## 4 A General Solving Methodology
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Deterministic Scenario**} & \multicolumn{3}{c}{**Stochastic Scenario**} & \multicolumn{3}{c}{**Fuzzy Scenario**} \\ \cline{2-7}
**Instance** & **BKS** & **OBD** & **GAP (\%)** & **Det** & **Stochastic** & **Stoch-Fuzzy** & **Fuzzy** \\  & **(1)** & **Sol. (2)** & **(1)–(2)** & **Sol. (3)** & **Sol. (4)** & **Sol (5)** & **Sol. (6)** \\ \hline p1.2.f & 80 & 80 & 0.0 & 78.9 & 79.3 & 77.4 & 76.8 \\ p1.2.i & 135 & 135 & 0.0 & 127.6 & 129.4 & 125.2 & 117.9 \\ p1.2.k & 175 & 175 & 0.0 & 169.3 & 174.4 & 164.6 & 143.2 \\ p1.2.n & 235 & 235 & 0.0 & 228.8 & 232.7 & 216.2 & 189.7 \\ p1.3.n & 190 & 190 & 0.0 & 182.5 & 189.6 & 180.0 & 174.1 \\ p1.4.j & 75 & 75 & 0.0 & 59.3 & 63.3 & 51.3 & 45.1 \\ p1.4.k & 100 & 100 & 0.0 & 98.3 & 99.9 & 99.9 & 95.8 \\ p1.4.l & 120 & 120 & 0.0 & 118.9 & 119.2 & 118.6 & 117.1 \\ p1.4.m & 130 & 130 & 0.0 & 98.2 & 102.9 & 91.5 & 86.3 \\ p1.4.n & 155 & 155 & 0.0 & 99.9 & 104.0 & 107.5 & 98.2 \\ p1.4.o & 165 & 165 & 0.0 & 159.4 & 164.2 & 148.6 & 143.6 \\ p1.4.p & 175 & 175 & 0.0 & 171.3 & 174.4 & 163.7 & 155.5 \\ p2.2.d & 160 & 160 & 0.0 & 150.6 & 150.6 & 150.0 & 137.5 \\ p2.2.i & 230 & 230 & 0.0 & 223.4 & 226.3 & 228.5 & 204.0 \\ p2.3.i & 200 & 200 & 0.0 & 191.5 & 195.2 & 186.7 & 177.9 \\ p3.2.c & 180 & 180 & 0.0 & 179.1 & 179.2 & 178.9 & 160.2 \\ p3.2.d & 220 & 220 & 0.0 & 212.3 & 217.5 & 197.7 & 179.8 \\ p3.2.g & 360 & 360 & 0.0 & 358.3 & 358.8 & 308.2 & 297.9 \\ p3.2.q & 760 & 760 & 0.0 & 748.5 & 755.2 & 663.4 & 630.6 \\ p3.2.r & 790 & 790 & 0.0 & 768.3 & 774.9 & 656.1 & 638.5 \\ p3.3.e & 200 & 200 & 0.0 & 198.2 & 199.0 & 195.7 & 187.6 \\ p3.4.g & 220 & 220 & 0.0 & 212.6 & 217.3 & 205.0 & 191.3 \\ p5.2.d & 80 & 80 & 0.0 & 75.5 & 77.4 & 73.7 & 70.7 \\ p5.2.k & 670 & 670 & 0.0 & 643.3 & 662.1 & 646.0 & 612.5 \\ p5.2.p & 1150 & 1150 & 0.0 & 1135.4 & 1138.1 & 1105.1 & 1073.0 \\ p5.3.f & 110 & 110 & 0.0 & 107.4 & 109.1 & 107.6 & 103.0 \\ p5.3.o & 870 & 870 & 0.0 & 856.2 & 865.1 & 836.9 & 806.9 \\ p5.4.g & 140 & 140 & 0.0 & 135.3 & 137.9 & 134.3 & 129.2 \\ p5.4.t & 1160 & 1160 & 0.0 & 1139.5 & 1148.4 & 1107.4 & 1068.1 \\ p5.4.u & 1300 & 1300 & 0.0 & 1279.5 & 1286.3 & 1239.2 & 1198.2 \\ p6.2.d & 192 & 192 & 0.0 & 185.4 & 188.1 & 177.8 & 164.2 \\ p6.2.e & 360 & 360 & 0.0 & 276.4 & 297.2 & 285.4 & 277.9 \\ p6.2.f & 588 & 588 & 0.0 & 577.4 & 580.0 & 519.5 & 501.2 \\ p6.2.g & 660 & 660 & 0.0 & 648.3 & 650.5 & 584.9 & 569.2 \\ \hline
**Average** & **362.8** & **362.8** & **0.0** & **349.9** & **354.3** & **333.3** & **318.3** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of results, in terms of collected reward, for the different TOP scenarios. Figure 8: Gaps of different optimization methods with respect to the OBD solution. **(a)** Results for the _VRP_ dataset. **(b)** Results for the _TOP_ dataset. Figure 9: Gaps of different optimisation methods with respect to the OBD solution. Figure 11: Best solution for VRP-Stochastic scenario. Figure 10: Best solution for VRP-Deterministic scenario. ## 7 Conclusions

This work has introduced the \"fuzzy simheuristic\" methodology to deal with NP-hard transportation problems under uncertainty scenarios, both probabilistic and fuzzy in nature. This uncertainty is tackled in a general way, since we consider that both stochastic and fuzzy uncertainty are present in many real-life transportation systems. Hence, pure

Figure 12: Best solution for VRP-Stochastic and Fuzzy scenario. Figure 13: Best solution for VRP-Fuzzy scenario. deterministic, pure stochastic, and pure fuzzy scenarios represent particular cases that can also be addressed by employing our fuzzy simheuristic methodology. Since our methodology combines metaheuristics with stochastic and fuzzy simulation, it takes the best characteristics of both worlds, i.e., (i) the metaheuristics component provides the efficiency necessary to explore the solution space in order to find near-optimal solutions in short computational times.