# Incorporating Epistemic Uncertainties in Ship Operability Study
## 2 Methods
### Risk-Based Perspective  
A review and analysis of general risk definitions, perspectives, and scientific approaches to risk analysis in the maritime transportation application area are provided in [37]. The results indicate that risk is strongly associated with probability, though alternative views exist. Additionally, various scientific approaches to risk analysis have been identified. More recently, a multi-attribute-based assessment model has been proposed in [38] to support decision making on risk control options. A discussion on risk assessment and key risk factors affecting ferry navigation safety is presented in [39]. The prevailing approach to risk analysis in maritime defines the risk (\(R\)) as the product of the probability of an event (\(P\)) and its associated consequence (\(C\)), e.g., [7]. The total risk for a combination of \(N\) uncorrelated serial events is given as:  
\[R=\sum_{i=1}^{N}R_{i}=\sum_{i=1}^{N}P_{i}C_{i} \tag{22}\]  
while the expression for the case of correlated events may be found in [7]. In addition, each individual risk \(R_{i}\) may be multiplied by the company-specific aversion factor \(A_{i}\). While the probabilities of an adverse event \(P_{i}\) are obtained through reliability analysis, the associated consequences are determined via consequence analysis, categorized as follows:  
* Safety of people (loss of life, injuries, and health);
* Risk to the maritime environment;
* Economic risk (damage to ship structure, equipment and cargo, including total loss, and loss of reputation). Quantifying the first two categories in monetary terms is controversial and typically beyond the scope of risk-based DSS. Events related to these risk categories generally have a low probability of occurrence (less than \(10^{-3}\)). Such extremely hazardous situations are classified as \"intolerable\" risks that must be mitigated regardless of cost. These risks may be considered at the ship design stage and addressed through proper masters' training or qualitative guidance [18]. Additionally, due to the well-known \"tail sensitivity\" issue in structural reliability analysis, the probabilities of such rare events are highly dependent on the uncertainty model used and cannot be applied universally [7]. In risk terminology, DSS is used in the \"As Low as Reasonably Practicable\" (ALARP) region, which lies between \"intolerable\" and \"negligible\" (broadly acceptable) risk levels [40]. The economic losses necessary for risk estimation can be expressed as:  
\[C_{tot}=C_{c}+C_{eq}+C_{s}+C_{d}+C_{rp} \tag{23}\]  
where  
* \(C_{c}\) represents damage to cargo;
* \(C_{eq}\) accounts for damage to equipment;
* \(C_{s}\) represents damage to ship;
* \(C_{d}\) is costs related to delays;
* \(C_{rp}\) represents costs associated with loss of reputation. The latter component, \(C_{rp}\), includes factors such as passenger seasickness, reduced crew performance, and the consequences of delays, all of which directly impact the economic viability and reputation of maritime services [18]. Estimates of economic losses should ideally be provided by the owner or operator, preferably supported by data from insurance companies [7]. An example of a consequence analysis for an oil tanker, including specific costs associated with potential hull failure or the exceedance of specific limit states, is presented in [36]. In that study, total consequences are divided into:  
* Direct consequences, i.e., losses due to damage or failure itself;
* Indirect consequences, i.e., losses related to system failures or malfunctions that result in external monetary impacts. The primary risk control parameters for a ship sailing in waves are forward speed and heading relative to waves. For each combination of these parameters, the probability of exceeding seakeeping criteria is determined through reliability analysis, while the total risk is calculated using Equation (22).