## 5 Examples and Results
### Algorithm Comparison Analysis
In order to compare the quantity of Pareto optimal solutions of different algorithms more accurately, the average running time (T), two-set coverage (SC), and hypervolume (HV) of the algorithms are selected to further compare the optimization performance of each algorithm from the perspectives including computational cost, accuracy, diversity, and convergence of Pareto optimal solution, respectively. Figure 8: Comparison of Pareto solution set distribution in different scenarios. 1. Average running time (T): it indicates the average time of algorithm running, which can be used to measure the time complexity of each algorithm, and the smaller the value, the higher the computational efficiency of the algorithm. \[\text{T}=\frac{1}{N}\sum_{i=1}^{N}\text{T}_{\text{i}} \tag{26}\]  
where T\({}_{i}\) is the time consumption of each independent operation and \(N\) is the number of independent runs of the algorithm. 2. Two-set coverage (SC) [43]: also known as C metric, it is used to compare the accuracy of the Pareto solutions obtained by each algorithm, and its lower value indicates the better quality of the Pareto solution set. \[C(\text{A},B)=\frac{1}{N}\sum_{i=1}^{N}\frac{|b\in\text{B}_{i}|\exists a\in \text{A}_{i}:a\prec b|}{|\text{B}_{i}|} \tag{27}\]  
where the set A\({}_{i}\) and the set B\({}_{i}\) are the sets of Pareto solutions obtained by different algorithms in each operation, respectively, and \(a\prec b\) indicates that the individual \(a\) is dominated by the individual \(b\). For comparison, the ERNS-DE is chosen as the benchmark algorithm to compare the accuracy of Pareto solutions obtained by the HDA-CR with the other three algorithms in this paper; when calculating the two-set coverage of ERNS-DE, the HDA-CR is chosen as the benchmark algorithm to calculate and compare the two-set coverage with that of HDA-CR in a two-by-two comparison. 3. Hypervolume (HV) [44]: It is used to measure the ratio of the size width of the Pareto solution set obtained by the algorithm in the area covered by the target space, and this index can measure the convergence and diversity of the algorithm at the same time. The larger the value, the more uniform the individual distribution and better the convergence of the Pareto solution set obtained by the algorithm. \[\text{HV}=\frac{1}{N}\sum_{i=1}^{N}\text{volume}(\bigcup_{i=1}^{|\Lambda_{i}|}v _{i}) \tag{28}\]  
where \(v_{i}\) is the hypervolume enclosed between the reference point and individual \(i\). After normalizing the fitness values of the individuals, the point (1,1) is selected as the reference point. According to the above multi-objective algorithm evaluation index, the values of each index of different algorithms under different scenarios are calculated, and the results are shown in Table 4.  
As can be seen from the comparison of the three indicators in Table 4, the HDA-CR proposed in this paper can effectively improve the quality of the solutions in the Pareto optimal solution set searched for at the expense of certain time resources, and outperforms other algorithms in terms of solution accuracy, diversity, and convergence; at the same time, it can effectively alleviate the resource conflict problem, reducing the cost of computing in  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline
**Experimental** & \multicolumn{3}{c}{**HDA-CR**} & \multicolumn{3}{c}{**MODE**} & \multicolumn{3}{c}{**ENNS-DE**} & \multicolumn{3}{c}{**NSGA-II**} & \multicolumn{3}{c}{**MOPSO**} \\ \cline{2-13}
**Scenarios** & \(\mathbf{T}\) & \(\mathbf{SC}\) & \  
resource-constrained scenarios. As for the formulation of multi-period allocation schemes for hazardous materials, the main purpose is to determine the optimal long-term allocation scheme, that is, to ensure the quality of the Pareto solution, so that the hazardous materials at each reserve point are consumed in an orderly and reasonable manner, and the running time of the algorithm can be disregarded.