# Incorporating Epistemic Uncertainties in Ship Operability Study
## 1 Introduction
It becomes evident from this review that operability studies are dependent not only on inherent (or aleatory) uncertainties of random sea surface elevation and wave environmental variations in ship operating zones, but also on the knowledge-based (or epistemic) uncertainties, encompassing modeling uncertainties of seakeeping theories and tools and uncertainties of seakeeping limiting values [16]. Obviously, both uncertainty types need to be included in the consistent probabilistic ship operability analysis [17]. The concept of the risk-based Navigation Decision Assistant (NDA) was proposed initially in [7]. The concept described the evaluation of probabilities and consequences, accounting for the uncertainties of the input variables. Another concept of the risk-based DSS was introduced, incorporating aleatory and epistemic uncertainties related to the parameters of the problem of ship motions in waves [18]. It is concluded that the system can be applied to both the ship's operational assessment and the design procedure, but there is a need to improve the efficiency of the probabilistic assessment for practical onboard applications [19]. A procedure to incorporate random variables and associated uncertainties in the calculations of the outcrossing rates that are the basis for risk-based DSS is suggested in [20]. The present study aims to continue this development of the risk-based DSS by formulating and solving the probabilistic problem of ship operability analysis for different seakeeping limiting criteria. Epistemic uncertainties are included in this operability study, resulting in probabilistic operability polar plots being more informative compared to the \"deterministic\" polar plots. While previous studies (e.g., [18; 19]) included seakeeping uncertainties in operability analysis, the present work extends these studies by incorporating uncertainties in operability limiting values. The limit state equations are formulated and solved using Comrel 2021 software for structural reliability analysis. This study also proposes extending the conventional OI for ship design to the Probabilistic Operability Index (POI) as a more rational measure of ship operability. The potential application of probabilistic operability diagrams for tactical DSS is also shown. An examination of probabilistic operability is provided for a passenger ship traveling on the actual sailing route across the Adriatic Sea. Fuel consumption and corresponding CO\({}_{2}\) emissions along common ferry routes in the Adriatic have recently been studied in [21]. To demonstrate the concept of probabilistic operability analysis, a simplified seakeeping approach is used, where transfer functions of heave, pitch, and roll are calculated by closed-form expressions (CFE), formulated in [22]. The model uncertainty of CFE for heave and pitch is defined by comparison to the model-scale experiments for a relatively large number of ships [23]. In previous studies, model uncertainties in seakeeping methods were assumed. In contrast, this study consistently employs a seakeeping method while explicitly defining the corresponding modeling uncertainty. For the sake of simplicity, a single-parameter wave spectrum, which is representative for the Adriatic Sea, defined only by a significant wave height, is used for the description of short-term sea conditions. A histogram of sea states along the sailing route is obtained from the wave hindcast database, described in [24]. The seakeeping limiting criteria of roll, pitch, vertical acceleration on forward perpendicular, slamming, and propeller emergence are considered as random variables to illustrate the procedure. The second-order reliability method (SORM) is used to calculate probabilities of exceedance of the seakeeping criteria, and the results are presented in probabilistic polar plots. This paper is organized as follows. The seakeeping method, wave spectrum model, uncertainty model of CFE based on a frequency-independent model error, reliability analysis, and risk-based decision making are described in Section 2.