## 2 Methodology of Research
### Solution Method Under Limited Available Data
#### 2.2.2 Bayesian Optimization Problem  
As discussed in previous sections, our model integrates the XGBoost algorithm to predict the power consumption of two primary cooling systems near the breeding pools. This adds complexity to the optimization function, making it a computationally expensive, black-box, and derivative-free problem. In such cases, the Bayesian Optimization (BO) machine learning approach provides an efficient solution by using a surrogate model to approximate the objective function. This approach is particularly effective under the following circumstances:  
* Low-Dimensional Decision Variables: In our model, the decision variable dimension is 24, as we aim to determine the optimal water temperature entering the breeding pools. * Simple Feasible Set: The decision variables must be constrained within a defined range, such as a hyper-rectangle \(x_{min}\leq x_{t}\leq x_{max}\). In this case, the optimal temperature is bounded as \(T_{t}^{min}\leq T_{t}^{out}\leq T_{t}^{in}\). * Continuity of the Objective Function: The objective function should be continuous in order to allow for effective approximation using Gaussian Processes (GP). Our objective function meets this criterion, as it seeks to minimize total energy consumption across the three cooling systems used in the aquaculture industry (see Equation (1)). Indeed, the selected approach performs well in low-data black-box scenarios that are expensive-to-evaluate. It offers intuitive control over the exploration-exploitationtradeoff and provides transparent uncertainty estimates that are trusted by industrial partners. Its lightweight and CPU-efficient implementation along with minimal need for hyperparameter tuning make it significantly more practical for deployment in factory IT environments compared to data-intensive neural networks. In the first step of BO, a prior distribution is assigned to the unknown objective function using a Gaussian Process (GP). Gaussian Processes are flexible non-parametric probabilistic models that are ideal for approximating complex and expensive-to-evaluate functions. The main strength of GPs lies in their ability to provide a probabilistic measure of uncertainty in predictions, balancing exploration (sampling in uncertain regions) with exploitation (sampling promising regions). A GP is characterized by a mean function and a covariance function, defined as follows:  
\[f(x)\sim\mathcal{GP}\big{(}m(x),k(x,x^{\prime})\big{)} \tag{19}\]  
where \(m(x)\) is typically chosen as a simple mean function (often zero) and \(k(x,x^{\prime})\) is the covariance function (kernel) that encodes assumptions regarding the smoothness and continuity of the function. In this work, the Matern covariance is considered as the covariance function, formulated as follows:  
\[\mathcal{C}_{\
u}(d)=\sigma^{2}\frac{2^{1-\
u}}{\Gamma(\
u)}(\sqrt{(2\
u)} \frac{d}{\rho})^{\
u}K_{\
u}(\sqrt{(2\
u)}\frac{d}{\rho}) \tag{20}\]  
where \(d\) is the Euclidean distance between points, \(\
u\) is a positive smoothness parameter, \(\sigma^{2}\) controls the function variance, \(\rho\) is the length-scale parameter governing the correlation between points, and \(K_{\
u}\) denotes the modified Bessel function of the second kind.