550
G. Filippi and M. Vasile
of a realisation of x. When enough information is available, the distribution of x is
known and one can exactly quantify if a realisation of x belongs to A or not. In this
case all F Es collapse to singletons and the following condition holds true:
Bel = P l = P
(17.24)
The functions Bel and P l are monotonic non-additive measures and have
following properties:
1. Bel(∅) = 0;
2. Bel(U ) = 1;
3. For every positive integer n and every collection θ 1 , . . . θ n of subsets of Θ:
Bel(θ 1 ∪. . .∪θ n ) ≥
i
Bel(θ i )−
i Bel(θ i ∩θ j )+−. . .+(−1)
n+1 Bel(θ 1 ∩. . .∩θ n )
and
1. P l(∅) = 0;
2. P l(U) = 1;
3. For every positive integer n and every collection θ 1 , . . . , θ n of subsets of Θ:
P l(θ 1 ∩. . .∩θ n ) ≥
i
Bel(θ i )−
i P l(θ i ∪θ j )+− . . .+(−1)
n+1 P l(θ 1 ∪. . .∪θ n )
Due to conditions 3, Belief and Plausibility are also called monotone capacities
of order ∞.
17.3 Robust Optimisation with Evidence Theory
The central idea underneath Evidence-Based Robust Optimisation is to maximise
the belief in statement (17.20). This condition alone, however, is not enough to
qualify the realisations of f . In fact, imagine that f is a performance indicator, then
the condition f ∈ Φ alone would not say much on the optimality of the values of
f .
Consider now the simple case in which Φ = {f |f ≤ ν} and f : U × D → R
is a function of some decision vector d ∈ D ⊆ R n d and some uncertain vector
u ∈ U ⊆ R n u . If f is a performance index, it is now easy to define the optimality
condition:
min ν∈R ν
s.t.
f (d, u) ≤ ν
(17.25)