Engineering Optimization: Theory and Practice, Fourth Edition

652 Stochastic Programming

As the value of the standard normal variate (si) orresponding to the probability 0.99c

is 2.33 (obtained from Table 11.1), we can state the equivalent deterministic nonlinear

optimization problem as follows:

MinimizeF=k 1 ( 05 x 1 + 001 x 2 )+k 2

√

400 x 12 + 5002 x^22

subject to

10 x 1 + 5 x 2 + 2. 33

√

36 x 12 + 61 x 22 + 502 , 000 − 2500 ≤ 0

4 x 1 + 01 x 2 + 2. 33

√

16 x 12 + 94 x 22 + 601 , 000 − 2000 ≤ 0

x 1 + 1. 5 x 2 + 2. 33

√

4 x^21 + 9 x 22 + 5002 − 450 ≤ 0

x 1 ≥ 0 , x 2 ≥ 0

Thisproblem can be solved by any of the nonlinear programming techniques once the

values ofk 1 andk 2 are specified.

11.4 Stochastic Nonlinear Programming

When some of the parameters involved in the objective function and constraints vary

about their mean values, a general optimization problem has to be formulated as a

stochastic nonlinear programming problem. For the present purpose we assume that

all the random variables are independent and follow normal distribution. A stochastic

nonlinear programming problem can be stated in standard form as

FindXwhich minimizesf (Y) (11.83)

subject to

P[gj( Y)≥ 0 ]≥pj, j= 1 , 2 ,... , m (11.84)

whereYis the vector ofNrandom variablesy 1 , y 2 ,... , yNand it includes the decision

variablesx 1 , x 2 ,... , xn. The case whenXis deterministic can be obtained as a special

case of the present formulation. Equations (11.84) denote that the probability of real-

izinggj( greater than or equal to zero must be greater than or equal to the specifiedY)

probabilitypj. The problem stated in Eqs. (11.83) and (11.84) can be converted into

an equivalent deterministic nonlinear programming problem by applying the chance

constrained programming technique as follows.

11.4.1 Objective Function

The objective functionf (Y)can be expanded about the mean values ofyi,yi, as

f(Y)=f (Y)+

∑N

i= 1

(

∂f

∂yi

∣

∣

∣

∣Y

)

(yi−yi) +higher-order derivative terms (11.85)