## 3 Method: Unbalance Prevention Control in Dehydration Based on BO
### Preparation of the UBT Dataset
#### 3.1.3 Composition of the Data and Score Matrix and Definition of the DS  
The preprocessed data is organized into a multidimensional matrix called the Data and Score Matrix, which includes the UBT values according to each experimental condition, experimental results, and Dehydration Score (DS) values. In this matrix, each row represents an instance of the dehydration unbalance prevention control specification (UBT), and the DS is calculated through multiple experimental results for each instance. This enables a performance comparison based on the DS. The UBT and DS values are visually compared using color intensity to represent their magnitudes. Darker colors indicate higher values and better performance, while lighter colors correspond to lower values and poorer performance. This matrix is essential in the BO process as it is used to track the performance of each UBT through iterative experiments and select the optimal control specification.  
The Data and Score Matrix was constructed for each input value based on the results collected from repeated experiments for each instance shown in Figure 1a: UBT Dataset.  
\[Data\ \&\ Score\ Matrix=[[\ x_{1},f_{DS}(x_{1})],[\ x_{2},f_{DS}(x_{2})],\dots,[\ x_{t},f_{DS}(x_{t})]], \tag{1}\]  
To compute the value of the objective function, we selected the DS, which reflects the entry time and vibration amount. The DS function \(f_{DS}(x_{i})\) takes the Output Parameters as input to calculate the performance metric and is defined as follows. This function is used as the key performance metric for optimization in the subsequent BO algorithm.  
\[Max\ \ f_{DS}(x_{i})=\ \Phi(Z_{t})\times\Phi(Z_{v})\times 100,\Phi\sim N\left( \mu,\ \sigma^{2}\right), \tag{2}\]  
\[Subject\ to\ Time\ Fair\ Count=0 Max\ TV\ Fair\ Count=0 \tag{3}\]  
\[Z_{t}=\ \frac{(T_{t}-\mu_{t}\ )}{\sigma_{t}},\ \ Z_{v}=\frac{(T_{v}-\mu_{v}\ )}{ \sigma_{v}}, \tag{4}\]  
Here, \(\Phi\) is the cumulative distribution function (CDF) of the standard normal distribution. We compute the satisfaction levels of both entry time and vibration via the cumulative distribution function (\(\Phi\)). Since \(\Phi\) produces values between 0 and 1, we multiply it by 100 to express the results as percentages, enabling easier interpretation and comparison without affecting the optimization goal. \(T_{t}\) is the threshold for entry time, \(\mu_{t}\) is the mean of entry times, \(\sigma_{t}\) is the standard deviation of entry times, \(T_{v}\) is the threshold for vibration value, \(\mu_{v}\) is the mean of vibration values, and \(\sigma_{v}\) is the standard deviation of vibration values.  
When setting the thresholds for entry time (\(T_{t}\)) and vibration (\(T_{v}\)), we referred to product testing regulations, consumer requirements, and internal safety criteria. Specifically, the entry time threshold was chosen based on the goal that the user's laundry cycle should finish within a certain time frame, while the vibration threshold was established to comply with the product's mechanical stability needs, noise regulations, and the company's internal testing standards.  
The DS is calculated by considering the thresholds, means, and standard deviations of dehydration entry time and tub vibration amount, assuming they follow a normal distribution. To calculate the probability of meeting the pass criteria, we use the Z values \(Z_{t}\) for entry time and \(Z_{v}\) for vibration amount, applying the cumulative normal distribution function \(\Phi\). The constraints are that the number of times the dehydration entry time and maximum vibration value exceed the maximum allowable value must be zero. This means achieving stable results without dehydration test specification failures in more than 10 repeated tests while satisfying the target performance.