11.3 Stochastic Linear Programming 647
11.2.9 Central Limit Theorem
IfX 1 , X 2 ,... , Xnarenmutuallyindependent random variables with finite mean and
variance (they may follow different distributions), the sum
Sn=
∑n
i= 1
Xi (11.60)
tends to a normal variable if no single variable contributes significantly to the sum as
ntends to infinity. Because of this theorem, we can approximate most of the physical
phenomena as normal random variables. Physically,Snmay represent, for example, the
tensile strength of a fiber-reinforced material, in which case the total tensile strength
is given by the sum of the tensile strengths of individual fibers. In this case the ten-
sile strength of the material may be represented as a normally distributed random
variable.
11.3 Stochastic Linear Programming
A stochastic linear programming problem can be stated as follows:
Minimizef (X)=CTX=
∑n
j= 1
cjxj (11.61)
subjectto
ATiX=
∑n
j= 1
aijxj≤bi, i= 1 , 2 ,... , m (11.62)
xj≥ 0 , j= 1 , 2 ,... , n (11.63)
wherecj, aij, andbiare random variables (the decision variablesxj are assumed to
be deterministic for simplicity) with known probability distributions. Several methods
are available for solving the problem stated in Eqs. (11.61) to (11.63). We consider a
method known as thechance-constrained programming technique, in this section.
As the name indicates, the chance-constrained programming technique can be used
to solve problems involving chance constraints, that is, constraints having finite proba-
bility of being violated. This technique was originally developed by Charnes and Cooper
[11.5]. In this method the stochastic programming problem is stated as follows:
Minimizef (X)=
∑n
j= 1
cjxj (11.64)
subjectto
P
∑n
j= 1
aijxj≤bi
≥pi, i= 1 , 2 ,... , m (11.65)
xj≥ 0 , j= 1 , 2 ,... , n (11.66)
