11.5 Stochastic Geometric Programming 659
Thus the equivalent deterministic optimization problem can be stated as follows: Min-
imizeF (d, t)given by Eq. (E 3 ) subject to the constraints given by Eqs. (E 10 ) to (E 15 ).
The solution of the problem can be found by applying any of the standard nonlinear
programming techniques discussed in Chapter 7. In the present case, since the number
of design variables is only two, a graphical method can also be used to find the solution.
11.5 Stochastic Geometric Programming
The deterministic geometric programming problem has been considered in Chapter 8. If
the constants involved in the posynomials are random variables, the chance-constrained
programming methods discussed in Sections 11.3 and 11.4 can be applied to this
problem. The probabilistic geometric programming problem can be stated as follows:
FindX= {x 1 x 2 · · ·xn}Twhich minimizesf(Y)
subject to (11.99)
P[gj( Y)> 0 ]≥pj, j= 1 , 2 ,... , m
whereY= {y 1 , y 2 ,... , yN}Tis the vector ofNrandom variables (may include the
variablesx 1 , x 2 ,... , xn) and, f (Y)andgj( ,Y) j= 1 , 2 ,... , m, are posynomials. By
expanding the objective function about the mean values of the random variablesyi,
yi, and retaining only the first two terms, we can express the meanand variance of
f (Y)as in Eqs. (11.87) and (11.88). Thus the new objective function,F (Y), can be
expressed as in Eq. (11.89):
F (Y)=k 1 ψ+k 2 σψ (11.100)
The probabilistic constraints of Eq. (11.99) can be converted into deterministic form
as in Section 11.4:
gj−φj(pj)
[N
∑
i= 1
(
∂gj
∂yi
∣
∣
∣
∣Y
) 2
σy^2 i
] 1 / 2
≥ 0 , j= 1 , 2 ,... , m (11.101)
Thus the optimization problem of Eq. (11.99) can be stated equivalently as follows:
FindYwhich minimizesF (Y)given by Eq. (11.100) subject to the constraints of
Eq. (11.101). The procedure is illustrated through the following example.
Example 11.7 Design a helical spring for minimum weight subject to a constraint on
the shear stress (τ) induced in the spring under a compressive loadP.
SOLUTION By selecting the coil diameter (D)and wire diameter (d)of the spring
as design variables, we havex 1 = Dandx 2 = d.The objective function can be stated
in deterministic form as [11.14, 11.15]:
f (X)=
π^2 d^2 D
4
(Nc+ Q)ρ (E 1 )
