The document describes reliability-based design optimization (RBDO) using a cell evolution method. RBDO aims to find optimal designs that satisfy reliability constraints accounting for uncertainties. Traditional RBDO methods are either computationally expensive double-loop approaches or faster single-loop approaches with reduced accuracy. The proposed cell evolution method generates reliability-test cells using genetic algorithms to efficiently and accurately solve RBDO problems. Numerical examples demonstrate the method finds optimal designs matching other approaches but with improved computational efficiency.

1. Reliability-Based Design Optimization
via a Cell Evolution Method
陳奇中
ctchen@fcu.edu.tw
逢甲大學化工系

2. Outline
1. Introduction
2. Reliability-Based Design Optimization
(RBDO)
2.1 Problem formulation
2.2 Traditional solution methods for RBDO
- Double Loop
- Single Loop
3. A Cell Evolution Method for RBDO
3.1 Single objective optimization
3.2 Multi-objective optimization
4. Design Examples
5. Conclusions

3. Introduction
Deterministic Design Optimization
- no uncertainties involved in the design
m in f (d , p )
d
s.t.
g i (d , p ) 0, i 1,  , n1
h j (d , p ) 0, j 1,  , n 2
L U
d d d
w h ere
T
d d1 , d 2 , , d m : d ecisio n variab les
T
p p1 , p 2 ,  , p l : p aram eters

4. Uncertainties ? Uncertainty
is
everywhere.
 Sources of uncertainties
- modeling errors
- physical parameter variations
- change of environments
- unknown dynamics
…
uncertainties
Deterministic design Not reliable

5. Optimal Design Under Uncertainties
m in f ( x, d , p )
x,d
s.t.
g i ( x, d , p ) 0, i 1,  , n1
h j ( x, d , p ) 0, j 1,  , n 2
L U L U
x x x , d d d
w here
d d1 , d 2 , , d m : determ inist ic d ecision variable
x x1 , x 2 ,  , x n : u n certain decision variable
p p1 , p 2 ,  , p l : u n certain param eters

6. Deterministic solution vs. Reliable solution
Stochastic constraint
Reliable solution
Deterministic optimum
*
Deb et al. (2009)

7. Stochastic Programming frameworks
- Here and Now (1/2)
Optimal solution
Diwekar (2002)

8. Stochastic Programming frameworks
- Wait and See (2/2)
Distribution of optimal
design
Objective function and constraints
(Scenario)
Diwekar (2002)

9. Reliability-Based Design Optimization
(RBDO)
m in f (d , μ x , μ p )
d, μx
s.t.
P r G i (d , x , p ) 0 Ri , i 1, ..., n1
g j (d , μ x , μ p ) 0, j 1, ..., n 2
L U L U
d d d , μx μx μx
where x
n
R , x ~ N μx ,σx ,
q
p R , p ~ N μp ,σp ,
Pr( ) Probability function
Ri Design reliability

10. The failure probability and
reliability index
Pr Gi (d, x , p ) 0  x ,p
(x, p ) dxdp
Gi ( d , x ,p ) 0
x ,p
(x, p ) joint probability density function
Reliability level Ri 1 Pi
Pi Pr G i ( d , x , p ) 0
Failure probability
First-order approximation Pi i
Reliability index i Standard normal cumulative dist. Func.

11. Traditional solution methods for RBDO (1/2)
- Double-loop method
Reliability analysis
loop
Optimization
loop Reliability analysis
loop
Reliability analysis
loop
Shan and Wang (2008)

12. Reliability analysis loop (inner loop)
(1/2)
A. RIA (reliability index approach)
Gj > 0
m in U MPP
U
s.t.
Gj U 0
*
N O T E : fo r reliab ility , U j
NOTE: MPP denotes the “most probable point.”

13. Reliability analysis loop (inner loop)
(2/2)
B. PMA (performance measure approach)
Gj > 0
m in G j ( U )
s.t. U MPP
j
w here
1
j
Rj " reliability index "
standard norm al density function
U : U -space , ~ N (0, 1)
*
N O T E : fo r reliab ility , G i U 0.

14. Traditional solution methods for RBDO (2/2)
- Single-loop method
- convert inner reliability loop by using a deterministic
optimization problem KKT optimality conditions
m in f (d , μ x , μ p )
d ,μ x
s.t.
g i (d , x i , p i ) 0, i 1, 2,  , n
r x
Gi
xi x i
2 2
x
Gi p
Gi
Gi approximation
r p
pi p i
2 2
x
Gi p
Gi
L U
d d d
L U
μX μX μX

15. Comparisons of
RBDO Solution methods
Method Advantage Disadvantage
Double-loop accuracy long computation time
Single-loop computationally fast less accuracy
Motivation: accuracy and computational efficiency?
New solution method ?

16. PMA-based RBDO problem
m in f (d , μ x , μ p )
d, μx
s.t.
Gj > 0
* 1
Gi FG i i
0, i 1, ..., n1
MPP
g j (d , μ x , μ p ) 0, j 1, ..., n 2
L U L U
d d d , μx μx μx
where
FG i cumulative distribution function
Calculated from PMA reliability optimization problem
*
Gi

17. Reliability-test cells
- Determination of MPPs
x2
G3 =0
mpp 23
mpp 21
G1 =0
mpp33 mpp 22
mpp32
mpp31 mpp13
mpp11
mpp12
G2 =0
x1

18. A cell generation method
--- sampling method
Step 1: Sobol quasi-random sequence
(Sobol, 1967; Bratley and Fox, 1988)
Step 2: Spherical parameterization method
(Watson, 1983; Zayer et al., 2006)

19. Some template reliability-test
cells (1/2) 2D cells in U-space
β 
1, N 100 β 
1, N 1000
β 
3, N 100
β 
3, N 1000

20. Some template reliability-test
cells (2/2) 3D cells in U-space
 β 
1, N 10000
β 1, N 1000
β 
3, N 1000
β 
3, N 10000

21. A cell evolution
Start
Initialize cell population
algorithm
k = k+1
Yes
Std.( F(Ɵ ) ≤
) ε ?
No
Alleviate premature
Cell generation
stagnation
RS Operation
+
A real-coded genetic algorithm No For each paired
parents, r > λ ?
(Chuang and Chen, 2011) Yes
x2 DRM Operation DBX Operation
G3 =0
mpp 23
Replacement Operation
mpp 21
G1 =0
mpp33 mpp 22
mpp32
mpp31 mpp13
No
Stop criteria met?
mpp11
mpp12
Yes
G2 =0
x1 Stop

22. What is genetic algorithm (GA)?
 GA is a particular class of evolutionary algorithm
Initially developed by Prof. John Holland
"Adaptation in natural and artificial systems“, University of Michigan press, 1975
Based on Darwin’s theory of evolution
“Natural Selection” & “Survival of the fittest”
物競天擇 適者生存 不適者淘汰
 Imitate the mechanism
of biological evolution
- Crossover
- Mutation
- Reprodution

23. Evolution in biology (1/3)
 Organisms produce a number of offspring similar
to themselves but can have variations due to:
(a) Crossover (Sexual reproduction )
Parents offspring
IMG from http://www.tulane.edu/~wiser/protozoology/notes/images/ciliate.gif
Ref. :http://www.cas.mcmaster.ca/~cs777/presentations/3_GO_Olesya_Genetic_Algorithms.pdf

24. Evolution in biology (2/3)
(b) Mutations (Random changes in the DNA sequence)
Before After
IMG from http://offers.genetree.com/landing/images/mutation.png
IMG from http://www.tulane.edu/~wiser/protozoology/notes/images/ciliate.gif
Ref. :http://www.cas.mcmaster.ca/~cs777/presentations/3_GO_Olesya_Genetic_Algorithms.pdf

25. Evolution in biology (3/3)
 Some offspring survive, and produce next
generations, and some don’t:
Ugobe Inc. Pelo
http://www.ugobe.com/Home.aspx
Ref. :http://www.cas.mcmaster.ca/~cs777/presentations/3_GO_Olesya_Genetic_Algorithms.pdf

26. Traditional GA
- binary-coded
 All variables of interest must be encoded as binary
digits (genes) forming a string (chromosome).
Gene – a single encoding of part of the solution space.
Chromosome – a string of genes that represent a solution.
1 gene
1 1 0 1 0 chromosome
IMG from http://static.howstuffworks.com/gif/cell-dna.jpg

27. Real-coded GA (RCGA)
 All genes in chromosome are real numbers
- suitable for most systems.
- genes are directly real values during genetic
operations.
- the length of chromosomes is shorter than that in
binary-coded, so it can be easily performed.
1.1 gene
1.1 0.1 15 10 0.12
chromosome
IMG from http://static.howstuffworks.com/gif/cell-dna.jpg

28. The cell evolution method
- Survival and elimination of cells according to their
fitness

29. Illustrative examples
- Example 1 (Liang et al., 2004)
Results Comparison
Methods DLP/PMAa Single loopb The Proposed
m in f 1 2 Design variables
1 3.4391 3.4391 3.4391
Pr G i ( x ) 0 Ri , i 1, 2 , 3 2
3.2866 3.2864 3.2866
2
x1 x 2
G1 x 1
20 Objective function
2 2
x1 x2 5 x1 x 2 12
G2 x 1
30 120 f μ 6.7257 6.7255 6.7257
80
G3 x 1 2
Constraints
x1 8 x2 5
G1( x ) 0 0 0
0 i
10, i 1, 2 G2(x) 0 0 0
1 2
0.3, G3(x ) -0.5 -0.5097 -0.5096
1
j
Rj 3, j 1, 2, 3 CPU time (s) 138 8.89 11.76
aResults are from Du and Chen [8]. bResults are from Liang et al. [7].

30. MPP determination using
different sampling numbers
MPP points
Sampling
Number MPP1 MPP2 MPP3
50 (2.6173, 2.9168) (3.7578, 2.4438) (4.0812, 3.9152)
100 (2.6168, 2.9179) (3.7573, 2.4446) (4.0807, 3.9161)
500 (2.6179, 2.9182) (3.7581, 2.4450) (4.0819, 3.9165)
1000 (2.6179, 2.9182) (3.7581, 2.4450) (4.0819, 3.9165)
5000 (2.6179, 2.9182) (3.7581, 2.4450) (4.0819, 3.9165)
10000 (2.6179, 2.9182) (3.7581, 2.4450) (4.0819, 3.9165)

31. Obtained solution cells with
different reliability indices (0,1,2,3)
Example 4.1
10
9
8
7
6
2
5
4
3
2
1
0
0 2 4 6 8 10
1

32. Illustrative examples
- Example 2
Reliability index, β 1 2
m in f 1
0 (0%) 7.7883 1.7928
0.5 (69.146%) 7.4476 2.1224
Pr Gi (x) 0 Ri , i 1, 2 , 3
2
x x21
1 (84.134%) 7.1146 2.4269
G1 x 1
20
2 2
x1 x2 5 x1 x2 12
G2 x 1 1.5 (93.319%) 3.2346 2.6961
30 120
G3 x 1 2
80 2 (97.725%) 3.2949 2.8974
x 1
8 x2 5
2.5 (99.379%) 3.3634 3.0941
3 (99.875%) 3.4391 3.2866
0 i
10, i 1, 2
1 2
0.3,
1
j
Rj 3, j 1, 2, 3

33. Solution cells with different
reliability indices (0, 0.5, 1, 1.5, 3)
Example 4.2
10
9
8
7
6
2
5
4
3
2
1
0
0 2 4 6 8 10
1

34. The dramatic change of the reliable
solution with respect to reliability
indices
8 6
Reliability index
0 (0%)
7
0.5 (69.146%)
4 1 (84.134%)
1.5 (93.319%)
6
μ2
μ1 2 (97.725%)
5 2.5 (99.379%)
2
3 (99.875%)
4 4 (99.996%)
5 (99.999%)
3 0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
β

35. Multi-objective reliability-based
design optimization
m in f1 d , μ x , μ p , f 2 d , μ x , μ p ,  , f k d , μ x , μ p
s.t.
Pr ( G i ( d , x , p ) 0) Ri , i 1, 2,  , n1
g j (d , x
, p
) 0, j 1, 2,  , n 2
L U n
d d d x R , x ~ N μx ,σx ,
L U
μx μx μx p
q
R , p ~ N μp ,σp ,

36. Concept of multi-objective
optimization
90%
Comfort
40%
10 k Cost (US$) 100 k

37. Concept of Pareto-optimal
solutions: non-dominated
(Goldberg, 1989)
Feasible objective space
B dominate A
A C dominate A
B, C non-dominated
B
D, E non-dominated
f2
Second level
C E dominate A, B, C
D
D dominate A, B
E
Pareto-optimal front
f1

38. How does multi-objective cell
evolution algorithm work?
Non-dominated Crowding distance New
Parents sorting sorting for each front Population
1
2 Front 1 Front 1 Front 1
RCGA
 Front 2 Front 2 Front 2 N
N CAT Front 3 Front 3 Front 3
Offspring
1
2
Rejected

N

39. An illustrative example
- Multi-objective RBDO (Deb et al., 2009)
m in f 1 x1
1 x2
m in f 2
x1
s.t.
Pr ( G i ( d , x , p ) 0) Ri , i 1, 2
G1 x2 9 x1 6
G2 x2 9 x1 1
0.1 1
1,0 2
5
0.03 , 1.28 , 2.0 , 3.0

40. Pareto front for the RBDO problem
10
= 0
= 1.28
9
= 2
= 3
8
7
6
5
2
f
4
3
2
1
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
f
1

41. Solutions for the RBDO problem
2.5
= 0
= 3
= 1.28
= 2
2
1.5
2
X
1
0.5
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X
1

42. Reliability-based design optimization
Applications in Chemical Engineering
1. Steam pipe design
2. Design of a bio-process
3. Heat sink design

43. Steam pipe design (Ho and Chan, 2011)
Min. cost
2 2
( r2 r1 )
m in f
4
s.t. Surrounding temperature T
Pr G r1 , r2 0 Rj T2
h eq r1 , r2 , K 0
0 .0 4 r1 0 .0 6 5 m , 0 .0 7 5 r2 0 .1 2 m T1
2 K T1 T2 4 4 r1
h eq : h 2 r2 T2 T 2 r2 C T 2 T
ln r2 / r1 Steam
K
h NuD
2 r2
2 r2
1/ 6
0 .3 8 7 R a D
NuD 0 .6 8 / 27
9 /16
1 0 .5 5 9 /
3
g B (T 2 T )( 2 r2 )
RaD
v
2 8
B , C 5 .6 7 1 0
T2 T

44. Reliable solutions
-3
x 10
9.8
9.7
9.6
9.5
Optimal function value
9.4
9.3
9.2
9.1
9
8.9
0 0.5 1 1.5 2 2.5 3
Reliability

45. Design of a bio-process
(Holland, 1975)
m ax P f
m ax P f / t B S f
s.t.
Pr ( G i ( d , x, p ) 0) Ri , i 1~ 4
G1 : 5 tB 15
G 2 : 20 S0 50
C ells G lu cos e O xygen M ore cells
G 3 : 50 K La 300 C ells
G lu cos e O xygen G luconolactone
G 4 : 0.0 5 X0 1.0
G luconolactone W ater G luconic A cid

46. Reliable solutions

47. Design of cylindrical heat sinks
- in-line (Khan et al., 2004)
Thermal analysis
1
R hs Rm R fin s R fin
h fin A fin fin
tb
Rm tanh( m H )
kA
fin
1 mH
R fin s 1
N 1 R bp Nussult Number correlation
Rc R fin Rbp hbp Abp
h fin D
1 4 h fin N u fin
1/ 2
C1 R e D P r
1/ 3
Rc m kf
h c Ac kD 0.785 0.212
[0.2 exp( 0.55S T )]S T S L
C1 0.5
(S T 1)
Friction factor correlation
4 5 .7 8
f K 1{0 .2 3 3
(S T 1)
1 .1
ReD
} Mass balance
ST 1 0 .0 5 5 3
K1 1 .0 0 9 ( )
1 .0 9 / R e D

m U app N T S T HD
S L
1

48. Design of cylindrical heat sinks
- staggered (Khan et al., 2004)
Thermal analysis 1
R fin
R hs Rm R fin s h fin A fin fin
tb tan h ( m H )
Rm fin
kA mH
1 1
R fin s Rbp
N 1 hb p Ab p
Rc R fin Rbp
4 h fin
m
1 kD
Rc
h c Ac
Nussult Number correlation
Friction factor correlation Mass balance
h fin D 1/ 2 1/ 3
N u fin C1 R e D P r
1.29
kf
13.1/ S T 0.68 / S T
f K 1 (378.6 / S T ) / ReD 0 .5 9 1 0 .0 5 3

m U app N T S T HD 0.61S T S L
S L 0.0807
C1 0 .5
K1 1.175( 0.3124
) 0.5 R e D (S T 1) (1 2 exp( 1.09 S T ) )
S T Re D

49. Heat sink performance variations under
change of environmental temperature
(in-line arrangement)
-3 For in-line H=0.01m U app=2 m/s N=7x7
x 10 x 10
-3 For in-line H=0.01m D=0.001m N=7x7
2.5 5
Tamb=300 K Tamb=300 K
Tamb=320 K Tamb=320 K
Tamb=340 K 4.5 Tamb=340 K
4
2
Sgen (W/K)
3.5
Sgen (W/K)
3
1.5
2.5
2
1
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 1.5
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
D (m) x 10
-3
U (m/s)
app

50. Heat sink performance variations under
change of environmental temperature
(staggered arrangement)
-3 For staggered H=0.01m U app=2 m/s N=7x7
x 10 x 10
-3 For staggered H=0.01m D=0.001 m N=7x7
3.5 5
Tamb=300 K Tamb=300K
Tamb=320 K Tamb=320K
Tamb=340 K Tamb=340K
4.5
3
4
2.5
Sgen (W/K)
Sgen (W/K)
3.5
2
3
1.5
2.5
1 2
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
D (m) x 10
-3
U app (m/s)

51. Heat sink performance variations under
un-uniform heat transfer between fins
x 10
-3 For staggered H=0.006m N=5x5
x 10
-3 For in-line H=0.006m N=5x5 6.5
8 Uapp=2
Uapp=2
Uapp=4
U =4 6
app Uapp=6
Uapp=6
7
5.5
5
6
Sgen (W/K)
4.5
Sgen (W/K)
5
4
4
3.5
3
3
2.5
2 2
160 180 200 220 240 260 280 300 320 340 200 250 300 350 400 450
熱傳係數 NuDfin 熱 傳 係 數 NuDfin
in-line staggered

52. RBDO problem formulation
Single objective
Q 
m P
m in 
S gen (
2
) Rhs Entropy generation rate
T am b T am b
s.t. Pr Gi X 0 Ri , i 1~ 9
6 H (m m ) 12
1 D (m m ) 3
1 U app ( m / s ) 6
5 N 20
0.1
Cell population size 100、max. gen.100、
Sampling no. 10000

53. Reliable solutions
(in-line)
β 0 0.5 1 1.5 2 2.5 3 3.5
N 18 18 13 11 10 8 7 6
H(m) 0.0080 0.0072 0.0091 0.0097 0.0096 0.0119 0.0120 0.0120
D(m) 0.0010 0.0010 0.0013 0.0015 0.0016 0.0020 0.0022 0.0026
Uapp 1 1 1.1791 1.5281 1.8884 2.0699 2.4012 2.7829
(m/s)
Sgen(W/K) 0.0535 0.0555 0.0578 0.0696 0.0727 0.0830 0.0929 0.1060
X 100

54. Reliable solutions
(staggered)
β 0 0.5 1 1.5 2 2.5 3 3.5
N 17 17 17 17 13 11 9 9
H(m) 0.0080 0.0076 0.0073 0.0070 0.0091 0.0105 0.0120 0.0120
D(m) 0.0010 0.0010 0.0010 0.0010 0.0013 0.0016 0.0019 0.0019
Uapp 1 1 1 1 1 1 1 1.0824
(m/s)
Sgen(W/K) 0.0472 0.0479 0.0480 0.0495 0.0532 0.0567 0.0629 0.0646
X 100

55. Optimal entropy generation rate
with respect to reliability indices
-4 -4
x 10 x 10
12 6.5
11
6
10
9
Sgen (W/K)
Sgen (W/K)
5.5
8
7
5
6
5 4.5
0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5
in-line staggered

56. Heat dispersion comparisons
(in-line; air velocity 0.7m/s)
Deterministic design Reliable design with β=3
(322.2 < T< 329.9) (314.5 < T< 318.1)

57. Heat dispersion comparisons
(staggered; air velocity 0.7m/s)
Deterministic design Reliable design with β=3
(321.0 < T< 323.6) (312.3 < T< 315.9)

58. RBDO problem formulation
Multi-objective
 Q 2 
m P
S gen ( ) Rhs Entropy generation rate
m in T am b T am b
C ost V olu m e $ Cost
s.t. Pr G j
X 0 Rj , j 1~ 9
6 H (m m ) 12
1 D (m m ) 3
1 U app ( m / s ) 6
5 N 20
0.1
Cell population size 100、max. gen.100、
Sampling no. 10000

59. Obtained Pareto front of the reliable design
(in-line)
1.4
Deterministic
= 1.28
1.35
= 3
1.3
1.25
1.2
Cost (NTD)
1.15
1.1
1.05
1
0.95
0.9
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
Sgen (W /K)

60. Obtained Pareto front of the reliable design
(staggered)
2.2
D eterministic
= 1.28
= 3
2
1.8
1.6
Cost (NTD)
1.4
1.2
1
0.8
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
Sgen (W /K)

61. Results comparison
in-line
Deterministic design Reliable design (β=3)
Solutions min Sgen min. cost min Sgen min. cost
Sgen(W/K) 0.0040 0.0363 0.0101 0.0396
Cost (NTD) 1.31 0.93 1.05 0.90
staggered
Deterministic design Reliable design (β=3)
Solutions
min Sgen min. cost min Sgen min. cost
Sgen(W/K) 0.0018 0.0078 0.0035 0.0423
Cost (NTD) 2.07 1.09 1.49 0.90

62. Conclusions
 Single- and multi-objective cell evolution
methods have been developed for reliability-
based design optimization.
 Simulation results reveal that the proposed
method is able to achieve accurate solution for
RBDO without sacrificing computational
efficiency.
 Application examples indicate the proposed cell
evolution method is a promising approach to
chemical process design under uncertainties.