## 3 Heuristics Method to Calculate the Global Fitness of a Solution
### Computation of Non-Conflict VTWs  
We denote \(\mathbf{a}_{s}=<\!\!g_{s1},\ g_{s2},\ldots,\ g_{s\,|\,\mathbf{a}_{1}|}\!>\) for convenience and assume that at some time point, AEOS \(s\) has tried to observe a set of targets \(\mathbf{b}_{s}=<\!\!g_{s1},\ g_{s2},\ldots,\ g_{u}\!>\), where \(u<1_{\mathbf{a}_{s}}1\), and the start-observing time of each target in \(\mathbf{b}_{s}\) is known (owing to the conflicts of VTWs, the start time of some target in bs may be zero). For a new target \(h\in\mathbf{a}_{s}\backslash\mathbf{b}_{s}\), its VTWs set \(\mathbf{W}_{sh}\) may be overlapped with the scheduled duration \([\{t_{g^{\prime}},t_{g^{\prime}}+d_{g^{\prime}}\}\ |\ g^{\prime}\in\mathbf{b}_{s}\) and \(t_{g^{\prime}}\
eq 0\)), resulting in a VTW conflict. For example, consider \(\mathbf{b}_{s}=[g_{1}]\) and target \(g_{2}\). As illustrated in Figure 3, there are four possible situations regarding the overlap between the scheduled duration \([t_{g1},\ t_{g1}+d_{g1}]\) and a VTW \(W_{sg2}k=[\varpi,\ \varphi]\in\mathbf{W}_{sg2}\). Especially, note that in Figure 3c, the VTW \([\varpi,\ \varphi]\) becomes infeasible for \(g_{2}\) because \(s\) observes \(g_{1}\) during that time interval.  
Figure 3: The analysis on the non-conflict VTWs where (**a**) the non-conflict VTW is located after the scheduled duration; (**b**) the non-conflict VTW is located before the scheduled duration; (**c**) there is no non-conflict VTW; and (**d**) the non-conflict VTW is split by scheduled duration into two parts.  
Since AEOS \(s\) can observe, at most, one target at a time, at least one non-conflict VTW is required before observing a new target. Motivated by the analysis in Figure 3, we develop Algorithm 1, namely _compute non-conflict VTWs_ (CNV), to compute a non-conflict VTW for a new target \(h\
otin\mathbf{b}_{s}\).  
```
Input: AEOS \(s\), observed target sequence \(\mathbf{b}_{s}\), \(t_{g}\) (\(\forall g\in\mathbf{b}_{s}\)), \(h\
otin\mathbf{b}_{s}\), and \(\mathbf{W}_{sh}\); Output: The non-conflict VTW set \(\mathbf{V}_{sh}(\mathbf{b}_{s})\);
1. Let \(\Omega=\mathbf{W}_{sh}\);
2. for each \(g\in\mathbf{b}_{s}\) for each VTW [\(\varpi\), \(\varphi\)] \(\in\Omega\) if\(t_{g}\leq\varpi\), \(t_{g}+d_{g}<\phi\) and \(t_{g}+d_{g}>\varpi\) // see Figure 3a Set \(\mathbf{\sigma}=t_{g}+d_{g}\);
3. else if\(t_{g}>\varpi\), \(t_{g}+d_{g}\geq\varphi\) and \(t_{g}<\phi\) // see Figure 3b Set \(\phi=t_{g}\);
4. else if\(t_{g}\leq\varpi\) and \(t_{g}+d_{g}>\phi\) // see Figure 3c Delete [\(\varpi\), \(\varphi\)] from \(\Omega\);
5. else if\(t_{g}>\varpi\) and \(t_{g}+d_{g}<\varphi\) // see Figure 3d Divide [\(\varpi\), \(\varphi\)] into two segments [\(\varpi\), \(t_{g}\)] and [\(t_{g}+d_{g}\), \(\varphi\)];
6. end if
7. end for
8. for each [\(\varpi\), \(\varphi\)] \(\in\Omega\) if\(\phi-\varpi<d_{h}\)  Delete [\(\varpi\), \(\phi\)] from \(\Omega\);
9. end for
10. End Output \(\mathbf{V}_{sh}(\mathbf{b}_{s})=\Omega\);
```  
**Algorithm 1**Compute non-conflict VTWs (CNV).  
In Algorithm CNV, we begin by setting \(\Omega=\mathbf{W}_{sh}\). For each visited target \(g\in\mathbf{b}_{s}\), we \"prune\" the overlapped part between every VTW [\(\varpi\), \(\varphi\)] \(\in\Omega\) and the time interval [\(t_{g}\), \(t_{g}+d_{g}\)], based on four different situations illustrated by Figure 3 (Lines 2-14). Next, in Lines 15-19, we delete the VTWs from \(\Omega\) that have a length shorter than \(d_{h}\) and finally output the non-conflict VTWs \(\mathbf{V}_{sh}(\mathbf{b}_{s})=\Omega\).