Engineering Optimization: Theory and Practice, Fourth Edition

648 Stochastic Programming

wherecj,aij, andbiare random variables andpiare specified probabilities. Notice

thatEqs. (11.65) indicate that theith constraint,

∑n

j= 1

aijxj≤bi

has to be satisfied with a probability of at leastpiwhere 0≤pi≤. For simplicity, 1

we assume that the design variablesxjare deterministic andcj,aij, andbiare random

variables. We shall further assume that all the random variables are normally distributed

with known mean and standard deviations.

Sincecjare normally distributed random variables, the objective functionf (X)

will also be a normally distributed random variable. The mean and variance off are

given by

f=

∑n

j= 1

cjxj (11.67)

Var(f )=XTVX (11.68)

wherecj is the mean value ofcj and the matrixVis the covariance matrix ofcj

defined as

V=

     

Var(c 1 ) Cov(c 1 , c 2 ) ·· · Cov(c 1 , cn)

Cov(c 2 , c 1 ) arV (c 2 ) ·· · Cov(c 2 , cn)

..

.

Cov(cn, c 1 ) ovC (cn, c 2 ) ·· · Var(cn)

     

(11.69)

with Var(cj) andCov(ci, cj) enoting the variance ofd cjand covariance betweenci

andcj, respectively. A new deterministic objective function for minimization can be

formulated as

F (X)=k 1 f+k 2

√

Var(f ) (11.70)

wherek 1 andk 2 are nonnegative constants whose values indicate the relative importance

off and standard deviation off for minimization. Thusk 2 = 0 indicates that the

expected value off is to be minimized without caring for the standard deviation of

f. On the other hand, ifk 1 = , it indicates that we are interested in minimizing the 0

variability offabout its mean value without bothering about what happens to the mean

value off. Similarly, ifk 1 =k 2 = , it indicates that we are giving equal importance 1

to the minimization of the mean as well as the standard deviation off. Notice that the

new objective function stated in Eq. (11.70) is a nonlinear function inXin view of the

expression for the variance off.

The constraints of Eq. (11.65) can be expressed as

P[hi≤ ] 0 ≥pi, i= 1 , 2 ,... , m (11.71)