Stochastic Programming - Index of

26 STOCHASTIC PROGRAMMING
(e.g. rotten or too unripe), “cooking apples”and “low (medium, high) quality
eatable apples”. Having sorted out the “unusable” and the “cooking apples”,
for the remaining apples experts could be asked to judge on ripeness, flavour,
colour and appearance, by assigning real values between 0 and 1 to parameters
r, f, c and a respectively, corresponding to the “degree (or percentage) of
achieving” the particular criterion.
Now we can construct a scalar value for any particular outcome (quality)
ω, for instance as
⎧
⎨ 0 if ω ∈ “unusable”,
1
ṽ(ω) := 2
if ω ∈ “cooking apples”,
⎩
(1 + r)(1 + f)(1 + c)(1 + a) otherwise.
Obviously ṽ has the range ṽ[Ω] = {0}∪{ 1 2
}∪{[1, 16]}. Denoting the events
“unusable” by U and “cooking apples” by C, we may define the collection F
of events as follows. With G denoting the family of all subsets of Ω − (U ∪ C)
let F contain all unions of U, C, ∅ or Ω with any element of G. Assume that
after long series of observations we have a good estimate for the probabilities
P (A),A∈F.
According to our scale, we could classify the apples as
• eatable and
– 1st class for ṽ(ω) ∈ [12, 16] (high selling price),
– 2nd class for ṽ(ω) ∈ [8, 12) (medium price),
– 3rd class for ṽ(ω) ∈ [1, 8) (low price);
• good for cooking for ṽ(ω) = 1 2
(cheap);
• waste for ṽ(ω) =0.
Obviously the probabilities to have 1st-class apples in our lot is
Pṽ({[12, 16]}) =P (ṽ −1 [{[12, 16]}]), whereas the probability for having 3rdclass
or cooking apples amounts to
Pṽ({[1, 8)}∪{ 1 2 })=P (ṽ−1 [{[1, 8)}∪{ 1 2 }])
= P (ṽ −1 [{[1, 8)}]) + P (C),
using the fact that ṽ is single-valued and {[1, 8)}, { 1 2 } and hence ṽ−1 [{[1, 8)}],
ṽ −1 [{ 1 2
}]=C are disjoint. For an illustration, see Figure 10.
✷
If it happens that Ω ⊂ IR k and F⊂A(i.e. every event is a “naturally”
measurable set) then we may replace ω trivially by ˜ξ(ω) by just applying the
identity mapping ˜ξ(ω) ≡ ω, which preserves the probability measure P˜ξ
on
F, i.e.
P˜ξ(A) =P (A) forA ∈F