Chapter 6. Engineering design optimization examples

Chapter 6. Engineering design optimization examples 

Author: Tzu-Chi Liu (2006-08-04); recommended: Yeh-Liang Hsu (2006-08-12). 

Note: This article is Chapter 6 of Tzu-Chi Liu’s PhD thesis “Developing a Fuzzy 

Proportional-Derivative Controller Optimization Engine for Engineering Optimization 

Problems.” 


In this Chapter, the fuzzy PD controller optimization engine is applied to 6 

engineering design optimization problems commonly seen in literature to demonstrate the 

practicality of the fuzzy PD controller optimization engine. 

6.1 Three-bar truss design optimization problem 

6.1.1 Formulating the optimization model 

Figure 6.1 shows a design optimization problem of the three-bar truss with the 

cross-section areas of members A 1 (and A 3 ) and A 2 as design variables [Rao, 1987]. The 

objective of this problem is to find the optimal values of the cross-sectional areas of the 

bars A 1 , A 2 , A 3 , which minimize the weight of the structure, subject to maximize stress 

constraints on the three bars. 

Figure 6.1 Three bar truss design problem 

1 

http://mech.designer.yzu.edu.tw/


Equation (6.1) shows the optimization model of this problem. Note that the design 

variables have been changed to x 1 and x 2 . It is assumed that A 1 = A 3 , so there are only two 

design variables in Equation (6.1). 

min. 

f = 2 2x 

+ x 

1 

2 

( u ) 

( x , x ) σ 

s.t. g1 = σ 

1 1 2 

−1 

≤ 0 , 

g 

g 

( u ) 

( x x ) σ 

1, 

2 

−1 

2 

= σ 

2 

≤ 

0, 

() l 

σ ( x ) −1 

0, 

= σ 

3 1, x2 

3 

≤ 

() l 

g 

4 

= x1 

x1 

−1 

≤ 0, 

( u ) 

g 

5 

= x1 

x1 

−1 

≤ 0, 

() l 

g 

6 

= x2 

x2 

−1 

≤ 0, 

( u ) 

g = x x −1 

0, 

(6.1) 

7 2 2 

≤ 

where σ 1 , σ 2 and σ 3 are the stress induced in members 1, 2 and 3 respectively, σ (u) is the 

maximum permissible stress in tension, σ (l) is the maximum permissible stress in 

compression, x (l) 1 and x (l) 2 are the lower bound on x 1 and x 2 respectively, and x (u) (u) 

1 and x 2 

are the upper bound on x 1 and x 2 respectively. The stresses are given by: 

σ 

1 

( x , x ) 

1 

2 

= P 

2x 

2x 

1 

2 

1 

+ x 

2 

+ 2x 

x 

1 

2 

σ 

2 

( x , x ) 

1 

2 

( x , x ) 

= P 

x 

= P 

1 

+ 

1 

2x 

x 

2 

2 

σ 

3 1 2 


2 

2x1 

+ 2x1x2 

In this example, the parameters are: 

P = 20000lb, σ (u) = 20000psi, σ (l) = -15000psi, 

x (l) i = 0.1in 2 (i = 1, 2), x (u) i = 5in 2 (i = 1, 2) (6.3) 

2 



In the fuzzy PD controller optimization engine, the objective function and all 

constraints can be implicit functions, which cannot be written explicitly in terms of design 

variables. Only the monotonicities of the design variables with respect to the objective and 

constraint function are required. Therefore, the optimization model required by the fuzzy 

PD controller optimization engine is: 

min. 

f 

+ + 

( x x ) 

1 

, 

2 

− − 

( x , x ) 0 

s.t. g 

1 1 2 

≤ , 

− − 

( x x ) 0 

g , 

2 1 

, 

2 

≤ 

+ − 

( x x ) 0 

g , 

g 

g 

g 

3 1 

, 

2 

≤ 

− 

( x ) 0 

4 1 

≤ 

+ 

( x ) 0 

5 2 

≤ 

− 

( x ) 0 

6 2 

≤ 

+ 

( x ) 0 

g (6.4) 

7 2 

≤ 

6.1.2 Monotonicity Analysis 

Table 6.1 is the corresponding monotonicity table for Equation (6.4). Figure 6.2 

shows the complete output from MONO, including rigorous Monotonicity Analysis steps 

and global knowledge about the optimization model. Note that this conclusion is obtained 

using only the monotonicity signs of the design variables with respect to the objective 

function and constraints (the monotonicity table in Table 6.1). Explicit formulations of the 

objective function and the constraints are not required. 

3 



Table 6.1 Monotonicity table of the three-bar truss problem 

x 1 x 2 

f + + 

g 1 - - 

g 2 - - 

g 3 + - 

g 4 - . 

g 5 + . 

g 6 . - 

g 7 . + 

----------------- PC MONO V3.2 : C:\MONO\threebar.out ------------------ 

TABLE 0 . 

x1 x2 

F + + 

G1 - - 

G2 - - 

G3 + - 

G4 - . 

G5 + . 

G6 . - 

G7 . + 

****************************** 

* GLOBAL FACTS * 

4 



****************************** 

* Irrelevant Variables : 

None 

* Unbounded Variables : 

None 

* Directed Equality Constraints : 

None 

* Critical Constraints : 

None 

* Uncritical (Relaxed) Constraints : 

None 

* Conditionally Critical Sets : 

One of ( G1 G2 G4 ) must be critical for x1. 

One of ( G1 G2 G3 G6 ) must be critical for x2. 

Figure 6.2 The output of MONO program of the three-bar truss example 

As shown in Figure 6.2, two conditionally critical sets exist in the three-bar truss 

design optimization problem. One of the constraints in the conditionally critical set (g 1 , g 2 , 

g 4 ) must be critical for design variable x 1 , and one of the constraints in the other 

conditionally critical set (g 1 , g 2 , g 3 , g 6 ) must be critical for design variable x 2 . 

6.1.3 Defining the initial values and move limits of the design variables 

The inputs of the fuzzy PD controller optimization engine are the constraint function 

values and the change of the constraint function values. The quantitative definitions for the 

error inputs (e) are the values of the constraint functions (g 1 , g 2 , g 3 , g 4 and g 6 ). The 

quantitative definitions for the error change inputs (Δe) are the change of the values of the 

constraint functions (Δg 1 , Δg 2 , Δg 3 , Δg 4 and Δg 6 ). 

5 



The values of the error inputs at different quantized levels are defined by the initial 

values of the constraint functions. In this example, the initial values of the design variables 

are: 

x 1 = 3 in 2 , x 2 = 3 in 2 (6.5) 

The constraint function values can be evaluated at the initial design point, which give: 

initial 

g 1 

initial 

g 4 

initial 

initial 

= -0.7643, = -0.8620, = -1.0732, 

g 2 

initial 

=-0.9667, = -0.9667, (6.6) 

g 6 

g 3 

The designer can define the “move limits” according to the characteristics and 

physical properties of the design variables. In this example, the values of the move limits 

are: 

(Δx 1 ) max = 0.1, (Δx 2 ) max = 0.11. (6.7) 

Note that there is a slight difference in move limits between design variables x 1 and x 2 

to avoid premature convergence in this case. The values of the quantization level of error 

change inputs are evaluated by Equation (5.10) as follows: 

Δg 1 (x 1 , x 2 ) max = 0.0082, Δg 2 (x 1 , x 2 ) max = 0.0051 Δg 3 (x 2 ) max = 0.0044, 

Δg 4 (x 1 ) max = 0.0012, Δg 6 (x 2 ) max = 0.0013 (6.8) 

6.1.4 The optimization results 

Using the initial design point and move limits described in the previous section, the 

fuzzy PD controller optimization engine terminates after 75 iterations, when the change in 

objective function value in consecutive iterations is less than 0.01%, and all constraints are 

satisfied with a tolerance of 0.01% of the initial values of the constraints. Table 6.2 

compares the results obtained by the fuzzy PD controller optimization engine and that from 

the literature. Note that only constraint g 1 is active at the optimum design point, though 

there are two design variables. Figure 6.3 shows the iteration histories of the objective 

function. 

Note that in this process, the function value of each constraint is evaluated 77 times, 

and no sensitivity information required. The designers only define the initial values and the 

move limits of the design variables in the process of the fuzzy PD controller optimization 

engine. 

6 



Table 6.2 Comparison of the result for the three-bar truss design problem 

Fuzzy PD Rao [1987] 

Objective Function 2.6507 2.6335 

Iteration 75 - 

x 1 0.7511 0.7871 

x 2 0.5262 0.4074 

g 1 0.0000 0.0415 

g 2 -0.3312 -5.3280 

g 3 -1.2485 -20.3695 

g 4 -0.8669 -0.6871 

g 5 -0.8498 -4.2129 

g 6 -0.8100 -0.3074 

g 7 -0.8948 -4.5926 

Figure 6.3 The iteration histories of the objective function of the three-bar truss example 

7 



6.2 A tension/compression spring design optimization problem 


This problem is from Arora [1989] and Belegundu [1982], which minimizes the 

weight of a tension/compression spring as shown Figure 6.4. The tension/compression 

spring design optimization problem subject to constraints on minimum deflection (g 1 ), 

shear (g 2 ), surge frequency (g 3 ), limits on outside diameter (g 4 ) and on design variables. 

The design variables are the mean coil diameter D, the wire diameter d and the number of 

active coils N. The mathematical formulation of this design problem can be described as 

Equation (6.9). 

N 

D 

d 

Figure 6.4The tension/compression spring design optimization problem 

min. 

F 

= 

2 

( N + 2) Dd 

4 

71785xd 

s.t. g 

1 

= −1 

≤ 0 , 

3 

D N 

2 

4D 

− dD 1 

g 

2 

= 

+ −1 

≤ 0 , 

2 

12566 

5108d 

3 4 

( Dd − d ) 

2 

D N 

g 

3 

= −1 

≤ 0 , 

140.45d 

d + D 

g 

4 

= −1 

≤ 0 . (6.9) 

1.5 

8 



The optimization model required by the fuzzy PD controller optimization engine is: 

min. F 

+ + + 

( d , D , N ) 

+ − − 

( d , D N ) 

s.t. g 

1 

, 

, 

− + 

( d D ) 

g , 

2 , 

− + + 

( d , D N ) 

g , 

3 

, 

+ + 

( d D ) 

g 

4 

, 

. (6.10) 



shows the complete output from MONO. Design variable D has only one critical constraint 

g 1. One of the constraints in the conditionally critical set (g 2 and g 3 ) must be critical for 

design variable d. The monotonicity sign of design variable N is “indeterminate” in the 

objective function of the final monotonicity table in Figure 6.5. 

Table 6.3 Monotonicity table of the spring design example 

d D N 

f + + + 

g 1 + - - 

g 2 - + . 

g 3 - + + 

g 4 + + . 

----------------- PC MONO V3.2 : C:\MONO\spring02.out ------------------ 

TABLE 0 . 

d D N 

F + + + 

9 



G1 + - - 

G2 - + . 

G3 - + + 

G4 + + . 

1. G1 is critical for D by MP1. 

2. D and G1 are eliminated. 

3. Monotonicity of G2 wrt d becomes 'i'. 

4. Monotonicity of G3 wrt d becomes 'i'. 

5. Monotonicity of F wrt N becomes 'i'. 

6. Monotonicity of G2 wrt N becomes '-'. 

7. Monotonicity of G3 wrt N becomes 'i'. 

8. Monotonicity of G4 wrt N becomes '-'. 

TABLE 1 . 

d 

N 

F + i 

G2 i - 

G3 i 

i 

G4 + - 

****************************** 


****************************** 

10 




None 


None 


None 


D eliminated by MP1, G1 critical below. 


None 


One of ( G2 G3 ) must be critical for d. 

Figure 6.5 The output of MONO program of the spring design example 




error inputs (e) are the values of the all constraint functions. The quantitative definitions 

for the error change inputs (Δe) are the change of the values of the all constraint functions. 



are: 

d = 0.1, D = 0.25, N = 13. (6.11) 


11 



initial 

g 1 

initial 

g 3 

initial 

= 34.34, = -0.86, 

g 2 

initial 

g 4 

= -0.94, = -0.77, (6.12) 

In this example, the values of the move limits are: 

(Δd) max = 0.001, (ΔD) max = 0.01, (ΔN) max = 0.1 (6.13) 

Therefore, the values of the quantization level of error change inputs are evaluated by 

Equation (5.10) as follows: 

Δg 1 (x 0 ) max = 6.5482, Δg 2 (x 0 ) max = 0.0089, 

Δg 3 (x 0 ) max = 0.0058, Δg 4 (x 0 ) max = 0.0073. (6.14) 

6.2.4 Optimization Results 




satisfied with a tolerance of 0.01% of the initial values of the constraints. Table 6.4 shows 

the comparison of the results for the speed reducer design optimization problem. Note that 

only constraint g 1 , g 2 and g 4 are active at the optimum design point, though there are three 

design variables. Figure 6.6 shows the iteration histories of the objective function. 




engine. 

12 



Table 6.4 Comparison of the results for the spring design example 

Fuzzy PD 

Coello and 

Monts (2002) 

Arora (1989) 

Belegundu 

(1982) 

Objective 

function value 

0.0126503 0.0126810 0.0127303 0.0128334 

Iteration 225 - - - 

d 0.052362 0.051989 0.053396 0.050000 

D 0.373153 0.363965 0.399180 0.315900 

N 10.36486 10.89052 9.185400 14.25000 

g 1 0.001987 -0.000013 0.000019 -0.000014 

g 2 0.000084 -0.000021 -0.000018 -0.003782 

g 3 -0.803753 -4.061338 -4.123832 -3.938302 

g 4 -0.716237 -0.722698 -0.698283 -0.756067 

Figure 6.6 The iteration history of objective function of the spring design example 

13 



6.3 Tubular column design optimization problem 


Figure 6.7 shows an example for designing a uniform column of tubular section to 

carry a compressive load P = 2500 kgf for minimum cost [Rao, 1996]. The column is made 

up of a material that has a yield stress (σ y ) of 500 kgf/cm 2 , modulus of elasticity (E) of 

0.85×106 kgf/cm 2 , and density (ρ) of 0.0025 kgf/cm 3 . The length of the column (L) is 250 

cm. The stress included in the column should be less than the bucking stress as well as the 

yield stress. The mean diameter of the column is restricted between 2 and 14 cm, and 

columns with thickness outside the range 0.2 to 0.8 cm are not available in the market. The 

cost of the column includes material and construction costs and can be taken as 9.82dt+2d, 

where d is the mean diameter of the column in centimeters, and t is tube thickness. 

Figure 6.7 Tubular column under compression design problem 

Equation (6.9) is the optimization model of this problem. The design variables are the 

mean diameter (d) and tube thickness (t). The objective is to minimize the cost. Constraint 

g 1 and g 2 indicates that the stress induced is lower than the yield stress (g 1 ) and buckling 

stress (g 2 ). Constraints g 3 to g 6 are side constraints. This design optimization model can be 

summarized as in Equation (6.15). 

14 



( d, 

t) = 9.82dt 

d 

min . f + 2 

P 

s. t. 

g1 = −1 

≤ 0, 

π d tσ 

y 

2 

8PL 

g 

2 

= 

−1 

≤ 0, 

3 

π Edt 

2 2 

( d + t ) 

g 

3 

= 2.0 d −1 

≤ 0, 

g 

4 

= d 14 −1 

≤ 0, 

g 

5 

= 0.2 t −1 

≤ 0, 

g = t 0.8 −1 

0, 


6 

≤ 

In the fuzzy PD controller optimization engine, only the monotonicities of the design 

variables with respect to the objective and constraint function are required. Therefore, the 

optimization model required by the fuzzy PD controller optimization engine is: 

min. 

( d 

+ t + ) 

f , 

( d 

− , t − ) 0 

s.t. 

g1 ≤ 

g 

g 

g 

g 

( d 

− , t − ) 0 

2 

≤ 

− 

( d ) 0 

3 

≤ 

+ 

( d ) 0 

4 

≤ 

− 

( t ) 0 

5 

≤ 

+ 

( t ) 0 


6 

≤ 



shows the complete output from MONO. There are two conditionally critical sets. One of 

the constraints in the conditionally critical set (g 1 , g 2 , g 3 ) must be critical for design 

15 



variable d, and one of the constraints in the other conditionally critical set (g 1 , g 2 , g 5 ) must 

be critical for design variable t. 

Table 6.5 Monotonicity table of the tubular column problem 

d 

t 

f + + 

g 1 - - 

g 2 - - 

g 3 - . 

g 4 + . 

g 5 . - 

g 6 . + 

----------------- PC MONO V3.2 : C:\MONO\tubular.out ------------------ 

TABLE 0 . 

x1 x2 

F + + 

G1 - - 

G2 - - 

G3 - . 

G4 + . 

G5 . - 

G6 . + 

****************************** 


16 



****************************** 


None 


None 


None 


None 


None 




Figure 6.8 The output of MONO program of the tubular column example 




error inputs (e) are the values of the constraint functions (g 1 , g 2 , g 3 and g 5 ). The 


constraint functions (Δg 1 , Δg 2 , Δg 3 and Δg 5 ). 



are: 

17 



d = 14.0 cm, t = 0.8 cm (6.17) 


initial 

g 1 

initial 

g 3 

initial 

= -0.8579, = -0.9785, 

g 2 

initial 

= -0.8571, = -0.7500, (6.18) 

g 5 


(Δd) max = 1.0, (Δt) max = 0.1. (6.19) 



Δg 1 (d, t) max = 0.0328, Δg 2 (d, t) max = 0.0092, 

Δg 3 (d) max = 0.0110, Δg 5 (t) max = 0.0357. (6.20) 






compares the results obtained from the fuzzy PD controller optimization engine and that 

from the literature. Note that two stress constraints g 1 and g 2 are active constraints, and 

side constraint g 5 is close to active too. Figure 6.9 shows the iteration histories of the 

objective function. 




engine. 

18 



Table 6.6 Comparison of the result for the tubular column design problem 

Fuzzy PD Rao [1996] 

Objective Function 25.5316 26.5323 

Iteration 53 

d 5.4507 5.440 

t 0.2920 0.293 

g 1 -7.8×10 -5 -0.8579 

g 2 -7.8×10 -5 -0.9785 

g 3 -0.6331 -0.8571 

g 4 -0.6107 0.0000 

g 5 -0.3151 -0.7500 

g 6 -0.6350 0.0000 

Figure 6.9 The iteration histories of the objective function of the tubular column example 

19 



6.4 Welded beam design optimization problem 


Figure 6.10 shows the welded beam structure which consists of beam A and the weld 

required to hold the beam to member B [Ragsdell and Phillips, 1976]. The objective is to 

find a feasible set of dimensions h, l, t and b to carry a certain load (P) while maintaining 

the minimum total fabrication cost. The constraints are on shear stress (τ), bending stress in 

the beam (σ), bucking load on the bar (P c ), and end deflection on the beam (δ). The design 

optimization model can be summarized in Equation (6.21). 

Figure 6.10 Welded Beam design problem 

min. 

s.t. 

( L l) 

2 

f = 1.10471h 

l + 0. 04811tb 

+ 

( )/ 

−1 

0 

g = τ x τ , 

1 max 

≤ 

( ) −1 

0 

g = σ x σ , 

g 

2 

/ 

max 

≤ 

= h / b −1 

3 

≤ 

0, 

( L + ) −1 

0 

2 

g 

4 

= 0.02094h 

+ 0.00962tb 

l ≤ , 

g = 0.125/ h −1 

0, 

5 

≤ 

( ) −1 

0 

g = δ x δ , 

6 

/ 

max 

≤ 

20 


( ) −1 

0 


g 

7 

= P / x ≤ , (6.21) 

P c 

where x is a vector of the design variables (h, l, t, b), L is the overhang length if the beam 

(14 in), τ max is the allowable design shear stress of weld (136,000 psi), τ(x) is the weld 

shear stress, σ max is the allowable design yield stress for the bar material (36,600 psi), σ(x) 

is the bending stress, P c (x) is the bar bucking load, P is the loading condition (6,000 

lb), δ max is the allowable bar end deflection, and δ(x) is the bar end deflection. The terms 

τ(x), σ(x), P c (x) and δ(x) are expressed as follows: 

l 

τ ( x ) = ( τ ′) + 2τ 

′ τ ′′ + ( τ ′ 

) 2 

, 

R 

2 2 

6PL 

σ ( x) 

= , 

2 

t b 

3 

4PL 

δ ( x) 

= , 

3 

Et b 

2 6 

4.013E 

t b 36 ⎛ t E 

( ) 

⎟ ⎞ 

P = c 

x ⎜ 

1 

2 − 

L ⎝ 2L 

4 G , (6.22) 

⎠ 

where 

P 

τ ′ = , 

2hl 

MR ⎛ l ⎞ 

τ ′′ = , M = P⎜ 

L + ⎟ , 

J ⎝ 2 ⎠ 

2 

l ⎛ h + t ⎞ 

R = + ⎜ ⎟ , 

4 ⎝ 2 ⎠ 

2 

⎪⎧ 

⎡ 

2 

2 

l ⎛ h + t ⎞ ⎤⎪⎫ 

J = 2⎨ 

2hl⎢ 

+ ⎜ ⎟ ⎥⎬ 


⎪⎩ ⎢⎣ 

12 ⎝ 2 ⎠ ⎥⎦ 

⎪⎭ 

The optimization model required by the fuzzy PD controller optimization engine is: 

min. 

+ + + + 

( h , l , t b ) 

f , 

− − − 

( h , l t ) 

s.t. g 

1 

, 

, 

( t 

− b − ) 

g , 

2 , 

( h 

+ b − ) 

g , 

3 

, 

21 



+ + + + 

( h , l , t b ) 

g , 

4 , 

− 

( h ) 

g 5 

, 

( t 

− b − ) 

g , 

6 

, 

( t 

− b − ) 

g 

7 

, 




shows the complete output from MONO. Constraint g 1 is critical for design variable l. One 

of the constraints in the conditionally critical set (g 2 , g 3, g 6 , g 7 ) must be critical for design 

variable b. The monotonicity sign of design variables h and t are “indeterminate” in the 

objective function of the final monotonicity table in Figure 6.11. 

Table 6.7 Monotonicity table of the welded beam problem 

h l t b 

f + + + + 

g 1 - - - . 

g 2 . . - - 

g 3 + . . - 

g 4 + + + + 

g 5 - . . . 

g 6 . . - - 

g 7 . . - - 

----------------- PC MONO V3.2 : C:\MONO\Welded.out ------------------ 

TABLE 0 . 

h l t b 

F + + + + 

G1 - - - . 

22 



G2 . . - - 

G3 + . . - 

G4 + + + + 

G5 - . . . 

G6 . . - - 

G7 . . - - 

1. G1 is critical for l by MP1. 

2. l and G1 are eliminated. 

3. Monotonicity of F wrt h becomes 'i'. 

4. Monotonicity of G4 wrt h becomes 'i'. 

5. Monotonicity of F wrt t becomes 'i'. 

6. Monotonicity of G4 wrt t becomes 'i'. 

TABLE 1 . 

h t b 

F i i + 

G2 . - - 

G3 + . - 

G4 i i + 

G5 - . . 

G6 . - - 

G7 . - - 

23 



****************************** 


****************************** 


None 


None 


None 


l eliminated by MP1, G1 critical below. 


None 


One of ( G2 G3 G6 G7 ) must be critical for b. 

Figure 6.11 The output of MONO program of the welded beam example 




error inputs (e) are the values of the constraint functions (g 1 , g 2 , g 3 , g 4 , g 5 , g 6 and g 7 ). The 


constraint functions (Δg 1 , Δg 2 , Δg 3 , Δg 4 , Δg 5 , Δg 6 and Δg 7 ). 

24 





are: 

h = 2 in, l = 10 in, t = 10 in, b = 2 in (6.25) 


initial 

g 1 

initial 

g 5 

g 2 

initial 

initial 

initial 

initial 

= -0.9620, = -0.9160, = 0.000, = 3.7023, 

g 6 

g 3 

initial 

g 7 

= -0.9375, = -0.9956, = -0.9990. (6.26) 

g 4 

The designer can define the “move limits” according to the characteristics and 

physical properties of the design variables. In this example, the values of the move limits 

are: 

(Δh) max = 0.1, (Δl) max = 0.1. (Δt) max = 0.1, (Δb) max = 0.1. (6.27) 



Δg 1 (h, l, t) max = -0.0962, Δg 2 (t, b) max = -0.0916, Δg 3 (h) max = 0.0000, 

Δg 4 (h, l, t, b) max = 0.3702, Δg 5 (h) max = -0.0938, Δg 6 (t, b) max = -0.0996, 

Δg 7 (t, b) max = -0.1000 (6.28) 






compares the results obtained from the fuzzy PD controller optimization engine and that 

from the literature. Note that constraint g 1 , g 2 , g 3 , g 5 and g 6 are active at the optimum 

design point. Figure 6.12 shows the iteration histories of the objective function. 

25 



Figure 6.12 The iteration history of the objective function of the welded beam example 

Note that this problem has many local minimums. Table 6.8 also shows the 

optimization results starting from 2 other different starting points: 

Fuzzy PD 02: h = 1.0, l = 5, t = 10, b = 2; move limits (Δh) max = 0.01, (Δl) max = 0.01, 

(Δt) max = 0.01, (Δb) max = 0.01. 

Fuzzy PD 03: h = 2, l = 5, t = 10, b = 2; move limits (Δh) max = 0.1, (Δl) max = 0.1, 

(Δt) max = 0.1, (Δb) max = 0.01. 

Table 6.9 shows the optimization results using different values of the quantization 

level of inputs and output. The cases of Fuzzy PD 04 to 06 use the same initial values and 

move limits to the case of Fuzzy PD 01 to 03. It can be observed that there is no significant 

difference using different initial values. 

26 



Table 6.8 Comparison of the result for the welded beam design problem 

Fuzzy PD 01 Fuzzy PD 02 Fuzzy PD 03 

Ragsdell and 

Phillips(1976) 

Coello (2000) 

Objective 

function 

2.0182 1.8052 1.8362 2.386 1.748 

Iteration 139 796 1157 - 900,000 

h 0.1740 1.6887 0.2326 0.2455 0.2088 

l 4.8075 4.4207 3.1795 6.1960 3.4208 

t 8.1836 9.1658 8.4355 8.2730 8.9975 

b 0.2508 2.0509 2.3609 0.2455 0.2100 

g 1 3.9×10 -5 8.2×10 -6 1.1×10 -4 -0.4223 -9.5×10 -5 

g 2 5.6×10 -5 -0.0250 4.9×10 -5 -1.6×10 -4 -0.0118 

g 3 -0.3062 -0.1766 -0.0144 0.0000 -0.0057 

g 4 -0.6279 -0.6662 -0.6697 -0.6041 -0.6824 

g 5 -0.2817 -0.2598 -0.4627 -0.4908 -0.4013 

g 6 -0.9361 -0.9444 -0.9380 -0.9368 -0.9426 

g 7 -0.4099 1.1×10 -4 -0.3069 -0.3753 -0.0571 

Table 6.9 Comparison of the result using different quantization factor 

Fuzzy PD 04 Fuzzy PD 05 Fuzzy PD 06 

The quantization level of inputs 4 4 4 

The quantization level of output 4 4 4 

Objective function 2.1498 1.8107 1.8571 

Iteration 160 958 876 

h 1.3866 1.6674 2.4131 

l 6.5110 4.4963 3.0785 

t 8.2413 9.1642 8.3195 

b 2.4735 2.0510 2.4271 

g 1 1.0×10 -4 9.1×10 -5 9.7×10 -5 

g 2 3.4×10 -5 -0.02467 1.3×10 -4 

g 3 -0.4394 -0.1871 -0.0057 

g 4 -0.5973 -0.6649 -0.6670 

g 5 -0.0985 -0.2503 -0.4820 

g 6 -0.9366 -0.9444 -0.9372 

g 7 -0.3875 1.7×10 -6 -0.3559 

27 



6.5 Speed reducer design optimization problem 


The design of the speed reducer [Golinski, 1973], shown in Figure 6.13, is considered 

with the face width (b), module of teeth (m), number of teeth on pinion (z), length of shaft 

1 between bearings (l 1 ), length of shaft 2 between bearings (l 2 ), diameter of shaft 1 (d 1 ), 

and diameter of shaft 2 (d 2 ). The objective is to minimize the total weight of the speed 

reducer. The constraints include limitations on the bending stress of gear teeth, surface 

stress, transverse deflections of shafts 1 and 2 due to transmitted force, and stresses in 

shafts 1 and 2. The design optimization model can be summarized in Equation (6.29). 

l 2 

z1 

l 1 

d 2 

d 1 

min. 

f 

− 

Figure 6.13 Speed reducer design problem 

2 

2 

( b, 

m, 

z, 

l1, 

l2 

, d1, 

d 

2 

) = 0.7854bm 

( 3.3333z 

+ 14.9334z 

− 43.0934) 

2 2 

3 3 

2 2 

1.508b( d + d ) + 7.477( d + d ) + 0.7854( l d + l d ) 

27 

s.t. g 

1 

= −1 

≤ 0, 

2 

bm z 

g 

g 

1 

397.5 

= −1 

2 2 

bm z 

2 

≤ 

1.93 

= 

3 

mzl d 

2 

−1 

3 

≤ 

4 

1 1 

1.93 

g 

4 

= −1 

≤ 

3 4 

mzl d 

2 

2 

0, 

0, 

0, 

1 

2 

1 

1 

2 

2 

28 


( 745l 

/ mz) 

2 

6 

1 

+ 16.9 × 10 

g 

5 

= 

−1 

≤ 0 , 

3 

110d 

( 745l 

/ mz) 

2 

1 

6 

1 

+ 157.5× 

10 

g 

6 

= 

−1 

≤ 0, 

3 

85d 

g 

7 

= mz / 40 −1 

≤ 0, 

g = 5m 

/ b −1 

0, , 

8 

≤ 

g 

9 

= b /12m −1 

≤ 0, 

g 

10 

= 2.6 / b −1 

≤ 0, 

g 

11 

= b / 3.6 −1 

≤ 0, 

g 

12 

= 0.7 / m −1 

≤ 0, 

g 

13 

= m / 0.8 −1 

≤ 0, 

g = 17 / z −1 

0 , 

14 

≤ 

g = z / 28 −1 

0, 

15 

≤ 

g = .3/ l −1 

0, 

16 

7 

1 

≤ 

g = l / 8.3 −1 

0, 

17 1 

≤ 

g = .3/ l −1 

0, 

18 

7 

2 

≤ 

g = l / 8.3 −1 

0, 

19 2 

≤ 

g = .9 / d −1 

0, 

20 

2 

1 

≤ 

g = d / 3.9 −1 

0, 

21 1 

≤ 

g = .0 / d −1 

0 , 

22 

5 

2 

≤ 

2 


29 



g = d / 5.5 −1 

0, 

23 2 

≤ 

1.5d1 

+ 1.9 

g 

24 

= −1 

≤ 0 , 

l 

1 

1.1d 

2 

+ 1.9 

g 

25 

= −1 

≤ 0 . (6.29) 

l 

2 

There are 7 design variables and 25 constraints. The optimization model required by 

the fuzzy PD controller optimization engine is: 

min. 

F 

+ + + + + + + 

( b , m , z , l , l , b b ) 

− − − 

( b , m z ) 

s.t. g 

1 

, , 

− − − 

( b , m z ) 

g , 

2 

, 

− − + − 

( m , z , l b ) 

3 1 

, 

1 2 1 

, 

g , 

− − + − 

( m , z , l b ) 

g , 

4 2 

, 

− − + − 

( m , z , l b ) 

g , 

5 1 

, 

− − + − 

( m , z , l b ) 

g , 

6 2 

, 

+ + 

( m z ) 

g , 

7 

, 

( b 

− m + ) 

g , 

8 

, 

+ − 

( b m ) 

g , 

9 

, 

− 

( b ) 

g 10 

, 

+ 

( b ) 

g 11 

, 

− 

( m ) 

g 12 

, 

+ 

( m ) 

g 13 

, 

1 

1 

2 

2 

2 

30 



− 

( z ) 

g 14 

, 

+ 

( z ) 

g 15 

, 

− 

( ) 

g , 

16 l 1 

+ 

( ) 

g , 

17 l 1 

− 

( ) 

g , 

18 l 2 

+ 

( ) 

g , 

19 

l 2 

− 

( ) 

g , 

20 b 1 

+ 

( ) 

g , 

21 b 1 

− 

( ) 

g , 

22 

b 2 

+ 

( ) 

g , 

23 b 2 

+ 

( l ) 

− 

b 

g , 

24 1 − 

, 

− + 

( l ) 

g . (6.30) 

25 2 

, b2 



shows the complete output from MONO. Note that there are 7 conditionally critical sets in 

this problem. 

31 



Table 6.10 Monotonicity table of the speed reducer design problem 

b m z l 1 l 2 b 1 b 2 

f + + + + + + + 

g 1 - - - . . . . 

g 2 - - - . . . . 

g 3 . - - + . - . 

g 4 . - - . + . - 

g 5 . - - + . - . 

g 6 . - - . + . - 

g 7 . + + . . . . 

g 8 - + . . . . . 

g 9 + - . . . . . 

g 10 - . . . . . . 

g 11 + . . . . . . 

g 12 . - . . . . . 

g 13 . + . . . . . 

g 14 . . - . . . . 

g 15 . . + . . . . 

g 16 . . . - . . . 

g 17 . . . + . . . 

g 18 . . . . - . . 

g 19 . . . . + . . 

g 20 . . . . . - . 

g 21 . . . . . + . 

g 22 . . . . . . - 

g 23 . . . . . . + 

g 24 . . . - . + . 

g 25 . . . . - . + 

32 



----------------- PC MONO V3.1 : speed03.out ------------------ 

TABLE 0 . 

b m z l1 l2 b1 b2 

F + + + + + + + 

G1 - - - . . . . 

G2 - - - . . . . 

G3 . - - + . - . 

G4 . - - . + . - 

G5 . - - + . - . 

G6 . - - . + . - 

G7 . + + . . . . 

G8 - + . . . . . 

G9 + - . . . . - 

G10 - . . . . . . 

G11 + . . . . . . 

G12 . - . . . . . 

G13 . + . . . . . 

G14 . . - . . . . 

G15 . . + . . . . 

G16 . . . - . . . 

G17 . . . + . . . 

33 



G18 . . . . - . . 

G19 . . . . + . . 

G20 . . . . . - . 

G21 . . . . . + . 

G22 . . . . . . - 

G23 . . . . . . + 

G24 . . . - . + . 

G25 . . . . - . + 

****************************** 


****************************** 


None 


None 


None 


None 


None 

34 




One of ( G1 G2 G8 G10 ) must be critical for b. 

One of ( G1 G2 G3 G4 G5 G6 G9 G12 ) must be critical for m. 

One of ( G1 G2 G3 G4 G5 G6 G14 ) must be critical for z. 

One of ( G16 G24 ) must be critical for l1. 

One of ( G18 G25 ) must be critical for l2. 

One of ( G3 G5 G20 ) must be critical for b1. 

One of ( G4 G6 G9 G22 ) must be critical for b2. 

Figure 6.14 The output of MONO program of the speed reducer example 




error inputs (e) are the values of the constraint functions (g 1 , g 2 , g 3 , g 4 , g 5 , g 6 , g 8 , g 9 , g 10 , g 12 , 

g 14 , g 16 , g 18 , g 20 , g 22 , g 24 , and g 25 ). The quantitative definitions for the error change inputs 

(Δe) are the change of the values of the constraint functions (Δg 1 , Δg 2 , Δg 3 , Δg 4 , Δg 5 , Δg 6 , 

Δg 8 , Δg 9 , Δg 10 , Δg 12 , Δg 14 , Δg 16 , Δg 18 , Δg 20 , Δg 22 , Δg 24 , and Δg 25 ). 



are: 

b = 3.6, m = 0.8, z = 28, l 1 = 8.3, 

l 2 = 8.3, b 1 = 3.9, b 2 , = 5.5. (6.31) 


initial 

g 1 

initial 

g 5 

initial 

g 10 

g 2 

initial 

initial 

initial 

initial 

= -0.58, = -0.78, = -0.79, = -0.95, 

g 6 

initial 

initial 

= -0.37, = -0.11, = -0.44, = -0.11, 

g 12 

g 3 

g 8 

initial 

g 14 

g 4 

initial 

g 9 

initial 

= -0.28, = 0.00, = -0.40, = -0.12, 

g 16 

35 



initial 

g 18 

initial 

g 25 

initial 

initial 

initial 

= -0.12, = -0.26, = -0.09, = -0.07, 

g 20 

g 22 

= -0.04. (6.32) 

g 24 


(Δb) max = 0.1, (Δm) max = 0.01, (Δz) max = 1.0, (Δl 1 ) max = 0.1, 

(Δl 2 ) max = 0.01, (Δb 1 ) max = 0.01, (Δb 2 ) max = 0.01. (6.33) 



Δg 1 (x 0 ) max = 0.04, Δg 2 (x 0 ) max = 0.03, Δg 3 (x 0 ) max = 0.02, 

Δg 4 (x 0 ) max = 5.2×10 -3 , Δg 5 (x 0 ) max = 5.1×10 -3 , Δg 6 (x 0 ) max = 4.9×10 -3 , 




Δg 24 (x 0 ) max = 1.3×10 -2 , Δg 25 (x 0 ) max =1.3×10 -2 . (6.34) 





less than the initial constraints with a tolerance of 10 -4 . Table 6.11 shows the comparison of 

the results for the speed reducer design optimization problem. Figure 6.15 shows the 

iteration history of the objective function. 

36 



Table 6.11 Comparison of the results for the speed reducer example 

Fuzzy PD Golinski [1973] 

Objective function value 3007.8 2985.2 

Iteration 302 - 

x 1 3.5197 3.5 

x 2 0.7039 0.7 

x 3 17.3831 17.0 

x 4 7.3000 7.3 

x 5 7.7152 7.3 

x 6 3.3498 3.35 

x 7 5.2866 5.29 

g 1 -0.1095 -0.0739 

g 2 -0.2458 -0.1980 

g 3 -0.5127 -0.4990 

g 4 -0.9073 -0.9194 

g 5 0.0000 0.0001 

g 6 0.0000 -0.0020 

g 7 -0.6941 -0.7025 

g 8 0.0000 0.0000 

g 9 0.5833 -0.5833 

g 10 -0.2613 -0.2571 

g 11 -0.0223 -0.0278 

g 12 -0.0056 0.0000 

g 13 -0.1201 -0.1250 

g 14 -0.0220 0.0000 

g 15 -0.3792 -0.3929 

g 16 0.0000 0.0000 

g 17 -0.1205 -0.1205 

g 18 -0.0538 0.0000 

g 19 -0.0705 -0.1205 

g 20 -0.1343 -0.1343 

g 21 -0.1411 -0.1410 

g 22 -0.0542 -0.0548 

g 23 -0.0388 -0.0382 

g 24 -0.0514 -0.0514 

g 25 0.0000 0.0574 

37 



Figure 6.15 The iteration history of objective function of the speed reducer example 

6.6 Heat exchange design optimization example 

6.6.1 Formulating the optimization model 

A fluid having a given flow rate W and specific heat C p is heated from temperature T 0 

(100 o F) to T 3 (500 o F) by passing three heat exchangers in series and WC p = 10 5 

(B.t.u/hr- o F), as shown in Figure 6.16 [Avriel and Williams, 1971]. In each heat exchanger 

(stage) the cold stream is heated by a hot fluid having the same flow rate W and specific 

heat Cp as the cold stream. The temperatures of the hot fluid entering the heat exchangers, 

t 11 (300 o F), t 21 (400 o F) and t 31 (600 o F) the overall heat transfer coefficients U 1 (120 

B.t.u/hr-ft 2 - o F), U 2 (80 B.t.u/hr-ft 2 - o F), U 3 (40 B.t.u/hr-ft 2 - o F) of the exchangers are known 

constants. Optimal design involves minimizing the sum of the heat transfer areas of the 

three exchangers (A 1 +A 2 +A 3 ). The design optimization model can be summarized in 

Equation (6.35). 

38 



T 0 T 1 T 2 T 3 

Stage 1 

Stage 2 

Stage 3 

A 1 A 2 A 3 

t 12 

t 11 

t 22 

t 21 

t 32 

t 31 

Figure 6.16 Three stage heat exchanger system 

min. 

( A1 

, A2 

, A3 

) = A1 

+ A2 

A3 

f + 

( T + ) −1 

0 

s.t. g = 0.0025 

1 

t12 

1 

≤ 

( − T + T + ) −1 

0 

g 

2 

= 0.0025 

1 2 

t22 

≤ , 

( − T + ) −1 

0 

g 

3 

= 0.01 

2 

t32 

≤ , 

, 

g 

4 

= 100A1 

83333.333 − A1t 

12 

83333.33 + 0.00999T1 

−1 

≤ 0 , 

g 

5 

= A2 

1250 − A2t 

22 

1250T1 

+ T2 

T1 

−1 

≤ 0 , 

g 

6 

= A3T2 

1250000 − A3t 

32 

1250000 − T2 

500 −1 

≤ 0 , 

g 

7 

= 100 / A1 

−1 

≤ 0, 

g = A /10000 −1 

0, 

8 1 

≤ 

g 

9 

= 1000 / A2 

−1 

≤ 0, 

g = A /10000 −1 

0, 

10 2 

≤ 

g 

11 

= 1000 / A3 

−1 

≤ 0, 

g = A /10000 −1 

0, 

12 3 

≤ 

g 

13 

= 10 / T1 

−1 

≤ 0, 

g = T /1000 −1 

0 , 

14 1 

≤ 

g 

15 

= 10 / T2 

−1 

≤ 0, 

39 



g = T /1000 −1 

0, 

16 2 

≤ 

g = / t −1 

0, 

17 

10 12 

≤ 

g = t /1000 −1 

0, 

18 12 

≤ 

g = / t −1 

0, 

19 

10 22 

≤ 

g = t /1000 −1 

0, 

20 22 

≤ 

g = / t −1 

0, 

21 

10 32 

≤ 

g = t /1000 −1 

0 


22 32 

≤ 

There are 8 design variables, 5 of them do not appear in the objective function. There 

are 22 constraints, mostly side constraints. The optimization model required by the fuzzy 

PD controller optimization engine is: 

min. 

s.t. 

f 

+ + + 

( A , A A ) 

1 2 

, 

+ + 

( T t ) 

g , 

1 1 

, 

12 

− + + 

( T , T t ) 

g , 

2 1 2 

, 

− + 

( T ) 

3 2 

, t32 

3 

22 

g , 

− + + 

( A , T t ) 

g , 

4 1 1 

, 

12 

− + + − 

( A , T , T t ) 

g , 

5 2 1 2 

, 

− + − 

( A , T t ) 

g , 

6 3 2 

, 

− 

( ) 

g , 

7 A 1 

g 

+ 

8 

( A 1 

) 

− 

( ) 

9 A 2 

32 

22 

g http://mech.designer.yzu.edu.tw/ 

40

g 

+ 

10 

( A 2 

) 

g 

− 

11( A 3 

) 

g 

+ 

12 

( A 3 

) 

− 

g 

13 

( T 1 

) 

+ 

g 

14 

( T 1 

) 

− 

g 

15 

( T 2 

) 

+ 

g 

16 

( T 2 

) 

g 

− 

17 

( t 12 

) 

g 

+ 

18 

( t 12 

) 

g 

− 

19 

( t 22 

) 

g 

+ 

20 

( t 22 

) 

g 

− 

21( t 32 

) 

+ 

( ) 

22 t 32 



6.6.2 Monotonicity Analysis 


shows the complete output from MONO. Note that there are 3 conditionally critical sets. 

The monotonicity sign of design variables T 1 T 2 , t 12 , t 22 and t 32 do not appear in the 

objective function. 

41 



Table 6.12 Monotonicity table of the heat exchange problem 

A 1 A 2 A 3 T 1 T 2 t 12 t 22 t 32 

F + + + . . . . . 

g 1 . . . + . + . . 

g 2 . . . - + . + . 

g 3 . . . . - . . + 

g 4 - . . + . - . . 

g 5 . - . + + . - . 

g 6 . . - . + . . - 

g 7 - . . . . . . . 

g 8 + . . . . . . . 

g 9 . - . . . . . . 

g 10 . + . . . . . . 

g 11 . . - . . . . . 

g 12 . . + . . . . . 

g 13 . . . - . . . . 

g 14 . . . + . . . . 

g 15 . . . . - . . . 

g 16 . . . . + . . . 

g 17 . . . . . - . . 

g 18 . . . . . + . . 

g 19 . . . . . . - . 

g 20 . . . . . . + . 

g 21 . . . . . . . - 

g 22 . . . . . . . + 

----------------- PC MONO V3.1 : Heat02.out ------------------ 

TABLE 0 . 

1 2 3 4 5 6 7 8 

F + + + . . . . . 

G1 . . . + . + . . 

G2 . . . - + . + . 

42 



G3 . . . . - . . + 

G4 - . . + . - . . 

G5 . - . + + . - . 

G6 . . - . + . . - 

G7 - . . . . . . . 

G8 + . . . . . . . 

G9 . - . . . . . . 

G10 . + . . . . . . 

G11 . . - . . . . . 

G12 . . + . . . . . 

G13 . . . - . . . . 

G14 . . . + . . . . 

G15 . . . . - . . . 

G16 . . . . + . . . 

G17 . . . . . - . . 

G18 . . . . . + . . 

G19 . . . . . . - . 

G20 . . . . . . + . 

G21 . . . . . . . - 

G22 . . . . . . . + 

****************************** 

43 




****************************** 


None 


None 


None 


None 


None 


One of ( G4 G7 ) must be critical for 1. 



Figure 6.17 The output of MONO program of the heat exchange example 

6.6.3 Defining the initial values and move limits of the design variables 



error inputs (e) are the values of the all constraint functions. The quantitative definitions 

for the error change inputs (Δe) are the change of the values of the all constraint functions. 

44 





are: 

A 1 = 1000, A 2 = 2500, A 3 = 5000, T 1 = 100, 

T 2 = 400, t 12 = 200, t 22 = 300 t 32 = 500. (6.37) 


initial 

g 1 

initial 

g 5 

initial 

g 9 

initial 

g 13 

initial 

g 17 

initial 

g 21 

g 2 

initial 

initial 

initial 

initial 

= -0.25, = 0.50, = 0.00, = -1.20, 

initial 

= -1.00, = -0.20, = -0.90, g = -0.90, 

g 6 

initial 

initial 

= -0.60, = -0.75, = -0.80, g = -0.50, 

g 10 

initial 

initial 

initial 

= -0.90, = -0.90, = -0.98, g = -0.60, 

g 14 

initial 

initial 

initial 

= -0.95, = -0.80, = -0.97, g = -0.70, 

g 18 

g 22 

g 3 

initial 

g 7 

initial 

g 11 

g 15 

g 19 

initial 

= -0.98, = -0.50. (6.38) 

g 4 

8 

12 

16 

20 


(ΔA 1 ) max = 1, (ΔA 2 ) max = 1, (ΔA 3 ) max = 1, (ΔT 1 ) max = 1, 

(ΔT 2 ) max = 1, (Δt 12 ) max = 1, (Δt 22 ) max = 1, (Δt 32 ) max = 1. 








Δg 13 (x 0 ) max = 1.0×10 -3 , Δg 14 (x 0 ) max = 1.0×10 -3 , Δg 15 (x 0 ) max = 6.3×10 -5 , 



Δg 22 (x 0 ) max = 1.0×10 -3 . (6.40) 

6.6.4 The optimization results 


fuzzy PD controller optimization engine terminates after 1963 iterations, when the change 

in objective function value in consecutive iterations is less than 0.01%, and all constraints 

45 



are satisfied with a tolerance of 0.01% of the initial values of the constraints.. Table 6.13 

shows the comparison of the results for the speed reducer design optimization problem. 

Figure 6.18 shows the iteration history of the objective function. 

Table 6.13 Comparison of the results for the heat exchange example 

Fuzzy PD Avriel and Williams [1971] 

Objective function value 7288.8 7049 

Iteration 1963 

A 1 951.8 567 

A 2 1529.5 1357 

A 3 4807.3 5125 

T 1 206.6 181 

T 2 307.9 295 

t 12 193.4 219 

t 22 298.7 286 

t 32 407.8 395 

g 1 2.4×10 -5 0.0000 

g 2 1.4×10 -7 0.0000 

g 3 1.1×10 -3 0.0000 

g 4 -1.2×10 -4 3.2×10 -4 

g 5 -5.5×10 -2 6.6×10 -5 

g 6 0.0000 0.0000 

g 7 -0.8949 -0.8236 

g 8 -0.9048 -0.9433 

g 9 -0.3462 0.2631 

g 10 -0.8470 -0.8643 

g 11 -0.7920 -0.8049 

g 12 -0.5193 -0.4875 

g 13 -0.9516 -0.9448 

g 14 -0.7934 -0.8190 

g 15 -0.9675 -0.9661 

g 16 -0.6921 -0.7050 

g 17 -0.9483 -0.9543 

g 18 -0.8066 -0.7810 

g 19 -0.9665 -0.9650 

g 20 -0.7013 -0.7140 

g 21 -0.9755 -0.9747 

g 22 -0.5922 -0.6050 

46 



Figure 6.18 The iteration history of objective function of the heat exchange example 

Reference 

Arora, J. S., 1989, “Introduction to optimum design,” McGraw-Hill, New York. 

Avriel, M., and Williams, A. C., 1971, “An extension of geometric programming with 

application in engineering optimization,” Journal of Engineering Mathematics, Vol. 5, pp. 

187-194. 

Belegundu, A. D., 1982, “A study of mathematical programming methods for 

structural optimization,” Department of Civil and Environmental Engineering, University 

of Iowa City, Iowa. 

Coello, C. A. C., and Montes, E. M., 2002, “Constraint-handling in genetic algorithm 

through the use of dominance-based tournament selection,” Advanced Engineering 

Informatics, Vol. 16, pp. 193-203. 

Golinski, J., 1973, “An adaptive optimization system applied to machine synthesis,” 

Mechanism and Machine Synthesis, Vol. 8, pp. 419-436. 

47 



Ragsdell, K. M. and Phillips, D. T., 1976, “Optimal design of a class of welded 

structures using geometric programming,” ASME Journal of Engineering for Industry, Vol. 

98, pp. 1021-1025. 

Rao, S. S., 1996, “Engineering Optimization: Theory and Practice, 3rd Ed.”, John 

Wiley & Sons, Inc. 

Rao, S. S., 1987, “Multiobjective optimization of fuzzy structure system,” 

International Journal for Numerical Methods in Engineering, Vol. 24, pp. 1157-1171. 

48

Chapter 6. Engineering design optimization examples

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