2.882
Complexity
April 20, 2005
Taesik Lee © 2005
Complexity in AD
• Complexity
“Measure of uncertainty in achieving the desired functional
requirements of a system”
– Difficulty
– Relativity
– Information
– Ignorance
Taesik Lee © 2005
Four types of complexity in AD
Complexity
Measure of uncertainty in achieving FR
Does uncertainty
change with time?
Time-independent
Complexity
Real
Imaginary
complexity complexity
Time-dependent
Complexity
Combinatorial
complexity
Taesik Lee © 2005
Periodic
complexity
Complexity: A measure of uncertainty in achieving the desired set of FRs of a system
• Time-independent real complexity
“Measure of uncertainty when the probability of achieving the functional
requirements is less than 1.0 (because the common range is not identical to the
system range)”
•
Time-independent imaginary complexity
“Uncertainty that arises because of the designer’s lack of knowledge and
understanding of a specific design itself”
• Time-dependent combinatorial complexity
“Time -dependent combinatorial complexity arises because in many situations,
future events cannot be predicted a priori. … This type of time-dependent
complexity will be defined as time-dependent combinatorial complexity.”
•
Time-dependent periodic complexity
“Consider the problem of scheduling airline flights. … it is periodic and thus
uncertainties created during the prior period are irrelevant. … This type of timedependent complexity will be defined as time-dependent periodic complexity.”
Taesik Lee © 2005
Time-independent Real Complexity
• Time-independent real complexity
– caused by system range’s being outside of the design range.
– Real complexity ~ Information content
– Take ui as a random variable
ui =
1 (success) with P(FRi = success) 0 (failure)
1-P(FRi = success) – Information content: I(ui= 1) ” - log2P(FRi =success)
|dr|
Design Range
Common
Range,
AC
System Range,
p.d.f. f(FR)
6
I(p) [bit]
p.d.f.
f(FR)
7
5
4
3
2
1
0
drl
dru
FR
0
0.2
0.4
0.6
0.8
P(FRi =success)
|sr|
Taesik Lee © 2005
1
Time-independent Imaginary complexity
• Imaginary complexity ~ Ignorance
• Ignorance causes complexity.
• Types of ignorance
– Functional requirement
– Knowledge required to synthesize(or identify) design parameters
– Ignorance about the interactions between FRs and DPs
• p (probability of selecting a right sequence)
– For uncoupled design, p= 1
– For decoupled design, p= z/n!
– For coupled design, p = 0
Taesik Lee © 2005
Time-dependent complexity
•
Time-dependency
– Complexity ” Uncertainty in achieving a set of FR
– Complexity is time-dependent if
1) uncertainty (probabilistic) is time-dependent
Time-varying system range
2) behavior of FR is time-dependent
FR = FR(t)
•
Combinatorial / Periodic complexity
– Uncertainty increases indefinitely : combinatorial complexity
– Uncertainty in one period is irrelevant to the next period :
periodic complexity
Taesik Lee © 2005
Origins of complexity and reduction
Taesik Lee © 2005
Time-independent
• Minimize Real complexity by
– Eliminating source of variation
– Desensitizing w.r.t. variation
– Compensating error
• Eliminate Imaginary complexity by
– Achieving uncoupled design
– Identifying design matrix
Taesik Lee © 2005
Time-dependent complexity
Combinatorial complexity
�
Taesik Lee © 2005
Periodic complexity
Time-varying system range
•
•
•
Detect changes in system range
Prevent system range deterioration by design
Bring the system range back into design range by re-initialization
Taesik Lee © 2005
Prevent system range deterioration by design
By eliminating coupling between ‘turn’ and ‘grasp’,
one can effectively delay system range deterioration.
Milled Flat end
of the shaft
Metal
shaft
Slot
Milled Flat end
of the shaft
A
A
Injection molded
nylon Knob
(a)
(a)
(b)
(b)
Section view AA
N. P. Suh, Axiomatic Design: Advances and Applications, 2001
Taesik Lee © 2005
Bring the system range back into design range: Re-
initialization
Example: Design of Low Friction Surface
• Dominant friction mechanism: Plowing by wear debris
• System range (particle size) moves out of the desired design range
� Need to re-initialize
Figure removed for copyright reasons.
Figure removed for copyright reasons.
N. P. Suh and H.-C. Sin, Genesis of Friction, Wear, 1981
Taesik Lee © 2005
S. T. Oktay and N. P. Suh, Wear debris formation and
Agglomeration, Journal of Tribology, 1992
Design of Low Friction Surface
• Periodic undulation re-initializes the system range
Figures removed for copyright reasons.
S. T. Oktay and N. P. Suh, Wear debris formation and agglomeration, Journal of Tribology, 1992
Taesik Lee © 2005
Example: Scheduling of mfg system
Periodicity should be introduced & maintained to prevent
the system from developing chaotic behavior
Subsystem X
Taesik Lee © 2005
Subsystem Y
Problem description
B
C
A
Subsys
X
Station
PTi or CTY
(sec)
Number of
machines
MvPki
(sec)
MvPli
(sec)
IN
-
1
5
-
30
40
50
80
60–5
1
1
1
2
1
5
5
5
5
-
5
5
5
5
5
a
b
c
d
X
D
Y
IN
• Fastest speed *
70 seconds when X determines the
system speed
65-75 seconds when Y determines
the system speed
Inter­
face
Subsystem Y
• Objective
Maximum utilization rate for the machine Y
• Constraint
Transport from C to D must be immediate
* Speed is measured by throughput time: shorter time means faster speed
Taesik Lee © 2005 6th part
ready at X
5th part
ready at X
4th part
ready at X
3rd part
ready at X
FPX
1st part
ready at X
2 FPX
2nd part
ready at X
3 FPX
Taesik Lee © 2005 4 FPX
Time, t
Y process
starts
5th demand
from Y
D5
z4
z3
1st demand
from Y
Y process
strats
z2
z6
FPY,5
FPY,4
6th part
ready at X
FPY,3
5th part
ready at X
FPY,2
4th part
ready at X
FPY,1
3rd part
ready at X
FPX
1st part
ready at X
2 FPX
2nd part
ready at X
3 FPX
Taesik Lee © 2005
4 FPX
Time, t
CTY : … 60sec - 60sec - 60sec - 60sec - 55sec - …
Y FINISH
Y START
Y FINISH
CT Y = 60 sec
1 2
Transport
(next period)
Y START
Max[ CT Y ] = 6 5 sec
Machine d1
1 2
Min[ CT Y ] = 55 sec
Machine c
Machine d2
3 4
Machine b
Machine a
5 6
6
Machine c
7 8
SP=70sec
3 4
4
Machine b
9 10
Machine d1
5 6
Machine a
Machine c
7 8
SP=70sec
Machine b
9 10 Machine a
New panel
New panel
Figure 10. Steady state operation with 70 seconds sending period
From Lee, Taesik. "Complexity Theory in Axiomatic Design." MIT PhD Thesis, 2003.
Taesik Lee © 2005 CTY : … 60sec - 60sec - 55sec - 65sec - 65sec …
Y FINISH
Max[CT ] = 6 5 sec
Y START
Y
Panel #
2
Machine d1
3
3
1
4
Y FINISH
Y START
Max[CT ] = 65 sec
2
Y
Machine d2
4
5
5
6
1
Machine c
7
8
3
4
Machine b
7
SP = 95 sec
Machine d1
5
3
9 10
X X X X
No Transport
6
9 10
2
6
Machine c
Machine a
New panel
7
8
Machine b
X X X X
9 10
9 10
SP = 75 sec
Figure 11 (b)
Y FINISH
Max[ CT ] = 6 5 sec
Y START
Y
Panel #
4
Machine d1
5
6
7
3
1
4
Y FINISH
Y START
Max[ CT ] = 6 5 sec
2
Y
Machine d2
5
Machine a
6
1
Machine c
7
8
8
4
Machine d1
Machine b
9 10
X X
SP = 90 sec
3
5
9 10
X
X
6
Machine a
Machine c
7
8
X X X X
No Transport
New panel
Figure 11 (c)
From Lee, Taesik. "Complexity Theory in Axiomatic Design." MIT PhD Thesis, 2003.
Taesik Lee © 2005 2
7
Average throughput time = (75+75+75+100)/4 = 81.25
From Lee, Taesik. "Complexity Theory in Axiomatic Design." MIT PhD Thesis, 2003.
Taesik Lee © 2005
• Single perturbation from subsystem Y causes incomplete
period in downstream
• The system regains periodicity after the perturbation is
removed but with undesirable performance
• Throughput time is 81.25 seconds in average
– Slower than the system capability
Taesik Lee © 2005
Re-initialization scheme in scheduling
• Define a “renewal” event that imposes period
u(0)
= { u0(t), u1(t), …, uk-1(t), uk(t), uk+1(t), …, uN(t)}
= {0, 0, …, 0, 0, 0, …, 0}
:
u(T-D) = {1, 1, …, 1, 0, 1, …, 1}
u(T)
= {1, 1, …, 1, 1, 1, …, 1}
u(T+e) = {0, 0, …, 0, 0, 0, …, 0} = u(0)
• Scheduling activity is confined within such a period with a goal of
maintaining “periodicity”
– Conditional renewal event
tini = trequest
if trequest ‡ 70 sec ( FPX )
if trequest < 70 sec
tini = 70sec
Taesik Lee © 2005
3rd demand
from Y
2nd demand
from Y
1st demand
from Y
Y process
strats
Y process
strats
FPY,1
Y process
strats
FPY,2
t
5th demand
from Y
Y
4th
process
demand
starts
from Y
Y process
starts
D5
D4
FPY,3
t4
t3
t5
5th part
ready at X
4th part
ready at X
2
t1
Time, t
z1
z2
2nd part
ready at X
•
3rd part
ready at X
Each period is independent (memoryless)
Taesik Lee © 2005
z6
Conclusion
• Breakdown of functional periodicity results in sub-optimal
throughput rate
• Periodicity should be introduced & maintained to prevent
the system from developing chaotic behavior
Taesik Lee © 2005
Example: Cell division
• A cell has a mechanism to coordinate cycles of two
subsystems such that the overall periodicity is
maintained
• Break-down of functional periodicity leads to anomaly of cell division and further chaotic behavior of the system
• Maintaining functional periodicity in the cell cycle is an
important functional requirement for cell division
Taesik Lee © 2005
Overview of the Cell Cycle
Figure removed for copyright reasons.
* Figure taken from Molecular Biology of the Cell, Alberts, Garland Science
Taesik Lee © 2005
Chromosome cycle & Centrosome cycle
Figure removed for copyright reasons.
* Figure taken from Molecular Biology of the Cell, Alberts, Garland Science
Taesik Lee © 2005
Importance of the correct number of chromosomes and centrosomes
• Centrosomal abnormalities
– Chromosome missegregation
– Aneuploidy
Figure removed for copyright reasons.
Figure 2 in Nigg, E. A. "Centrosome abberation: cause or
consequence of cancer progression?" Nature Reviews Cancer 2 (2002): 815-825.
Figure removed for copyright reasons.
* Figure taken from http://www.sivf.com.au/chromosomes.htm
Taesik Lee © 2005
Functional periodicity
B
Figure removed for copyright reasons.
See Figure 1 in Nigg, E. A. "Centrosome abberation: cause or consequence of cancer progression?" Nature Reviews Cancer 2 (2002): 815-825.
C
A
Machine
X
D
IN
Inter­
face
Machine Y
Taesik Lee © 2005
Mechanism
Cdk2 initiates both cycle
ensuring one level of
synchronization
G1Cdk
Cdk2 (S-Cdk)
promote
accumulation
Mitogen-dependent
mechanism
(Extracellular signal)
inactivate
Rb
inactivate
inactivate
G1/Scyclin
G1/SCdk
E2F
Firing origin of
replication
Centriole
split
Replicating DNA
Duplicate
centrosome
Check for completion
of DNA replication
DNA replication
checkpoint ensures
completion of
chromosome
duplication
S-cyclin
S-Cdk
inactivate
Check for
completion
Centrosome cycle lacks a
mechanism to check its completion
Prophase
Prometaphase
Metaphase
Check for
spindle
attachment
Taesik Lee © 2005
Anaphase
Telophase
Hct1­
APC
mutually
inhibits
Centrosome
duplication
Chromosome
duplication
gene
transcription
mutually
inhibits
Cytokinesis
mutually
inhibits
CKI:p27
Conclusion
• A cell has a mechanism to coordinate cycles of two
subsystems such that the overall periodicity is
maintained
• Maintaining functional periodicity in the cell cycle is an
important functional requirement for cell division
– Can pose questions with new perspective
Taesik Lee © 2005
Time-dependent FR
• Functional periodicity
• u(t) = {u1(t), u2(t), … , uN(t)}
T1
(a) Periodic
T2
t
u5
u4
u3
u2
u1
u0
FR State
u5
u4
u3
u2
u1
u0
FR State
FR State
Periodic – There exist Ti s.t. u(Ti ) = u(Tj) with regular transition pattern
Semi-periodic – There exist Ti s.t. u(Ti) = u(Tj) without regular transition
pattern
Aperiodic – None of the above
T1
(b) Semi-periodic
Taesik Lee © 2005 T2
u5
5
u4
4
u3
3
u2
2
u1
1
u0
t
t
(c) Aperiodic
Uncertainty and functional periodicity
• Ps(t) = P(u(t) = u*(t))
– For periodic & semi-periodic FR(t), Ps returns to one at the
beginning of a new period
• Predictability of FR
– (Periodicity) fi (Predictability)
– (Unpredictability) fi (Aperiodicity) �
~ (Aperiodicity) fi ~ (Unpredictability) • Uncertainty in current period is independent of a prior period only if the initial state is properly established Taesik Lee © 2005