Stochastic Programming - Index of

18 STOCHASTIC PROGRAMMING
✻
%
❜ 15
❜
❜
❜
❜
12
9
❜
❜
∼N(0, 12)
❜ ∼N(0, 9)
❜
❜
❜
3
❜
❜
❜
❜
✲
-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30
Figure 7 Discrete distribution generated from N (0, 12), N (0, 9); (r 1,r 2)=
(15, 15).
6
❜
Obviously, these discrete distributions with 15 realizations each can
only be rough approximations of the corresponding normal distributions.
Therefore approximating probabilities of particular events using these discrete
distributions can be expected to cause remarkable discretization errors. This
will become evident in the following numerical examples.
Using these latter distributions, with 15 realizations each, we get 15 2 =
225 realizations for the joint distribution, and hence 225 blocks in our
decomposition problem. This yields as an optimal solution for the linear
program (3.11) (with γ(·) the total objective of (3.11) and γ I (x) =2x raw1 +
3x raw2 )
˜x =(˜x 1 , ˜x 2 )=(38.539, 20.539), γ(˜x) = 140.747, (3.12)
with corresponding first-stage costs of
γ I (˜x) = 138.694.
Defining ρ(x) as the empirical reliability (i.e. the probability to be feasible) for
any production plan x, we find—with respect to the approximating discrete
distribution—for our solution ˜x that
ρ(˜x) =0.9541,
whereas using our original linear program’s solution ˆx =(36, 18) would yield
the total expected cost
γ(ˆx) = 199.390