## 3 Method: Unbalance Prevention Control in Dehydration Based on BO
### Application of the Acquisition Function  
Using the surrogate model's predicted values, the acquisition function proposes UBT values for the unbalance prevention control specification, aiming to discover the maximum DS in untested regions. In this study, we used Expected Improvement (EI) as the acquisition function. As the experiment progresses, we recommend the UBT for the unbalance control specification through the EI method, experimentally verify whether it meets the target value, and perform the optimization. Figure 2b: Suggest Candidate Sampling is the sampling process used to select candidate input values for the next stage in the BO process, with the aim being to select input value candidates likely to maximize the objective function value. We generated candidate values using a uniform distribution sampling method, and these candidate values were used as inputs for the acquisition function. EI is an acquisition function that balances exploration and exploitation. Exploitation acts to improve performance further in regions already known to perform well, while exploration seeks new possibilities in regions not yet explored. EI predicts the probability and degree of improvement over the current best observed performance to suggest the optimization direction. EI is calculated using the predicted mean performance \(\mu_{DS}(x)\) and standard deviation \(\sigma_{DS}(x)\) for input value x. Here, \(\mu_{DS}(x)\) represents the predicted performance and \(\sigma_{DS}(x)\) represents the uncertainty of that prediction. EI combines these two values to evaluate the potential for performance improvement at that point. Additionally, EI measures the potential for improvement at a given point by comparing the current best observed performance, \(f_{DS}(x^{+})\) with predicted outcomes. A greater difference suggests a higher likelihood that exploring that point will lead to better performance, guiding the optimization process efficiently. Figure 2c: Suggest Possible Maximum shows the process of deriving a new UBT using the acquisition function and calculating the score based on it. This process aims to generate a new UBT (unbalance prevention control specification), evaluate its performance, and derive the optimal control specification. The acquisition function predicts the optimal control parameters based on existing data to generate a new UBT. The various colored rectangles displayed on the washing machine model represent the UBT derived through the acquisition function, with each color indicating specific ranges of control values. This visualizes the control specifications that minimize unbalance. The newly calculated DS is  
based on key test results such as entry time and vibration amount, and we can confirm that the performance has improved compared to previous table values. Figure 2c: Suggest Possible Maximum visually shows the new UBT and the corresponding DS matrix, highlighting instances where a high DS has achieved optimal performance. The EI formula is as follows: [18]  
\[EI(x)=\left\{ \begin{array}{cc}(\mu_{DS}(x)-f_{DS}(x^{+})-\xi)\;\Phi(Z_{DS})+ \sigma_{DS}(x)\;\varphi(Z_{DS}),&\sigma(x)>0\\ 0,&\sigma(x)=0\end{array} \right. \tag{7}\]  
where  
\[Z_{Dehy\;Score}=\left\{ \begin{array}{cc}\frac{\mu_{DS}(x)-f_{DS}\left(x^{+} \right)-\xi}{\sigma_{DS}(x)},&\sigma(x)>0\\ 0,&\sigma(x)=0\end{array} \right. \tag{8}\]  
Here, \(\mu_{DS}(x)\) is the predicted mean performance at input \(x\); \(\sigma_{DS}(x)\) is the standard deviation of that prediction, representing the uncertainty. \(f_{DS}(x^{+})\) is the current best observed performance value, and \(\xi\) is a parameter that adjusts the degree of exploration; larger values promote exploration. \(\Phi(Z_{DS})\) is the cumulative distribution function (CDF) of the standard normal distribution of the DS, and \(\varphi(Z_{DS})\) is the probability density function (PDF) of the standard normal distribution. \(Z_{Dehy\;Score}\) is a standardized indicator that considers both the mean and standard deviation of the performance improvement potential of the DS.