## 3 Algorithm Description
### Amending
When a token flows into the sink place \(p_{q,e}\), it signifies the complication of each task in \(w_{q}\). Step 2 constructs the subnet (\(\beta_{k}\), \(M_{k}\)), which represents the task assignments of edge servers in \(s_{k}\). Each transition \(t_{q,h}\in T_{k}\) is associated with a resource place \(p_{j}\in P_{k}\) through a pair of directed arcs (\(t_{q,h}\), \(p_{j}\)) and (\(p_{j}\), \(t_{q,h}\)). This connection indicates that task \(m_{h}\) is assigned to edge server \(u_{j}\). Each resource places \(p_{j}\) in \(P_{k}\) always contains one token. **Example 2**. : _Given the individual in Figure 4 with solution \(\theta=\{\theta_{1},\ldots,\theta_{7}\}\), where \(\theta_{1}=\{m_{3}\}\), \(\theta_{2}=\{m_{S}\), \(m_{4}\}\), \(\theta_{3}=\{m_{1}\}\), \(\theta_{4}=\{m_{2}\}\), \(\theta_{5}=\{m_{5}\), \(m_{7}\}\), \(\theta_{6}=\emptyset\), and \(\theta_{7}=\{m_{6}\}\). Figure 6a shows the Petri net (\(\alpha_{q}\), \(M_{q}\)) for each \(w_{\theta}\in\mathbf{W}\). Then we have \(T_{1}=\{t_{1,3}\}\), \(T_{2}=\{t_{2,8}\), \(t_{1,4}\}\), \(T_{3}=\{t_{1,1}\}\), \(T_{4}=\{t_{1,2}\}\), \(T_{5}=\{t_{1,5}\), \(t_{2,7}\}\), and \(T_{7}=\{t_{1,6}\}\) according to \(\theta\), and the Petri net (\(\beta_{k}\), \(M_{k}\)) for each satellite \(s_{k}\in\mathbf{S}\) is shown in Figure 6b. After composing all above (\(\alpha_{q}\), \(M_{q}\)) and (\(\beta_{k}\), \(M_{k}\)), the Petri-net-based amended (\(N\), \(M_{0}\), \(\theta\)) is shown in Figure 6c._  
In (\(N\), \(M_{0}\), \(\Theta\)), each transition can be fired exactly once under initial marking \(M_{0}\) or the state \(M\) reached from \(M_{0}\). The eventually marking, where all tokens are collected in \(p_{q,e}\) for all \(\forall w_{q}\in\mathbf{W}\), is denoted as \(M_{E}\). From \(M_{0}\) to \(M_{E}\), each transition in (\(N\), \(M_{0}\), \(\Theta\)) is fired once, resulting in the completion of all tasks. For an individual \(\Delta=\{N;\,Y\}\) and its amended (\(N,M_{0}\), \(\Theta\)), transition \(t_{q,h}\) corresponds to task \(m_{h}\), where \(m_{h}\in w_{q}\). The transition sequence \(\pi_{\Delta}\) can be obtained by replacing each task in \(\aleph\) with the corresponding transition. For instance, the transition sequence for individual in Figure 4 is \(\pi_{\Delta}=t_{1,1}t_{2}s_{1}t_{4}t_{1}s_{1}t_{2}t_{3}t_{2}t_{1,6}\). The sequence \(\pi_{\Delta}\) is _feasible_ if \(M_{E}\) can be reached from \(M_{0}\) through \(\pi_{\Delta}\), which is represented as \(M_{0}[\pi_{\Delta}>M_{E})\). Then, firing transitions in the sequence \(\pi_{\Delta}\) sequentially can accomplish all tasks in \(\mathbf{W}\), and the solution is feasible. Thus, a feasible individual can be obtained by finding a transition sequence \(\pi_{\Delta}\) that satisfies \(M_{0}[\pi_{\Delta}>M_{E})\). After the aforementioned analysis, we can propose an algorithm to amend individuals that violate precedence constraints using amended (\(N,M_{0}\), \(\theta\)). ```
Input: a candidate individual \(\Delta=\{\aleph;\ \Upsilon\}\); Output: a feasible individual \(\Delta^{*}=\{\aleph^{*};\ \Upsilon^{*}\}\);
1: Obtain solution \(\theta\) from \(\Delta\) by algorithm BD;
2: Construct Petri-net-based amended (\(N,M_{0}\), \(\theta\));
3: Generate the transition sequence \(\pi_{\Delta}\) from \(\Delta\);
4:for\(n=1\) to \(v\)
5:while (\(\pi_{\Delta}[n]\) is disabled under \(M_{n-1}\))
6: Move \(\pi_{\Delta}[n]\) to the end of \(\pi_{\Delta}\);
7: Move \(\Upsilon[n]\) to the end of \(\Upsilon\);
8:end
9: Let \(M_{n-1}[\pi_{\Delta}[n]>M_{n}\);
10:end
11: Let \(\pi_{\Delta}{}^{*}=\pi_{\Delta}\) and \(\Upsilon^{*}=\Upsilon\);
12: Obtain a permutation of tasks N* from \(\pi_{\Delta}{}^{*}\);
13: Output \(\Delta^{*}=\{\aleph^{*};\ \Upsilon^{*}\}\);
```  
**Algorithm AM (Amending method)**  
First, we create amended (\(N,M_{0}\), \(\theta\)) and transition sequence \(\pi_{\Delta}\) for candidate individual \(\Delta=\{\aleph;\ \Upsilon\}\) and its corresponding solution \(\theta\). Next, we check whether \(\pi_{\Delta}[n]\) is disabled under \(M_{n-1}\) in sequential order.