# Incorporating Epistemic Uncertainties in Ship Operability Study
## 4 Discussion  
### Application in Ship Design  
The probabilistic approach to the operability diagrams may be used in ship design as the decision-making tool when acquiring a new ship for a specific service. For that purpose, the Operability Index (OI) is proposed, representing the ratio, expressed as a percentage, between the number of sea states (for all available peak periods) with significant wave heights that do not exceed the limiting significant wave height (\(n_{ss}\), \(\beta\)) over the total number of sea states (\(N\)) in a certain wave scatter diagram. The OI reads [1]:  
\[\text{OI }(\%)=\frac{\sum_{H_{1/3},T_{p}}n_{ss,\beta}\left(H_{s}<H_{s}^{lim} \right)}{N}.100, \tag{25}\]  
In the present study, the Probabilistic Operability Index (POI (\(\%\))) is proposed as:  
\[\text{POI}(\%)=\sum_{H_{1/3},T_{p}}f_{H_{1/3},T_{p}}P\left(H_{s}<H_{s}^{lim} \right)\cdot 100, \tag{26}\]  
where \(f_{H_{s},T_{p}}\) is the relative frequency of the occurrence of the sea state, described by \(H_{s}\) and \(T_{p}\), given as  
\[f_{H_{s},T_{p}}=\frac{n_{H_{s},T_{p}}}{N}, \tag{27}\]  
where \(n_{H_{s},T_{p}}\) is the number of sea states with \(H_{S}\) and \(T_{p}\). \(P\left(H_{s}<H_{s}^{lim}\right)\) is the probability that the seakeeping limiting value is not exceeded at sea states defined by \(H_{S}\) and \(T_{p}\), calculated by the approach outlined in Section 2. Expressions (25)-(27) require a wave scatter diagram relevant to the ship's operational area to determine the number of sea states defined by \(H_{S}\) and \(T_{p}\). Additionally, it should be clarified that the application of these expressions assumes a constant ship speed. Consequently, different operability indices are calculated for varying ship speeds.  
The process is illustrated by calculating the POI (\(\%\)) and OI (\(\%\)) for the histogram of significant wave heights (Figure 3) for the shipping route in the Adriatic Sea (Figure 2). Thus, POI = 98.7% and OI = 98.9% for a speed of 17 kn are obtained.  
The POI, as defined by Equation (26), is well-suited for application in risk-based decision making. To incorporate this into decision making, the cost of the non-fulfillment of ship operations should be included by estimating the total risk associated with a given sailing route. However, estimating the cost of non-fulfillment is challenging, as it is company-specific and lacks publicly available references.  
A rough estimate can be made by considering the primary cost categories associated with delays, namely:  
* The operational costs of delay, including increased fuel consumption due to rerouting, which can be approximately USD 3000-USD 5000 per additional hour of voyage time. A similar cost may apply for crew overtime;
* Port and logistics costs related to late arrivals, rescheduling, and supply chain disruptions, estimated at USD 5000-USD 25,000;
* Passenger compensation, including partial refunds, food, and accommodation expenses, which can exceed USD 25,000.  
According to a response generated by ChatGPT (GPT-4-turbo, OpenAI, April 2024; retrieved 31 March 2025), for a 3 to 6 h delay--a realistic scenario previously observed on the Ancona-Split route--the estimated cost could be anywhere from USD 10,000 to USD 100,000+, depending on fuel adjustments, port penalties, and passenger compensation. Longer delays (overnight cancellations) could push costs beyond USD 250,000. For a lifetime risk assessment, this value should be scaled based on the total number of voyages.