112
D. Krpelík et al.
Reliability theory is a field at the interface of mathematics and engineering which
primary interest is to evaluate whether a system (a device, policy, treatment, etc.)
will behave as desired. Historically, since it cannot be predicted with certainty,
a lot of interest was allocated into assessing the validity of the logical statement
the system will work. The natural choice of the validity measure seems to be
probability, since probability theory offers a consistent reasoning apparatus in which
one can, deductively, from a set of basic assessments, derive probabilities of related
statements similarly as in the familiar system of binary logic. But care must be taken
to properly interpret the actual numerical values, probabilities, which this approach
allocates to propositions. Probability theory is commonly used for assessing statements about both the frequency of events and the likeliness of occurrence of specific
outcome in the next conducted trial. The first interpretation focuses on describing the
sampling process, the anticipated relative ratio of occurrences of any particular trait
of outcome and the aleatory uncertainty. If the predictive model is well-calibrated,
the relative frequency tends to be close to the numerical value assigned to it by
the model, where the closeness is understood in a limit of infinitely many trials,
as in the law of large numbers. The second interpretation is also often termed
epistemic uncertainty, since for any particular separate observation, an assigned
numerical value of probability does not correspond to any observable quantity. The
outcome of the next observation is a precise value, and we will perceive it as such,
once the observation have been realised. Probability theory, here, serves as a tool
to describe our state of knowledge, perception of likeliness of occurrence of an
event, and allows us to reason about particular attributes of the future observation,
including what actions we might take in order to improve the chance that the future
observation will have desired properties, like “Is the system more likely to function
if we use component A instead of component B?”. In the epistemic interpretation,
the quality of a reasoning procedure manifests as an observable quantity through
relative frequency of correct decisions in a series of repeated applications and the
quality of the analytic methods rather than of the constructed models.
Both interpretations are relevant for applications of the reliability theory.
Aleatory interpretation plays a role, e.g., for planning processes in which we
assume that components will need to be replaced over time, and we need to
schedule the maintenance and inspection policies (Sect. 4.5.2) or for the statistical
quality control (Sect. 4.4.4). With good enough models, we can assess the longtime costs associated with operating our systems and also optimise the policies
addressing their manufacture, maintenance and the logistical issues associated
with the replacements. In such scenarios, failures are anticipated. Sometimes, we
may discover that using a lower quality component may be beneficial from an
economical perspective, leading to overall lower costs of the operation.
On the other hand, with some systems, usually the one-of-a-kind ones, we simply
cannot afford them to fail. Some examples are the nuclear power plants, airplanes,
residential buildings and many others. In these cases, we need to “ensure” that either
the system works perfectly or we can detect an upcoming failure in time to mitigate
its consequences. The issue is addressed by the so-called risk analysis which
focuses on enlisting possible undesirable events and constructs measures aimed
D. Krpelík et al.
Reliability theory is a field at the interface of mathematics and engineering which
primary interest is to evaluate whether a system (a device, policy, treatment, etc.)
will behave as desired. Historically, since it cannot be predicted with certainty,
a lot of interest was allocated into assessing the validity of the logical statement
the system will work. The natural choice of the validity measure seems to be
probability, since probability theory offers a consistent reasoning apparatus in which
one can, deductively, from a set of basic assessments, derive probabilities of related
statements similarly as in the familiar system of binary logic. But care must be taken
to properly interpret the actual numerical values, probabilities, which this approach
allocates to propositions. Probability theory is commonly used for assessing statements about both the frequency of events and the likeliness of occurrence of specific
outcome in the next conducted trial. The first interpretation focuses on describing the
sampling process, the anticipated relative ratio of occurrences of any particular trait
of outcome and the aleatory uncertainty. If the predictive model is well-calibrated,
the relative frequency tends to be close to the numerical value assigned to it by
the model, where the closeness is understood in a limit of infinitely many trials,
as in the law of large numbers. The second interpretation is also often termed
epistemic uncertainty, since for any particular separate observation, an assigned
numerical value of probability does not correspond to any observable quantity. The
outcome of the next observation is a precise value, and we will perceive it as such,
once the observation have been realised. Probability theory, here, serves as a tool
to describe our state of knowledge, perception of likeliness of occurrence of an
event, and allows us to reason about particular attributes of the future observation,
including what actions we might take in order to improve the chance that the future
observation will have desired properties, like “Is the system more likely to function
if we use component A instead of component B?”. In the epistemic interpretation,
the quality of a reasoning procedure manifests as an observable quantity through
relative frequency of correct decisions in a series of repeated applications and the
quality of the analytic methods rather than of the constructed models.
Both interpretations are relevant for applications of the reliability theory.
Aleatory interpretation plays a role, e.g., for planning processes in which we
assume that components will need to be replaced over time, and we need to
schedule the maintenance and inspection policies (Sect. 4.5.2) or for the statistical
quality control (Sect. 4.4.4). With good enough models, we can assess the longtime costs associated with operating our systems and also optimise the policies
addressing their manufacture, maintenance and the logistical issues associated
with the replacements. In such scenarios, failures are anticipated. Sometimes, we
may discover that using a lower quality component may be beneficial from an
economical perspective, leading to overall lower costs of the operation.
On the other hand, with some systems, usually the one-of-a-kind ones, we simply
cannot afford them to fail. Some examples are the nuclear power plants, airplanes,
residential buildings and many others. In these cases, we need to “ensure” that either
the system works perfectly or we can detect an upcoming failure in time to mitigate
its consequences. The issue is addressed by the so-called risk analysis which
focuses on enlisting possible undesirable events and constructs measures aimed