36
D. Krpelík and T. Basu
design process. Here, we identify two major aspects that should be included in such
a procedure.
The first aspect is the ability to assess consequences of various actions—making
predictions of the future behaviour. Science, per se, is a field that explores relations
between various aspects of reality and constructs models upon which we may base
our predictions. But there is no guarantee that these models are totally accurate.
Mathematical models are usually simplifications of the occurring phenomena,
additional simplifications often various need to be employed to make the computation tractable, and the numerical evaluation itself may introduce additional error
(e.g. when simulating processes described by differential equations). Furthermore,
another type of error is introduced when providing numerical inputs for the models,
i.e. their parameters. These also come from scientific inference and, therefore, suffer
from similar issues to those of the models themselves. Their usual sources are
measurements with finite resolution, statistical inference from finite dataset and
expert elicitation. All these are subjected to uncertainty. Therefore, regardless of
the chosen model, its predictions are always subject to uncertainty, and this fact
needs to be considered.
The second aspect is how to choose a single design from a set of admissible
possibilities. The usual way to tackle this problem is to describe what it means
when one solution is preferred over the other and then search for a solution which is
preferred over all alternatives. Imagine for a while (and please drop this assumption
later) that we can predict the consequences without a doubt. What would then
constitute an “optimal” design? Most importantly, we would like the system to
provide the service it has been designed for. Designs which ensure this are preferred
over those which do not (see Chap. 4). But such a definition of preference generally
fails to identify a single design because there are usually many ways to ensure the
desired service. One could then introduce other desirable properties; e.g. a system is
preferred over another if it is cheaper to realise, or when it is more environmentally
friendly, or when it is easier to maintain, or when it produces greater volume of
outputs in less time, or due to some other criteria. With such a definition, one can
formulate, mathematically, a constrained optimisation problem and use standard
algorithms to find its solution. But the preference criteria may be contradictory
(e.g. cost vs performance), so the optimisation problem could generally not have
a unique solution, and the procedure would yield a set of incomparable designs.
This happens, for example, during a multi-objective optimisation. The solution to
the multi-objective optimisation problem is a so-called Pareto set, which consists
of solutions which are better than those excluded, but none is strictly preferred over
the others in the set (see Chap. 8). Besides, once we drop the assumption that we can
make perfect predictions, the optimum yield based on the erroneous model might
not be the true optimal design we were searching for.
The main focus of this chapter lies in the first mentioned aspect—how to
model the uncertainty associated with our assessments. Nowadays, this field is
dominated by two complementary theories, probability theory and interval arithmetic. Although these two allow us to model many scenarios, they suffer from
D. Krpelík and T. Basu
design process. Here, we identify two major aspects that should be included in such
a procedure.
The first aspect is the ability to assess consequences of various actions—making
predictions of the future behaviour. Science, per se, is a field that explores relations
between various aspects of reality and constructs models upon which we may base
our predictions. But there is no guarantee that these models are totally accurate.
Mathematical models are usually simplifications of the occurring phenomena,
additional simplifications often various need to be employed to make the computation tractable, and the numerical evaluation itself may introduce additional error
(e.g. when simulating processes described by differential equations). Furthermore,
another type of error is introduced when providing numerical inputs for the models,
i.e. their parameters. These also come from scientific inference and, therefore, suffer
from similar issues to those of the models themselves. Their usual sources are
measurements with finite resolution, statistical inference from finite dataset and
expert elicitation. All these are subjected to uncertainty. Therefore, regardless of
the chosen model, its predictions are always subject to uncertainty, and this fact
needs to be considered.
The second aspect is how to choose a single design from a set of admissible
possibilities. The usual way to tackle this problem is to describe what it means
when one solution is preferred over the other and then search for a solution which is
preferred over all alternatives. Imagine for a while (and please drop this assumption
later) that we can predict the consequences without a doubt. What would then
constitute an “optimal” design? Most importantly, we would like the system to
provide the service it has been designed for. Designs which ensure this are preferred
over those which do not (see Chap. 4). But such a definition of preference generally
fails to identify a single design because there are usually many ways to ensure the
desired service. One could then introduce other desirable properties; e.g. a system is
preferred over another if it is cheaper to realise, or when it is more environmentally
friendly, or when it is easier to maintain, or when it produces greater volume of
outputs in less time, or due to some other criteria. With such a definition, one can
formulate, mathematically, a constrained optimisation problem and use standard
algorithms to find its solution. But the preference criteria may be contradictory
(e.g. cost vs performance), so the optimisation problem could generally not have
a unique solution, and the procedure would yield a set of incomparable designs.
This happens, for example, during a multi-objective optimisation. The solution to
the multi-objective optimisation problem is a so-called Pareto set, which consists
of solutions which are better than those excluded, but none is strictly preferred over
the others in the set (see Chap. 8). Besides, once we drop the assumption that we can
make perfect predictions, the optimum yield based on the erroneous model might
not be the true optimal design we were searching for.
The main focus of this chapter lies in the first mentioned aspect—how to
model the uncertainty associated with our assessments. Nowadays, this field is
dominated by two complementary theories, probability theory and interval arithmetic. Although these two allow us to model many scenarios, they suffer from