17 Introduction to Evidence-Based Robust Optimisation
543
particular, the work in [23], that is focused on RBDO, suggests a sequential method
that incorporates mixed aleatory-epistemic uncertainty. The computational cost
remains, however, exponentially complex with the problem dimension. Other works,
as [25], make the assumptions of normally distributed epistemic parameters. [24]
also could be intractable with high order problems.
The chapter introduces different formulations of Evidence-Based Robust Optimisation both with single and multiple objectives. It then proceeds with some
techniques to efficiently compute robust solutions and with the definition of a particular model, presented in Sect. 17.4.4.1, called Evidence Network Model (ENM),
that translates a complex engineering system into a non-directed graph. In this graph
each node represents a sub-system or component and the associated uncertainty
is quantified with basic belief assignments. Exploiting the ENM properties, a
decomposition procedure is explained that reduces the computational cost to a
polynomial complexity with respect to the number of interacting components.
The framework presented in this chapter finds applicability to the design and optimisation of complex aerospace systems, composed of a number of interconnected
components, the behaviour of which cannot be inferred only by the behaviour of
each of its parts.
17.1.1 A Classification of Uncertainty
Uncertainty comes in different forms and the nature of uncertainty suggests how
it should be modelled and treated. It is, therefore, useful to classify the types of
uncertainty:
• Aleatory uncertainties are non-reducible uncertainties that depend on the very
nature of the phenomenon under investigation. They can generally be captured
by well-defined probability distributions as one can apply a frequentist approach,
e.g. measurement errors.
• Epistemic uncertainties are reducible uncertainties and are due to a lack of
knowledge. Generally they cannot be quantified with a well-defined probability
distribution and a more subjectivist approach is required. Two classes: a lack of
knowledge on the distribution of the stochastic variables or a lack of knowledge
of the model used to represent the phenomenon under investigation.
• Structural (or model) uncertainty is a form of epistemic uncertainty on our
ability to correctly model natural phenomena, systems or processes. If we accept
that the only exact model of Nature is Nature itself, we also need to accept that
every mathematical model is incomplete. One can then use an incomplete (and
often much simpler and tractable) model and account for the missing components
through some model uncertainty.
• Experimental uncertainty is generally aleatory but if one consider the uncertainty associated to measurements it can be considered epistemic as it incorporates the possible lack of knowledge on the performance of the sensor.
543
particular, the work in [23], that is focused on RBDO, suggests a sequential method
that incorporates mixed aleatory-epistemic uncertainty. The computational cost
remains, however, exponentially complex with the problem dimension. Other works,
as [25], make the assumptions of normally distributed epistemic parameters. [24]
also could be intractable with high order problems.
The chapter introduces different formulations of Evidence-Based Robust Optimisation both with single and multiple objectives. It then proceeds with some
techniques to efficiently compute robust solutions and with the definition of a particular model, presented in Sect. 17.4.4.1, called Evidence Network Model (ENM),
that translates a complex engineering system into a non-directed graph. In this graph
each node represents a sub-system or component and the associated uncertainty
is quantified with basic belief assignments. Exploiting the ENM properties, a
decomposition procedure is explained that reduces the computational cost to a
polynomial complexity with respect to the number of interacting components.
The framework presented in this chapter finds applicability to the design and optimisation of complex aerospace systems, composed of a number of interconnected
components, the behaviour of which cannot be inferred only by the behaviour of
each of its parts.
17.1.1 A Classification of Uncertainty
Uncertainty comes in different forms and the nature of uncertainty suggests how
it should be modelled and treated. It is, therefore, useful to classify the types of
uncertainty:
• Aleatory uncertainties are non-reducible uncertainties that depend on the very
nature of the phenomenon under investigation. They can generally be captured
by well-defined probability distributions as one can apply a frequentist approach,
e.g. measurement errors.
• Epistemic uncertainties are reducible uncertainties and are due to a lack of
knowledge. Generally they cannot be quantified with a well-defined probability
distribution and a more subjectivist approach is required. Two classes: a lack of
knowledge on the distribution of the stochastic variables or a lack of knowledge
of the model used to represent the phenomenon under investigation.
• Structural (or model) uncertainty is a form of epistemic uncertainty on our
ability to correctly model natural phenomena, systems or processes. If we accept
that the only exact model of Nature is Nature itself, we also need to accept that
every mathematical model is incomplete. One can then use an incomplete (and
often much simpler and tractable) model and account for the missing components
through some model uncertainty.
• Experimental uncertainty is generally aleatory but if one consider the uncertainty associated to measurements it can be considered epistemic as it incorporates the possible lack of knowledge on the performance of the sensor.