## 5 Numerical Experiments and Analysis
### Computational Results and Analysis
#### 5.2.3 Comparison Between Deterministic and Robust Model  
In order to verify the risk-averse nature of the proposed robust optimization model, we next compare the results obtained from our model, deterministic model with unit energy consumption taking the expected value, and the traditional Min-Max robust optimization model which focuses only on the worst situation of the uncertainty (i.e., unit energy consumption taking the largest value). Instances with different size are tested, and the results are listed in Table 7, where TC (D), TC (R), TC (W) are the objective values of total cost achieved by deterministic model, robust model, and the model which optimizes the situation with the largest energy consumption. The column of \"D@W\" means the total cost for the optimal solution obtained by the deterministic model but the worst situation occurs (i.e., the unit energy consumption takes the upper bound of the uncertainty set). The maximum number of iterations for NHALNS is set to 200.  
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline \multirow{2}{*}{**Inst**} & \multicolumn{4}{c}{**TC (NHALNS)**} & \multicolumn{4}{c}{**TC (Hybrid VNS/TS)**} & \multirow{2}{*}{**Gap (\%)**} \\ \cline{2-3} \cline{5-10}  & **N. V** & **Best** & **Avg** & **SD** & **N. V** & **Best** & **Avg** & **SD** \\ \hline c106\_21 & 28 & 35,333.22 & 35,334.12 & 0.94 & 38 & 47,795.51 & 47,796.31 & 1.66 & \(-\)26.07 \\ c107\_21 & 12 & 15,838.29 & 15,838.89 & 1.28 & 31 & 39,197.80 & 39,200.1 & 6.90 & \(-\)59.59 \\ c108\_21 & 18 & 22,834.47 & 22,835.37 & 1.92 & 32 & 40,324.11 & 40,329.61 & 10.37 & \(-\)43.38 \\ c109\_21 & 16 & 20,429.54 & 20,429.64 & 0.30 & 22 & 28,183.96 & 28,184.56 & 1.50 & \(-\)27.51 \\ r106\_21 & 16 & 20,435.07 & 20,435.77 & 2.10 & 30 & 37,832.91 & 37,832.91 & 0.00 & \(-\)45.98 \\ r107\_21 & 14 & 17,953.09 & 17,953.79 & 2.10 & 30 & 37,563.01 & 37,564.21 & 3.60 & \(-\)52.21 \\ r108\_21 & 13 & 16,637.11 & 16,637.41 & 0.90 & 22 & 27,880.78 & 27,882.88 & 4.35 & \(-\)40.33 \\ r109\_21 & 19 & 24,158.10 & 24,158.5 & 1.20 & 26 & 32,842.60 & 32,844 & 4.20 & \(-\)26.44 \\ r110\_21 & 14 & 17,948.89 & 17,949.79 & 1.81 & 23 & 29,107.66 & 29,107.66 & 0.00 & \(-\)38.33 \\ r111\_21 & 14 & 17,911.49 & 17,912.89 & 2.33 & 23 & 29,068.03 & 29,068.13 & 0.30 & \(-\)38.38 \\ r112\_21 & 15 & 19,177.41 & 19,177.41 & 0.00 & 19 & 24,255.26 & 24,256.66 & 4.20 & \(-\)20.94 \\ r106\_21 & 17 & 22,046.55 & 22,046.75 & 0.60 & 41 & 51,917.51 & 51,920.71 & 6.94 & \(-\)57.54 \\ r107\_21 & 15 & 19,426.81 & 19,426.81 & 0.00 & 27 & 34,152.59 & 34,153.49 & 1.92 & \(-\)43.12 \\ r108\_21 & 14 & 18,526.17 & 18,528.77 & 7.80 & 26 & 33,007.91 & 33,019.71 & 34.41 & \(-\)43.89 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison results between NHALNS and VNS/TS.  
Figure 3: Convergence of NHALNS and Hybrid VNS/TS.  
It can be concluded from Table 7 that the robust optimization model of this paper can achieve a satisfactory tradeoff between the cost performance and risk aversion. In other words, our model effectively considers both cost saving and reasonable risk aversion simultaneously. From the view of risk analysis, although the optimal solution obtained by the deterministic model seems to outperform all the others, it will be dramatically degraded (poorer than the robust solution or even infeasible) when an unfavorable environment is experienced. With respect to optimization performance, the total cost of our robust solution is significantly smaller than that obtained by the traditional Min-Max robust optimization model. In fact, risk is not avoidable under an uncertain environment, whereas it is almost impossible that all the arcs encounter the worst case in practice. Hence, it would be overly conservative to make a decision based upon the traditional Min-Max robust optimization model. But on the other hand, big losses may be incurred if risk is ignored.