Engineering Optimization: Theory and Practice, Fourth Edition

650 Stochastic Programming

=

∑n

k= 1

[

x^2 kVar (aik)+ 2

∑n

l=k+ 1

xkxlCov (aik, ail)

]

+Var(bi)− 2

∑n

k= 1

xkCov (aik, bi) (11.77)

Thus the constraints in Eqs. (11.71) can be restated as

P

[

hi−hi

√

Var(hi)

≤

−hi

√

Var(hi)

]

≥pi, i= 1 , 2 ,... , m (11.78)

where [(hi−hi)]/

√

Var(hi) epresents a standard normal variable with a mean valuer

of zero and a variance of 1.

Thus ifsidenotes the value of the standard normal variable at which

φ(si)=pi (11.79)

the constraints of Eq. (11.78) can be stated as

φ

(

−hi

√

Var(hi)

)

≥φ(si), i= 1 , 2 ,... , m (11.80)

These inequalities will be satisfied only if the following deterministic nonlinear inequal-

ities are satisfied:

−hi

√

Var(hi)

≥si, i= 1 , 2 ,... , m

or

hi+si

√

Var(hi) ≤ 0 , i= 1 , 2 ,... , m (11.81)

Thus the stochastic linear programming problem of Eqs. (11.64) to (11.66) can be

stated as an equivalent deterministic nonlinear programming problem as

MinimizeF (X)=k 1

∑n

j= 1

cjxj+k 2

√

XTVX , k 1 ≥ 0 , k 2 ≥ 0 ,

subjectto

hi+si

√

Var(hi) ≤ 0 , i= 1 , 2 ,... , m

xj≥ 0 , j= 1 , 2 ,... , n (11.82)

Example 11.5 A manufacturing firm produces two machine parts using lathes, milling

machines, and grinding machines. If the machining times required, maximum times

available, and the unit profits are all assumed to be normally distributed random vari-

ables with the following data, find the number of parts to be manufactured per week

to maximize the profit. The constraints have to be satisfied with a probability of at

least 0.99.