Engineering Optimization: Theory and Practice, Fourth Edition

11.3 Stochastic Linear Programming 649

wherehiis a new random variable defined as

hi=

∑n

j= 1

aijxj−bi=

∑n+^1

k= 1

qikyk (11.72)

where

qik=aik, k= 1 , 2 ,... , n qi,n+ 1 =bi

yk=xk, k= 1 , 2 ,... , n, yn+ 1 = − 1

Noticethat the constantyn+ 1 is introduced for convenience. Sincehi is given by a
linear combination of the normally distributed random variablesqik, it will also follow
normal distribution. The mean and the variance ofhiare given by

hi=

n∑+ 1

k= 1

qikyk=

∑n

j= 1

aijxj−bi (11.73)

Var(hi)=YTViY (11.74)

where

Y=











y 1

y 2

..

.

yn+ 1











(11.75)

Vi=

     

Var(qi 1 ) Cov(qi 1 , qi 2 ) ·· · Cov(qi 1 , qi,n+ 1 )

Cov(qi 2 , qi 1 ) Var(qi 2 ) ·· · Cov(qi 2 , qi,n+ 1 )

..

.

Cov(qi,n+ 1 , qi 1 ) ovC (qi,n+ 1 , qi 2 ) ·· · Var(qi,n+ 1 )

     

(11.76)

This can be written more explicitly as

Var(hi)=

n∑+ 1

k= 1

[

yk^2 Var (qik)+ 2

∑n+^1

l=k+ 1

ykylCov (qik, qil)

]

=

∑n

k= 1

[

yk^2 Var (qik)+ 2

∑n

l=k+ 1

ykylCov (qik, qil)

]

+yn^2 + 1 Var (qi,n+ 1 )+ 2 y^2 n+ 1 Cov (qi,n+ 1 , qi,n+ 1 )

+

∑n

k= 1

[2ykyn+ 1 Cov (qik, qi,n+ 1 )]