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G. Filippi and M. Vasile
1. The conditional expectation E of a utility function φ(f ):
F φ (d) = E[φ(f )|d]
(17.6)
where the value of the expectation is conditional to the choice of the design (or
decision) vector d. Different definitions of the function φ have been proposed in
the literature, see [37] and [38] for some examples. When the utility function is
simply f one can account for both the expected value of f and its variance σ
with the two-objective problem:
min d E(f |d)
min d σ (f |d)
.
(17.7)
Methods for solving problem (17.7) can be found in [38–42].
2. The probabilistic threshold, where, for a given threshold q, the conditional
probability that the function f assumes values lower than q is maximised:
max
d
P r(f < q|d).
(17.8)
This approach can be easily extended by adding q as an objective function to be
minimised.
Note that these two approaches are in fact equivalent if one takes the utility function
φ = f < q. In this case the utility function is the indicator function and the
expectation of the indicator is the probability that f < q.
When there is not enough information to quantify uncertainty with a probability
distribution, alternative theories have been used to derive a proper quantification.
In this chapter we will focus on the use of Evidence Theory (or DST) already
introduced by previous authors in the context of engineering applications [43].
However, it is here worthwhile to briefly mention another approach using
Fuzzy Set Theory. Fuzzy Set Theory describes the feasibility of a solution by
the membership function μ c (in the interval [0 1]) instead of using the extremes
values 0 or 1 as with the crisp sets. Simoes [44] and Campos et al. [45] propose
to use μ c and an other membership function μ f for the objective function f ,
considering μ f = 1 for the unconstrained optimum and μ f = 0 for the worst
case. Finally the robust solution can be found maximising the minimum of the two
defined membership functions:
max(min[μ c , μ f ])
(17.9)
17.2 Evidence Theory
Evidence Theory, known also as Dempster–Shafer Theory (DST) and as Theory of
Belief functions, was developed by Shafer (1976) in [19] starting from Dempsters
G. Filippi and M. Vasile
1. The conditional expectation E of a utility function φ(f ):
F φ (d) = E[φ(f )|d]
(17.6)
where the value of the expectation is conditional to the choice of the design (or
decision) vector d. Different definitions of the function φ have been proposed in
the literature, see [37] and [38] for some examples. When the utility function is
simply f one can account for both the expected value of f and its variance σ
with the two-objective problem:
min d E(f |d)
min d σ (f |d)
.
(17.7)
Methods for solving problem (17.7) can be found in [38–42].
2. The probabilistic threshold, where, for a given threshold q, the conditional
probability that the function f assumes values lower than q is maximised:
max
d
P r(f < q|d).
(17.8)
This approach can be easily extended by adding q as an objective function to be
minimised.
Note that these two approaches are in fact equivalent if one takes the utility function
φ = f < q. In this case the utility function is the indicator function and the
expectation of the indicator is the probability that f < q.
When there is not enough information to quantify uncertainty with a probability
distribution, alternative theories have been used to derive a proper quantification.
In this chapter we will focus on the use of Evidence Theory (or DST) already
introduced by previous authors in the context of engineering applications [43].
However, it is here worthwhile to briefly mention another approach using
Fuzzy Set Theory. Fuzzy Set Theory describes the feasibility of a solution by
the membership function μ c (in the interval [0 1]) instead of using the extremes
values 0 or 1 as with the crisp sets. Simoes [44] and Campos et al. [45] propose
to use μ c and an other membership function μ f for the objective function f ,
considering μ f = 1 for the unconstrained optimum and μ f = 0 for the worst
case. Finally the robust solution can be found maximising the minimum of the two
defined membership functions:
max(min[μ c , μ f ])
(17.9)
17.2 Evidence Theory
Evidence Theory, known also as Dempster–Shafer Theory (DST) and as Theory of
Belief functions, was developed by Shafer (1976) in [19] starting from Dempsters