## 4 Experiments
### Experimental Results and Analysis
#### 4.4.2 Experiment 2
\[Z^{\prime}{}_{\text{PRO}}=\sum_{p\in SP}o_{p}\times\min\{c^{\prime}{}_{m}\} \tag{30}\]  
Lastly, assuming that maximum loading capacities of all manufacturers' vehicles are set to the maximal value of \(u_{mk}\), and the unit delivery costs and distances are set the lowest value between manufacturers and warehouses, then the lowest delivery cost of manufacturers' vehicles are shown as follows:  
\[Z^{\prime}{}_{\text{MTS}} =2\times\left(\Bigg{[}\sum_{p\in SP}o_{p}/\max_{m,k}(u_{mk}) \Bigg{]}\right)\times\min\{d^{\prime}{}_{mp}\}\times\min\{b^{\prime}{}_{mk}\} \tag{31}\] \[+\left(P-\Bigg{[}\sum_{p\in SP}o_{p}/\max_{m,k}(u_{mk})\Bigg{]} \right))\times\min\{d^{\prime}{}_{pp^{\prime}}\}\times\min\{b^{\prime}{}_{mk}\}\]  
From Formula (1) to Formula (5), a lower bound \(S\_L\) for each instance is obtained, denoted as Formula (32). \[S\_L{=}Z^{\prime}{}_{\text{PUR}}{+}Z^{\prime}{}_{\text{STM}}{+}Z^{\prime}{}_{ \text{PRO}}{+}Z^{\prime}{}_{\text{MTS}} \tag{32}\]  
Then, the optimization rate \(OP\) is calculated as Formula (33) which asses the quality of the solutions of the algorithm. The experiment results are shown in Tables 12 and 13 and Figure 9.  
\[OP=\frac{\text{Best solution}{-}S\_L}{S\_L}\times 100\% \tag{33}\]  
It can be seen from Tables 12 and 13 that: when time consumption of all the algorithms are close, all the best solutions, average solutions, and worst solutions of 20 instances solved by HCDBAVNS are not worse than those solved by VNS and PSOVNS. On the other hand, there are seven and nine instances of HCDBAVNS obviously better than HDBAVNS in both the best solutions and average solutions, accounting for 35% and 45% of the total instances, respectively. Then, although DCIS made the optimization capacity of HCDBAVNS not increase significantly (all the examples were less than 1%), it made HCDBAVNS more steady. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multirow{2}{*}{**MIT**} & \multirow{2}{*}{\(S\_L\)} & \multicolumn{6}{c}{**VNS**} & \multicolumn{6}{c}{**PSOVNS**} \\ \cline{3-14}  & & & \multicolumn{2}{c}{**Best**} & \multicolumn{2}{c}{**Average**} & \multicolumn{2}{c}{**Wost**} & \multicolumn{2}{c}{**OPs(\%)**} & \multicolumn{2}{c}{**Time**} & \multicolumn{2}{c}{**Best**} & \multicolumn{2}{c}{**Average**} & \multicolumn{2}{c}{**Wost**} & \multicolumn{2}{c}{**OPs(\%)**} & \multicolumn{2}{c}{**Time**} \\  & & & **Solution** & **Solution** & **solution** & **Solution** & **Solution** & **Solution** & **Solution** & **Solution** & **Solution** & **Solution** & **O  
In Figure 9, it can be sees that HCDBAVNS has a higher possibility of finding a better solution than VNS. Especially, the \(OP\) of HCDBAVNS, which is 4% less than that of VNS in P09. In addition, HDBAVNS is also better than PSOVNS. It proves that BA has better global optimization ability than PSO. Further, the paired two-sample mean analysis method in the \(t\)-Test is used to test whether the three algorithms are significantly different. Supposing that the significance level is 0.05 and mean deviation is 0, The experimental results are shown in Table 14. In the \(t\)-Test, \(p>0.05\) indicates no significant difference; \(0.01<p<0.05\) indicates a significant difference; \(p<0.01\) indicates an extreme significant difference. In Table 14, it can be seen from W1 test that the \(p\) values of the best solution and average solution are 0.01149 and 0.01514 (both less than 0.05). In other words, both the best solution and average solution of HCDBAVNS show significant differences to HDBAVNS. In addition, compared with VNS, HDBAVNS, and HCDBAVNS have shown extreme significant differences. Furthermore, HDBAVNS shows a significant difference to PSOVNS. This also proves that HCDBAVNS finally exhibits strong optimization capability and stability.