## 3 The Algorithm
### SaNSDE Algorithm
\right.\) End For Step 1.4: \(\overline{u}_{i}\gets T(u_{i})\)  Individual update  If \(G_{1}(\overline{u}_{i})\geq G_{1}(z_{i})\) \(z_{i}=u_{i}\)  Record \(cr_{i}\) in \(CR_{rec}\) \(n_{s}=n_{s}+1\) Else \(m_{s}=m_{s}+1\) End If  End For  If \(g>LP\) \(f_{p}=\frac{n_{i}(n_{i}+m_{s})}{n_{s}(x_{i}+m_{s})+n_{i}(x_{i}+m_{s})}\) \(CR_{m}=\frac{\sum_{k=1}^{K}CR_{m}(k)}{|CR_{rec}|}\) End If  End For

## 4 Results

We conducted several experiments to assess the performance and efficiency of the SaNSDE algorithm. We will first present the results of a small example in Section 4.1. In Section 4.2, we will compare the performance/efficiency of the proposed algorithm with other ones. In Section 4.3, we will present the results on the influence of the learning period parameter on the performance/efficiency of the proposed algorithm. The statistical significance of the results will be presented in Section 4.5. ### A Small Example

We applied the SaNSDE algorithm proposed in Section 3.2 to find the solutions for several cases. Comparison with other algorithms based on results of experiments will be presented in the next subsection. We first illustrate the proposed method by applying it to the motivating example in Section 2. In this example, we consider a planning problem for a process agent \(n\) with five operations to be performed by resource agents. Suppose the upper bound of the processing time of the process is \(\omega_{n}=200\). The DTPN models, \(\Omega_{n}^{1},\Omega_{n}^{2},\Omega_{n}^{3},\Omega_{n}^{4}\) and \(\Omega_{n}^{5}\) for the first operation, the second operation, the third operation, the fourth operation and the fifth operation, respectively, are shown in Figure 1a-e. As there are five operations to be performed, the requirements of the process agent \(n\) are described by \(\Re eq_{n}=(a_{n1},d_{n2},d_{n3},\ldots,d_{nK_{n}},\omega_{n})\). In this case, \(K_{n}=5\), \(d_{n1}=1\), \(d_{n2}=1\), \(d_{n3}=1\), \(d_{n4}=1\) and \(d_{n5}=1\). The model of the process agent \(n\) is shown in Figure 2.

Suppose there are nine types of resource agents in the CPS to perform the five operations. Each type of resource agent can only perform a subset of the five operations. The capability of each type of resource agents is described by the resource activity models in Figure 3. Based on the resource activity model, each resource agent submits a bid to indicate the operations it can perform. For example, the resource activity model \(A_{na_{1}}^{1}\) for resource agent \(a_{1}\) consists of transition \(t_{1}\) and transition \(t_{2}\). As transition \(t_{1}\) is the start transition of Cyber World model \(\Omega_{n}^{1}\) and transition \(t_{2}\) is the end transition of Cyber World model \(\Omega_{n}^{1}\), resource agent \(a_{1}\) can perform the first operation. Therefore, resource agent \(a_{1}\) submits a bid \(B_{a_{1j}}=(a_{a_{1j}1},a_{a_{1j}2},a_{a_{1j}3},\ldots,a_{a_{1j}K_{n}},\tau_{a_ {1j}})=(1,0,0,0,0,25)\), where \(25\) is the processing time of resource agent \(a_{1}\) for the first operation. Similarly, the resource activity model \(A_{na_{2}}^{5}\) for resource agent \(a_{2}\) consists of transition \(t_{5}\) and transition \(t_{6}\). As transition \(t_{5}\) is the start transition of Cyber World model \(\Omega_{1}^{5}\) and transition \(t_{6}\) is the end transition of Cyber World model \(\Omega_{1}^{5}\), resource agent \(a_{2}\) can perform the fifth operation. Therefore, resource agent \(a_{2}\) submits a bid \(B_{a_{2j}}=(a_{a_{2j}1},a_{a_{2j}2},a_{a_{2j}3},\ldots,a_{a_{2j}K_{n}},\tau_{a_ {2j}})=(0,0,0,0,1,25)\), where \(25\) is the processing time of resource agent \(a_{2}\) for the fifth operation. The resource activity model \(A_{na_{3}}^{1}\|A_{na_{3}}^{5}\) for resource agent \(a_{3}\) consists of transitions \(t_{1}\), \(t_{2}\), \(t_{5}\) and \(t_{6}\). As transition \(t_{1}\) is the start transition of Cyber World model \(\Omega_{n}^{1}\) and transition \(t_{2}\) is the end transition of Cyber World model \(\Omega_{n}^{1}\), resource agent \(a_{3}\) can perform the first operation.