Engineering Optimization: Theory and Practice, Fourth Edition

722 Modern Methods of Optimization

Step 2: For any antk, the probability of selecting pathx 1 jin Fig. 13.4 is given by

p 1 j=

τ 1 j

∑^7

p= 1

τ 1 p

; j= 1 , 2 ,... , 7

whereτ 1 j= 0. 25 ;j= 1 , 2 , 4 , 5 , 6 ,7 andτ 13 = 8. 9 231. This gives

p 1 j= 100 5. 4231.^2 = 0. 0240 , j= 1 , 2 , 4 , 5 , 6 , 7 ;p 13 = 108 231.^9. 4231 = 0. 8561

To determine the discrete value or path selected by an ant using a random num-

ber selected in the range (0, 1), cumulative probabilities are associated with

different paths as (roulette-wheel selection process):

x 11 = ( 0 , 0. 0240 ), x 12 = ( 0. 0240 , 0. 0480 ), x 13 = ( 0. 0480 , 0. 9040 ),

x 14 = ( 0. 9040 , 0. 9280 ), x 15 = ( 0. 9280 , 0. 9520 ),

x 16 = ( 0. 9520 , 0. 9760 ), x 17 = ( 0. 9760 , 1. 0 )

With this information, we go to step 3 and then to step 4. Steps 2, 3, and 4

are repeated until the process converges (until all the ants choose the same

best path).

13.6 Optimization of Fuzzy Systems

In traditional designs, the optimization problem is stated in precise mathematical terms.

However, in many real-world problems, the design data, objective function, and con-

straints are stated in vague and linguistic terms. For example, the statement, “This beam

carries a load of 1000 lb with a probability of 0.8” is imprecise because of random-

ness in the material properties of the beam. On the other hand, the statement, “This

beam carries a large load” is imprecise because of the fuzzy meaning of “large load.”

Similarly, in the optimum design of a machine component, the induced stress(σ )is con-

strained by an upper bound value(σmax) sa σ≤σmax. Ifσmax= 0,000 psi, it implies 3

that a design withσ= 30 ,000 psi is acceptable whereas a design withσ= 30 , 001

psi is not acceptable. However, there is no substantive difference between designs with

σ= 30 ,000 psi andσ= 30 ,001 psi. It appears that it is more reasonable to have a

transition stage from absolute permission to absolute impermission. This implies that

the constraint is to be stated in fuzzy terms. Fuzzy theories can be used to model and

design systems involving vague and imprecise information [13.22, 13.26, 13.27].

13.6.1 Fuzzy Set Theory

LetXbe a classical crisp set of objects, called theuniverse,whose generic elements are

denoted byx. Membership in a classical subsetAofXcan be viewed as a characteristic

functionμAfrom Xto [0, 1] such that

μA(x)=

{

1 ifx∈A

0 ifx /∈A

(13.44)