13 Risk Measures in the Context of Robust and Reliability Based Optimization
413
some optimization problems might require maximization. Indeed, the maximization
of f (z) is always equivalent to the minimization of −f (z) [18].
However, in most of the engineering problems, unknowns or future states must be
considered. Moreover, they must account for the stochastic nature of the system and
processes to be designed. For example, industrial manufacturing processes and real
operating conditions inevitably introduce tolerances in the production and uncertainties in the working conditions, respectively, that will lead to deviations from the
considerations taken at design stage. Hence, random variables are introduced, and
a stochastic optimization problem is defined to correctly model the process under
investigation.
A random variable is defined as a measurable function X : Ω → R that maps
possible outcomes Ω to a measurable space R, with (Ω, F , P ) a properly defined
probability space, with ω ∈ Ω, F = 2 Ω , and P a probability measure [4].
Mathematically, the direct introduction of random variables into the optimization
problem (Eq. 13.1) introduces a functional dependency that has to be properly
treated to avoid inconsistencies. Indeed, the introduction of random variables leads
to the following problem formulation:
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
min
f (z, X)
z ∈ S
s.to :
c i (z, X) ≤ 0 i = 1, . . . , m
S ⊆ R n
(13.2)
where the objective function and the constraints are now functionals. Therefore, a
way to recast the problem into an optimization one must be searched. Herein, several
approaches are shown.
Best Estimate A particular outcome is chosen ¯
ω ∈ Ω as the best estimate of the
unknown status. As a consequence, the problem is reconstructed as a deterministic
optimization:
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
min
f (z, X( ¯
ω))
z ∈ S
s.to :
c i (z, X( ¯
ω)) ≤ 0 i = 1, . . . , m
S ⊆ R n
(13.3)
Although attractive for its simplicity, this kind of alternative is very risky, as the
choice of the typical outcome ¯
ω is somewhat arbitrary and might not reflect at all
what happens in the reality.
Worst Case Contrary to the best estimate strategy, the worst possible outcomes are
identified for the unknown status. This leads to the following minmax problem:
413
some optimization problems might require maximization. Indeed, the maximization
of f (z) is always equivalent to the minimization of −f (z) [18].
However, in most of the engineering problems, unknowns or future states must be
considered. Moreover, they must account for the stochastic nature of the system and
processes to be designed. For example, industrial manufacturing processes and real
operating conditions inevitably introduce tolerances in the production and uncertainties in the working conditions, respectively, that will lead to deviations from the
considerations taken at design stage. Hence, random variables are introduced, and
a stochastic optimization problem is defined to correctly model the process under
investigation.
A random variable is defined as a measurable function X : Ω → R that maps
possible outcomes Ω to a measurable space R, with (Ω, F , P ) a properly defined
probability space, with ω ∈ Ω, F = 2 Ω , and P a probability measure [4].
Mathematically, the direct introduction of random variables into the optimization
problem (Eq. 13.1) introduces a functional dependency that has to be properly
treated to avoid inconsistencies. Indeed, the introduction of random variables leads
to the following problem formulation:
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
min
f (z, X)
z ∈ S
s.to :
c i (z, X) ≤ 0 i = 1, . . . , m
S ⊆ R n
(13.2)
where the objective function and the constraints are now functionals. Therefore, a
way to recast the problem into an optimization one must be searched. Herein, several
approaches are shown.
Best Estimate A particular outcome is chosen ¯
ω ∈ Ω as the best estimate of the
unknown status. As a consequence, the problem is reconstructed as a deterministic
optimization:
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
min
f (z, X( ¯
ω))
z ∈ S
s.to :
c i (z, X( ¯
ω)) ≤ 0 i = 1, . . . , m
S ⊆ R n
(13.3)
Although attractive for its simplicity, this kind of alternative is very risky, as the
choice of the typical outcome ¯
ω is somewhat arbitrary and might not reflect at all
what happens in the reality.
Worst Case Contrary to the best estimate strategy, the worst possible outcomes are
identified for the unknown status. This leads to the following minmax problem: