## 3 The Algorithm
### SaNSDE Algorithm
As transition \(t_{5}\) is the start transition of Cyber World model \(\Omega_{n}^{5}\) and transition \(t_{6}\) is the end transition of Cyber World model \(\Omega_{n}^{5}\), resource agent \(a_{3}\) can perform the fifth operation. Therefore, resource agent \(a_{3}\) submits a bid \(B_{a_{3j}}=(a_{a_{3j}1},a_{a_{3j}2},a_{a_{3j}3},\ldots,a_{a_{3j}K_{n}},\tau_{a_ {3j}})=(1,0,0,0,1,45)\), where \(45\) is the processing time of resource agent \(a_{3}\) for the first operation and the second operation. Based on the corresponding resource activity models, resource agents \(a_{4}\) through \(a_{9}\) also submit their bids. The information on all the bids submitted by resource agents is shown in Tables 3 and 4. The algorithmic parameters used by the SaNSDE algorithm and NSDE algorithm are listed as follows:

The parameters of the SaNSDE algorithm are as follows. \[V_{\max}=4\]

\[CR=0.5\]

\[LP=50\]

The number of iterations: 10,000

Population size (\(NP\)): 30

The objective function used is \(G(x)=G(\Gamma(x,B),Eng(x,B))=w_{1}(\omega_{n}-\ \ \Gamma(x,B))-w_{2}Eng(x,B)\), where \(w_{1}=1\) and \(w_{2}=1\). The following solution is found by the proposed algorithm: \(x_{11}=0\), \(x_{21}=0\), \(x_{31}=1\), \(x_{41}=0\), \(x_{51}=0\), \(x_{61}=0\), \(x_{71}=1\), \(x_{81}=1\), \(x_{91}=1\). The configuration of our solution is the bids submitted by the set of resource agents {3, 7, 8, 9}. This solution corresponds to the Cyber World model for the configuration in Figure 4b. The processing time achieved for this solution is \(45+40+28+32=145\). The fitness function value found for this example is 42. For this example, there are several candidate configurations that can perform all the operations in the process. These candidate configurations include the bids submitted by the sets of resource agents { 1, 2, 7, 8, 9}, { 1, 5, 7, 8, 9}, { 3, 7, 8, 9}, { 4, 2, 7, 8, 9}, { 4, 5, 7, 8, 9} and {6, 7, 8, 9}. The values of the fitness function defined in Section 3.2 for these candidate configurations are 42, 32, 42, 32, 22 and 27, respectively. The configuration of the solution found by the proposed algorithm is the bids submitted by the set of resource agents {3, 7, 8, 9}, which is optimal as the value of the objective function is maximal. So, the solution found is consistent with our expectation. ### Comparison with Other Algorithms

To illustrate the effectiveness of the SaNSDE algorithm proposed in Section 3.2, the PSO algorithm, six standard DE algorithms (DE1 through DE6) and a variant of the neigh \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Agent (\(a\))** & \(j\) & **Bid (\(E_{a\beta}\))** & \(o_{a\beta 1}\) & \(o_{a\beta 2}\) & \(o_{a\beta 3}\) & \(o_{a\beta 4}\) & \(o_{a\beta 5}\) & **Energy Consumption (\(e_{a\beta}\))** \\ \hline
1 & 1 & \(E_{11}\) & 1 & 0 & 0 & 0 & 0 & 2 \\
2 & 1 & \(E_{21}\) & 0 & 0 & 0 & 0 & 1 & 2 \\
3 & 1 & \(E_{31}\) & 1 & 0 & 0 & 0 & 1 & 4 \\
4 & 1 & \(E_{41}\) & 1 & 0 & 0 & 0 & 0 & 3 \\
5 & 1 & \(E_{51}\) & 0 & 0 & 0 & 0 & 1 & 3 \\
6 & 1 & \(E_{61}\) & 1 & 0 & 0 & 0 & 1 & 6 \\
7 & 1 & \(E_{71}\) & 0 & 0 & 1 & 0 & 0 & 4 \\
8 & 1 & \(E_{81}\) & 0 & 1 & 0 & 0 & 0 & 2 \\
9 & 1 & \(E_{91}\) & 0 & 0 & 0 & 1 & 0 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The information of energy consumption of bids in \(E\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Agent (\(a\))** & \(j\) & **Bid (\(B_{a\beta}\))** & \(o_{a\beta 1}\) & \(o_{a\beta 2}\) & \(o_{a\beta 3}\) & \(o_{a\beta 4}\) & \(o_{a\beta 5}\) & **Processing Time (\(\tau_{a\beta}\))** \\ \hline
1 & 1 & \(B_{11}\) & 1 & 0 & 0 & 0 & 0 & 25 \\
2 & 1 & \(B_{21}\) & 0 & 0 & 0 & 0 & 1 & 25 \\
3 & 1 & \(B_{31}\) & 1 & 0 & 0 & 0 & 1 & 45 \\
4 & 1 & \(B_{41}\) & 1 & 0 & 0 & 0 & 0 & 30 \\
5 & 1 & \(B_{51}\) & 0 & 0 & 0 & 0 & 1 & 35 \\
6 & 1 & \(B_{61}\) & 1 & 0 & 0 & 0 & 1 & 60 \\
7 & 1 & \(B_{71}\) & 0 & 0 & 1 & 0 & 0 & 40 \\
8 & 1 & \(B_{81}\) & 0 & 1 & 0 & 0 & 0 & 28 \\
9 & 1 & \(B_{91}\) & 0 & 0 & 0 & 1 & 0 & 32 \\ \hline \hline \end{tabular}
\end{table}