## 3 IUWSNs Task Allocation Model
The length of the solution is composed of the assigned numbers of the sensor nodes arranged in the order of the target batch number, indicating a possible scheme. Here, the model assumes 4 sensor nodes and 6 tasks. It is \begin{table}
\begin{tabular}{l l} \hline
**Character** & **Description** \\ \hline \(y\) & Number of nodes of monitoring sensors \\ \(x\) & Number of tasks \\ \(A\) & Advantage matrix of sensor \\ R & Network revenue of IUWSNs \\ U & Urgency of the task \\ S & Solution of task assignment \\ S* & Solution in binary form \\ \hline \end{tabular}
\end{table} 
Table 2: Symbol description. Figure 1: Radio energy consumption model diagram. estimated that the estimated benefit of assigning the \(j\)th task to the \(i\)th sensor is denoted as \(A_{i,j}\), and it is represented by (6). \[A_{i,j}= \begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}&a_{15}&a_{16}\\ a_{21}&a_{22}&a_{23}&a_{24}&a_{25}&a_{26}\\ a_{31}&a_{32}&a_{33}&a_{34}&a_{35}&a_{36}\\ a_{41}&a_{42}&a_{43}&a_{44}&a_{45}&a_{46}\end{bmatrix} (i\in[1,4],j\in[1,6]) \tag{6}\]  
Each task has the following urgency, as shown below. \[U_{j}= \begin{bmatrix}e_{1}&e_{2}&e_{3}&e_{4}&e_{5}&e_{6}\end{bmatrix} (j\in \begin{bmatrix}1\,,\,6\end{bmatrix} ) \tag{7}\]  
Suppose that there is 1 individual, 4 sensor nodes and 6 tasks in the model. Then, a random generation function is used to randomly generate solutions to store the assigned results (8), where \(S\) represents the solution of task assignment. Section 4 describes the coding method. \[S= \begin{bmatrix}3&1&4&2&2&3\end{bmatrix}  \tag{8}\]  
In order to improve the efficiency of code execution, it is necessary to convert real number coding to binary coding method to carry out the rest operations. The allocation matrix of binary files is as follows:  
\[S^{*}= \begin{bmatrix}0&1&0&0&0&0\\ 0&0&0&1&1&0\\ 1&0&0&0&0&1\\ 0&0&1&0&0&0\end{bmatrix}  \tag{9}\]  
According to matrix (6) and matrix (9), we calculate the value of \(A^{*}_{i,j}\) and express it with (10). Next, we express \(A^{*}_{i,j}\), the advantage value of assigning the \(j\)th task to the \(i\)th sensor node. \(S^{*}_{i,j}\) represents the matrix after real number coding is converted to binary coding. \(S^{*}_{i,j}\) means to assign the \(j\)th task to the \(i\)th sensor node. \(A_{i,j}\) represents the dominance matrix. The value, which in the dominance matrix stands for the dominance value corresponding to the \(j\)th task, is assigned to the \(i\)th sensor. \[A^{*}_{i,j}=A_{i,j}\times S^{*} \tag{10}\]  
\[A^{*}_{i,j}= \begin{bmatrix}0&a_{12}&0&0&0\\ 0&0&0&a_{24}&a_{25}&0\\ a_{31}&0&0&0&a_{36}\\ 0&0&a_{43}&0&0&0\end{bmatrix} (i\in[1,4]\quad j\in[1,6]) \tag{11}\]  
The identity matrix with column \(i\) is generated in \(a_{i}\) according to the following formula. The number of sensors corresponds to the number of columns of \(q_{i}\). The calculation result of \(A_{j}\) is shown in (12). \[q_{i}= \begin{bmatrix}1&1&1&1\end{bmatrix}  \tag{12}\]  
\[A_{j}=q_{i}\times A^{*}_{i,j} \tag{13}\]  
\[A_{j}= \begin{bmatrix}a_{31}&a_{12}&a_{43}&a_{24}&a_{25}&a_{36}\end{bmatrix}  \tag{14}\]  
\(R_{j}\) expresses the network revenue value of a group of assigned tasks, which is calculated as follows. \[R_{j}=A_{j}\times E^{\prime}_{j} \tag{15}\]  
In IUWSNs, the central objective is to maximize the efficiency of all tasks. Task execution efficiency refers to the combined effectiveness of task completion across all sensor nodes. Within IUWSNs, sensor nodes are non-isomorphic, and the efficiency of each node in executing tasks varies. Consequently, the overall task efficiency accomplished by sensor nodes will consequently differ. In practical applications, task allocation consistently results in heightened and superior network performance advantages. Therefore, the gateway assumes the responsibility of task scheduling by assessing the workload magnitude of each task and the execution capability of individual sensor nodes.