# The Importance of Modeling Path Choice Behavior in the Vehicle Routing Problem
## 4 Vehicle Routing Problem
### Discussion
From the state of the transport system, it was assumed that the cost in the shortest path (first, \(\mathbf{G}^{(1)}\)), which connects the pair of nodes in the routing problem, was equal to:  
Figure 4: Test network. It was assumed that three paths were perceived in each pair of nodes (a different number of paths perceived in each pair of nodes could also be assumed). For the pair of nodes 1-4, it was assumed that the cost in the second path was equal to the minimum cost increased by 50% and the cost in the third path was equal to the minimum cost increased by 100%. In the other pairs of nodes, it was assumed that the cost in the second path was equal to the minimum cost increased by 10% and the cost in the third path was equal to the minimum cost increased by 20%. The costs in the second (**G\({}^{\text{(2)}}\)**) and in the third path (**G\({}^{\text{(3)}}\)**) were: \begin{tabular}{c c c c c c c c c c c} \hline \hline  & _Nodes_ & \(1\) & \(2\) & \(3\) & \(4\) & \(5\) & _Nodes_ & \(1\) & \(2\) & \(3\) & \(4\) & \(5\) \\ \hline  & \(1\) & - & 9.1 & 2.5 & 2.1 & 7.0 & & \(1\) & - & 9.9 & _2.7_ & _2.8_ & _7.7_ \\  & \(2\) & 10.0 & - & 9.9 & 10.0 & 7.4 & & \(2\) & 11.0 & - & 11.0 & 11.0 & 8.0 \\ \(\text{G}^{\text{(2)}}=\) & \(3\) & _2.7_ & _11.0 & - & _3.3_ & _9.3_ & **G\({}^{\text{(3)}}=\)** & \(3\) & _3.0_ & _12.0_ & - & _3.6_ & _10.0_ \\  & \(4\) & _1.7_ & _11.0_ & _3.6_ & - & _7.4_ & & \(4\) & _1.9_ & _13.0_ & _4.0_ & - & 8.0 \\  & \(5\) & _7.7_ & _8.1_ & _10.0_ & _8.1 & - & & \(5\) & _8.5_ & _8.9_ & _11.0_ & _8.9_ & - \\ \hline \hline \end{tabular}   
The cost vector \(\mathbf{g}_{\text{VRP}}\) specified in the paper was obtained starting from the three matrices: the pairs of nodes were reported in sequence and the three paths cost were reported for each pair:  
\[\mathbf{g}^{\text{(VRP)}}=[8.2;9.1;9.9;2.2;2.5;2.7;\ldots;9.3;10.0;11.0;7.4;8. 1;8.9]^{\prime} \tag{25}\]  
Considering the DUM for the PCP, the minimum cost for each pair of nodes was considered. The best route was with the sequence of nodes 1-4-5-2-3-1 with an expected travel cost for the route of 27.0, which was the sum of the cost in the minimum paths. Considering the RUM for the PCP, with Logit choice with a Gumbel parameter of 2.5 [6, 27], given a pair of nodes, the rate choice for path k was:  
\[\text{p}^{\text{(VRP)}}_{\text{k}}=\text{e}^{-\text{gk/\Theta}}/\Sigma_{\text{ h}}\,\text{e}^{-\text{gh/\Theta}} \tag{26}\]  
The path rate vector \(\mathbf{p}^{\text{(VRP)}}\) was:  
\[\mathbf{p}^{\text{(VRP)}}=[45\%;32\%;23\%;37\%;33\%;30\%;\ldots;46\%;32 \%;22\%;44\%;32\%;24\%]^{\prime} \tag{27}\]  
With the RUM for the PCP, the best route was with the sequence of nodes 1-5-2-3-4-1 with 29.6 as the estimated travel cost for the route given by the weighted sum (with probability) of the cost considering three routes in each pair of nodes. In this particular case, the optimal routing considering the DUM and the RUM for the PCP gave different results in terms of node sequences. The expected cost evaluated with the RUM was greater than the expected cost evaluated with the DUM. This case demonstrates that applying the DUM to solve the VRP can give a different solution than the case without the DUM. Considering the DUM is not a real case, it should be avoided in the VRP.