## 4 Sensor Allocation Model and Algorithm  
The optimal sensor-deployment problem is a \"minimum set covering problem\", meaning that the optimized sensor set will contain the minimum number of sensors to meet certain restrictions. To maximize the precision of the sensor network for the physical parameter estimation of the wet-grinding system, the model developed for sensor allocation design comprises a measurement of the estimation quality of physical parameters or state variables in the objective function. The precision of the sensor network for physical parameter estimation mainly depends on two factors: the first is the reasonable arrangement of the sensor, and the second is the data-acquisition accuracy of the sensor set. The latter is contingent on the precision of the sensor itself. Here, the detection accuracy of the sensor network is indicated by the data output deviation index, \(\gamma_{i\to j}\), which represents the data output precision of the \(i\)th type sensor used to observe the \(j\)th type physical parameters; that is,  
\[\gamma_{i\to j}=\frac{\sigma_{i}}{x_{i}}\Big{\{}1+(C_{i})^{d_{i,j}\times x_{i }}\Big{\}}, \tag{13}\]  
where \(\sigma_{i}\in\delta\) is the measurement deviation of the sensor itself, \(x_{i}\) is the number of the \(i\)th type sensor placed to monitor the system operations, and \(C_{i}\in\psi\) is the \(i\)th type sensor failure probability. \(d_{i,j}\) is a binary bipartite variable that indicates the cause-effect information between parameters and sensors, which has the following characteristics:  
\[\big{(}d_{i,j}\big{)}_{\mathcal{F}\times\mathcal{S}}\;=\;\left\{ \begin{array}[ ]{ll}1&S_{i}\;\;\text{to}\;\;\text{observe}\;\;F_{j}\\ 0&\;\text{otherwise}\end{array} \right. \tag{14}\]  
Equation (13) shows that the data output deviation index of the sensor set, \(\gamma_{i\to j}\), is decided by the measurement deviation of the sensor itself, and it considers the influence of the sensor failure rate. \(d_{i,j}\) = 1 means that the physical parameter \(F_{j}\) can be observed by sensor \(S_{i}\). In this case, the data output deviation index, \(\gamma_{i\to j}\), is the superposition of the measurement deviation of the sensor itself and the deviation caused by the sensor failure. Examples of this include the non-linearization of the sensor data output generated by the  
Figure 2: Sensitivity index to the bead–wall impact energy fluctuation in 30 boundary zones. external environment or sensor aging. Similarly, \(d_{i,j}=0\) means that no sensor is arranged at the position to observe the physical parameter \(F_{j}\), that is, \(x_{i}=0\). In this case, the data output deviation index, \(\gamma_{i\to j}\), is considered to be infinite, which becomes meaningless in industrial applications. Sensor deployment aims to achieve the minimum data output deviation index and sensor costs under the constraints of the mean shift detectability. Mathematically, this can be expressed as  
\[Min:max\big{(}\gamma_{i\to j}\big{)}=max\Big{\{}\frac{c_{i}}{x_{i}} \Big{(}1+(C_{i})^{d_{ij}\times x_{i}}\Big{)}\Big{\}} \tag{15}\] \[Min:\sum\limits_{i}\Bigg{(}C_{i}\times\sum\limits_{j}\big{(}d_{ i,j}\sum x_{i}\big{)}\Bigg{)}\] \[Subject\ to:\varepsilon_{j}\geq z_{a/2}-\psi^{-1}\bigg{(}1-\Big{(} \Phi^{L}_{1,\varepsilon_{i}}\Big{)}^{-1}\bigg{)}\] \[x_{i}\in Z^{+}\]  
The abovementioned formulation is based on minimizing the maximum data output deviation index \(\gamma_{i\to j}\) among all of the sensor sets. This outcome is based on the philosophy that a chain can be no stronger than its weakest link. Equation (15) achieves overall minimum data output deviation and sensor costs under the mean shift detectability of the physical variable. Sensor allocation was formulated as multiple objectives on the data output deviation index (primary) and sensor costs (secondary) under the constraints of the mean shift detectability. Under MOO conditions, the minimum sensor covering set is, in fact, a Pareto optimal set.