembly lines.

Page 5: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

5

Bernoulli linesBernoulli linesDefinitionDefinition

• 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%.

•The TDF model is simpler for analysis

Page 6: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

6

Bernoulli linesBernoulli linesDefinitionDefinition

• 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.

Page 7: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

7

Bernoulli linesBernoulli linesOperating rulesOperating rules

M1 B1 M2 B2 M3 B3 M4

• A Bernoulli line can be represented by a vector

(p1, ..., pM, N1, ..., NM-1)

of machine reliability parameters and buffer capacities.

• The time is slotted with the cycle time of the machines.

• The status of each machine is determined at the beginning and the state of the buffers at the end of each time slot.

• The status of a machine is UP with proba pi and DOWN with proba (1-pi) and it is independent of past history and the status of the remaining system

Page 8: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

8

Bernoulli linesBernoulli linesOperating rulesOperating rules

M1 B1 M2 B2 M3 B3 M4

• Blocking Before Service:

─ 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.

Page 9: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

9

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.

Page 10: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

10

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

Page 11: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

11

Transformation of Transformation of a failure-prone line into a Bernoulli linea failure-prone line into a Bernoulli line

Examples to work out:Line 1e = {0.867; 0.852; 0.925; 0.895; 0.943; 0.897; 0.892; 0.935; 0.903; 0.870};Tdown = {14.23; 16.89; 18.83; 16.08; 7.65; 11.09; 19.05; 18.76; 11.15; 18.42};N = {7.026; 17.350; 33.461; 5.345; 9.861; 12.097; 11.955; 26.133; 14.527};U = {1.950; 1.231; 1.607; 1.486; 1.891; 1.762; 1.457; 1.019; 1.822; 1.445}.

Line 2e = {0.945; 0.873; 0.911; 0.899; 0.939; 0.926; 0.896; 0.852; 0.932; 0.895};Tdown = {14.22; 16.89; 18.83; 16.08; 7.65; 11.09; 19.05; 18.76; 11.15; 18.42};N = {5.535; 31.138; 20.578; 37.614; 21.310; 19.653; 34.618; 23.380; 12.093};U = {1.672; 1.838; 1.020; 1.681; 1.380; 1.832; 1.503; 1.709; 1.429; 1.305}.

Line 3e = {0.869; 0.869; 0.918; 0.880; 0.904; 0.865; 0.920; 0.888; 0.936; 0.935};Tdown = {13.91; 12.45; 18.48; 17.33; 14.68; 17.27; 14.90; 10.13; 9.35; 10.12};N = {26.746; 32.819; 38.490; 23.291; 35.805; 11.054; 39.291; 14.501; 13.832};U = {1.534; 1.727; 1.309; 1.839; 1.568; 1.370; 1.703; 1.547; 1.445; 1.695}.

Page 12: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

12

Two-machine Bernoulli linesTwo-machine Bernoulli lines

Page 13: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

13

States of the system:

•Bernoulli machines are memoryless

•System state = Buffer state xn at the end of time slot n

•xn is a discrete time Markov chain

State transition diagram

M1 B M2

p1 p2N > 0

0 1 N-1 N

p01

…p12

pN-2,N-1 pN-1,N

p10 p21pN-1,N-2

pN,N-1

p00p11 pNN

pN-1,N-1

DTMC modelDTMC model

Page 14: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

14

Blockage of M1 in period n+1•xn = N•M1 is UP•M2 is DOWN

0 1 N-1 N

p01

…p12

pN-2,N-1 pN-1,N

p10 p21pN-1,N-2

pN,N-1

p00p11 pNN

pN-1,N-1

DTMC modelDTMC model

Starvation of M2 in period n+1•xn = 0•M2 is UP

M1 B M2

p1 p2N

Page 15: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

15

DTMC modelDTMC modelTransition probabilitiesTransition probabilities

p00 = 1 - p1

p01 = p1

p10 = (1 - p1)p2

M1 B M2

p1 p2N

0 1 N-1 N

p01

…p12

pN-2,N-1 pN-1,N

p10 p21pN-1,N-2

pN,N-1

p00p11 pNN

pN-1,N-1

pii = p1p2 + (1 - p1) (1 - p2)

pi,i+1 = p1(1 - p2), i = 1, ..., N-1

pi+1,i = (1 - p1)p2

pNN = p1p2 + (1 - p1) (1 - p2) + p1(1 - p2) = p1p2 + 1 - p2

Page 16: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

16

DTMC modelDTMC modelSteady state distributionSteady state distribution

0 1 N-1 N

p01

…p12

pN-2,N-1 pN-1,N

p10 p21pN-1,N-2

pN,N-1

p00p11 pNN

pN-1,N-1

Equilibrium equation

•states {0,1, ..., i} : i+1pi+1,i = ipi,i+1, i < N

Normalization equation

•0+ 1 + ... + N = 1

M1 B M2

p1 p2N

Page 17: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

17

DTMC modelDTMC modelSteady state distributionSteady state distribution

0 1 N-1 N

p01

…p12

pN-2,N-1 pN-1,N

p10 p21pN-1,N-2

pN,N-1

p00p11 pNN

pN-1,N-1

To be shown :

M1 B M2

p1 p2N

20 2

2

02

1 21 2

2 1

1

1 ...

, 01

1,

1

N

i

i

p

p

ip

p pp p

p p

Page 18: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

18

DTMC modelDTMC modelSteady state distributionSteady state distribution

0 1 N-1 N

p01

…p12

pN-2,N-1 pN-1,N

p10 p21pN-1,N-2

pN,N-1

p00p11 pNN

pN-1,N-1

Case of identical machines, p1 = p2 = p

M1 B M2

p1 p2N

0

1 2

1

1

1, 0

1

, 1

i

p

N p

iN p

p p

For practical case with p 1,0 0i 1/N, i > 0

Page 19: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

19

DTMC modelDTMC modelSteady state distributionSteady state distribution

0 1 N-1 N

p01

…p12

pN-2,N-1 pN-1,N

p10 p21pN-1,N-2

pN,N-1

p00p11 pNN

pN-1,N-1

Case of nonidentical machines, i.e. p1 ≠p2

M1 B M2

p1 p2N

10

1

2

1 1

1 N

p

p

p

Page 20: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

20

DTMC modelDTMC modelSteady state distributionSteady state distribution

p1 = 0.8, p2 = 0.82, N = 5 p1 = 0.82, p2 = 0.8, N = 5

p1 = 0.6, p2 = 0.9, N = 5 p1 = 0.9, p2 = 0.6, N = 5

Page 21: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

21

DTMC modelDTMC modelSteady state distributionSteady state distribution

Theorem: Function Q(x, y, N) defined below, with 0<x<1, 0<y<1, and N ≥ 1, takes values on (0,1) and is

•strictly decreasing in x,

•strictly increasing in y

•strictly decreasing in N

where

1 1 ,, if

1 ,, ,

1, if

1

N

x x yx y

xx y

Q x y N y

xx y

N x

Page 22: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

22

DTMC modelDTMC modelSteady state distributionSteady state distribution

Theorem:

•0 = Q(p1, p2, N)

•N = Q(p2, p1, N)/(1-p2)

•(y, x) = 1/(x, y)

Meaning of Q(p1, p2, N) :

The intermediate buffer is empty

Implication : M2 is starved if it is UP

Meaning of Q(p2, p1, N) :

The intermediate buffer is full & its downstream machine does not produce

Implication : M1 is blocked if it is UP

Page 23: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

23

DTMC modelDTMC modelPerformance measuresPerformance measures

Production rate (PR)

•PR = p2(1 - 0)

•PR = p1(1 - N(1-p2))

•PR = p2(1 - Q(p1, p2, N))

•PR = p1(1 - Q(p2, p1, N))

Page 24: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

24

DTMC modelDTMC modelPerformance measuresPerformance measures

Work In Process (WIP)

1 2

1 1 211 2

1 22 1 1 2

1, if

2 1

1 ,, , otherwise

1 ,,

N

i Ni N

N

N Np p p

N pWIP i

p ppN p p

p pp p p p

Page 25: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

25

DTMC modelDTMC modelBlockage and StarvationBlockage and Starvation

Blocking probability of M1 (BL1)

BL1 = p1N(1-p2) = p1 Q(p2, p1, N)

Starvation probability of M2 (ST2)

ST2 = p20 = p2 Q(p1, p2, N)

Relation with PR

PR = p1 - BL1

PR = p2 - ST2

Page 26: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

26

DTMC modelDTMC model

L1: p1 = p2 = 0.9 L2: p1 = 0.9, p2 = 0.7

L3: p1 = 0.7, p2 = 0.9

Page 27: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

27

DTMC modelDTMC model

Theorem:

1 2

1 2

1 11 2

2 1

1 2

1 1 2

2 2 1

lim min( , )

,

1lim ,

lim ,2

lim

lim

N

N

N

N

N

PR p p

p p

p pWIP p p

p p

Np p

BL p p

ST p p

Page 28: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

28

Long Bernoulli LinesLong Bernoulli Lines

Page 29: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

29

DTMC model

• The vector of buffer states

(x1(n), x2(n), ..., xM-1(n))

is a discrete time Markov chain.

• Unfortunately, the state space is large with (N1+1) (N2+1)... (NM-1+1) states.

• Analytical formula are not available for performance measures of long Bernoulli lines.

• Focus on an aggregation approach.

M1 B1 M2 B2 M3 B3 M4

p2 p3 p4N2 N3p1 N1

Page 30: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

30

Idea of the aggregationBackward aggregration

• pb3 = production rate of the 2-machine

line (M3, B3, M4)

• Repeating the aggregation process

• pbi = production rate of the 2-machine

line (Mi, Bi, Mbi+1)

• Drawback : is quite different from the production rate of the M-machine line

M1 B1 M2 B2 M3 B3 M4

M1 B1 M2 B2 Mb3

M1 B1 Mb2

Mb1

Page 31: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

31

Idea of the aggregationForward aggreation

• Forward aggreation is introduced to improve the aggregration.

• pfi is determine to take into account

the starvation of Bi-1 in the 2-machine line (Mf

i-1, Bi-1, Mbi)

• The whole process repeats to futher improved the aggregation

Mf2 B2 Mb

3

M1 B1 Mb2

Mf3

B3 Mb4

Mf4

Page 32: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

32

Aggregation procedureFormal definition

The recursive aggregation procedure is as follow (Why?)

with initial condition

and boundary conditions

11 1 1 , , , 1,...,1b b fi i i i ip s p Q p s p s N i M

1 11 1 1 , 1 , , 2,...,f f bi i i i ip s p Q p s p s N i M

0 , 1,...,fi ip p i M

1 1f

bM M

p s p

p s p

1 1 ,, if

1 ,, ,

1, if

1

N

x x yx y

xx y

Q x y N y

xx y

N x

Page 33: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

33

Aggregation procedureExample to workout with Excel

A 3-machine line L = (0.9, 0.9, 0.9, 2, 2)

2 2 3 2 2

1 1 2 1 1

2 2 1 2 2

3 3 2 3 2

2 2

1 1 1 , 0 , 0.9 1 0.9,0.9,2 0.8571

1 1 1 , 0 , 0.9 1 0.8571,0.9,2 0.8257

1 1 1 , 1 , 0.9 1 0.9,0.8571,2 0.8670

1 1 1 , 1 , 0.9 1 0.8670,0.9,2 0.8333

2 1

b b f

b b f

f f b

f f b

b

p p Q p p N Q

p p Q p p N Q

p p Q p p N Q

p p Q p p N Q

p p Q p

3 2 2

1 1 2 1 1

2 2 1 2 2

3 3 2 3 2

2 , 1 , 0.9 1 0.9,0.8670,2 0.8650

2 1 2 , 1 , 0.9 1 0.8650,0.9,2 0.8318

2 1 2 , 2 , 0.9 1 0.9,0.8650,2 0.8654

2 1 2 , 2 , 0.9 1 0.8654,0.9,2 0.8321

...

b f

b b f

f f b

f f b

p N Q

p p Q p p N Q

p p Q p p N Q

p p Q p p N Q

Page 34: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

34

Aggregation procedureConvergence

Theorem.

Both sequence pfi(s) and pb

i(s) are converging, i.e. the following limits exist :

For each i, the sequence pfi(s) is monotonically decreasing

and the sequence pfi(s) is monotonically increasing.

Moreover,

: lim , : limb b f fi i i is s

p p s p p s

1b f

Mp pInterpretation

the downstream subline of buffer Bi-1

the upstream subline of buffer Bi

bipf

ip

Page 35: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

35

Aggregation procedureExercice

L1 : (0.9, 0.9, 0.9, 0.9, 0.9; 3, 3, 3, 3)

L2 : (0.7; 0.75; 0.8; 0.85; 0.9; 3, 3, 3, 3)

L3: (0.7; 0.85; 0.9; 0.85; 0.7; 3, 3, 3, 3)

L4: (0.9; 0.85; 0.7; 0.85; 0.9; 3, 3, 3, 3)

How the production capacity is distributed in above lines?

Page 36: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

36

Aggregation procedurePerformance measures

Production rate estimation:

WIP estimation

estimated directly for the corresponding 2-machine line

Blockage estimation

Starvation estimation

1 or b fMp p

i 1,B , f bi iM M

1, , ,b f bi i i i i i i iBL p Q p p N p p BL

1 1, , ,f b fi i i i i i i iST p Q p p N p p ST

Page 37: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

37

Aggregation procedureNumerical evidence on the accuracy of the estimates

• In general, the PR estimate is relatively accurate with the error within 1% for most cases and 3% for the largest error

• The accuracy of WIP, ST and BL estimates is typically lower

• The highest accuracy of all estimates is for the uniform machine efficiency pattern

• The lowest accuracy is for the inverted bowl and "oscillating" pattern

Page 38: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

38

Aggregation procedureHome work examples

Eight 5-machines with with identical buffer capacity Ni = N varying from 1 to 20

L1 : p = [0.9; 0.9; 0.9; 0.9; 0.9] :uniform patternL2 : p = [0.9; 0.85; 0.8; 0.75; 0.7] : decreasing efficiencyL3 : p = [0.7; 0.75; 0.8; 0.85; 0.9] : increasing efficiencyL4 : p = [0.9; 0.85; 0.7; 0.85; 0.9] : bowl patternL5 : p = [0.7; 0.85; 0.9; 0.85; 0.7] : inverted bowl patternL6 : p = [0.7; 0.9; 0.7; 0.9; 0.7] : oscillatingL7 : p = [0.9; 0.7; 0.9; 0.7; 0.9] : oscillatingL8 : p = [0.75; 0.75; 0.95; 0.75; 0.75] : single bottleneck

Page 39: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

39

Aggregation procedureProperties

Static law of production systems

11 , ,b b fi i i i ip p Q p p N

1 11 , ,f f bi i i i ip p Q p p N

Monotonicity :

The production rate PR(p1, ..., pM, N1, ..., NM-1) is

•strictly increasing in Ni

•strictly increasing in pi

Page 40: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

40

Aggregation procedureProperties

Reversibility : Consider a line L and its reverse Lr with opposite flow direction. Then,

1,L Lr L Lri M iPR PR BL ST

Implications:

1.More capacity at the end of line is not appropriate for buffer capacity assignment

2.If only one buffer is possible and all machines are identical, then it should be in the middle of the line

3.If all machines are identical and a total buffering capacity N* must be

allocated, reversibility implies "symmetric assignment".

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.

Page 41: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

41

Continuous Improvement of Bernoulli Lines

Page 42: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

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?

Page 43: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

43

Constrained Improvability

Page 44: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

44

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

Page 45: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

45

Definitions

Definition: A Bernoulli line is

•improvable wrt BC if there exists a buffer assignment N'i such that iN'i = N* and

PR(p1, ..., pM, N'1, ..., N'M-1) > PR(p1, ..., pM, N1, ..., NM-1)

•improvable wrt WF if there exists a workforce assignment p'i such that i p'i = p* and

PR(p'1, ..., p'M, N1, ..., NM-1) > PR(p1, ..., pM, N1, ..., NM-1)

•improvable wrt BC and WF simultaneously if there exist sequences N'i and p'i such that i N'i = N*, i p'i = p* and

PR(p'1, ..., p'M, N'1, ..., N'M-1) > PR(p1, ..., pM, N1, ..., NM-1)

Page 46: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

46

Improvability with respec to WF

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

Page 47: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

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.

Page 48: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

48

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

Page 49: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

49

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

Page 50: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

50

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)

Page 51: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

51

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

Page 52: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

52

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.

Page 53: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

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

Page 54: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

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

Page 55: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

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).

Page 56: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

Unconstrained Improvability

Page 57: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

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).

Page 58: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

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

0.9 6 0.7 6 0.8 1 0.7 1 0.75 4 0.6 6 0.7 2 0.85

PR/p 0.05 0.06 0.28 0.38 0.31 0.17 0.06 0.05WIPi 5.59 5.39 0.87 0.69 1.68 1.18 0.66

0.8 2 0.83 2 0.77 3 0.8

PR/p 0.369 0.452 0.443 0.022

Page 59: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

Bottleneck buffer

Definition:

•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

Page 60: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

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

Page 61: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

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

Page 62: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

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.

Page 63: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

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.

Page 64: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

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

Page 65: 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding the mathematical models of production lines Understanding.

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.