# The Importance of Modeling Path Choice Behavior in the Vehicle Routing Problem
## 4 Vehicle Routing Problem
### Definitions and Notations  
The VRP provides the best routes (\(\mathbf{\Omega}\)) to follow in the context of transport problems (i.e., freight distribution, dial and ride for passengers, etc.). The route to follow is constrained by external and technical constraints and behavioral constraints relating to the path (k) chosen between two consecutive nodes (r, s) of the route.  
The following variables are defined in the VRP:  
* r and s are two pick-up and delivery points to be served by the VRP;
* \(\mathbf{\Omega}=[\begin{array}{c}\ldots&\omega_{\mathrm{v,rs}}\ldots&\end{array}]\) is the matrix of routes defined in term of sequences of nodes to be visited by a set of vehicles; the entry \(\omega_{\mathrm{v,rs}}\) of the matrix \(\mathbf{\Omega}\) is equal to one if the vehicle \(\mathrm{v}\) connects the nodes r and s, and it is zero otherwise;  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline _Utility Model_ & _Name_ & _Perceived Utility_ & _DUM_, _RUM_, _FUM_, _QUM_ & _Choice Rate_ & _Starting from Random Utility_ \\ _(UM)_ & & _Function_\(v_{\mathrm{k}}\)(\(\chi_{\mathrm{k}}\), \(\theta_{\mathrm{k}}\)) & _Evaluation_ & \(\mathbf{p}_{\mathrm{k}}\) & \\ \hline \multirow{3}{*}{\begin{tabular}{c} Deterministic \\ (DUM) \\ \end{tabular} } & \multirow{3}{*}{\(\mathrm{P}^{\mathrm{d}}_{\mathrm{k}}\)} & \multirow{3}{*}{\
\begin{tabular}{c} Deterministic \\ utility function \\ \end{tabular} } & \multirow{3}{*}{\(\mathrm{p}^{\mathrm{d}}_{\mathrm{k}}\geq 0\) for the alternative \\  & & & of maximum utility \\  & & & with \(\Sigma_{\mathrm{k}}\)\(\mathrm{p}^{\mathrm{d}}_{\mathrm{k}}=100\%\) \\ \hline \multirow{3}{*}{
\begin{tabular}{c} Random \\ (RUM) \\ \end{tabular} } & \multirow{3}{*}{\(\mathrm{P}^{\mathrm{r}}_{\mathrm{k}}\)} & Continuous \\  & & Continuous \\  & & random utility \\  & & function & (\(pr\) means probability) \\ \cline{1-1} \cline{3-5} Fuzzy & & Continuous \\ \cline{1-1} \cline{3-5} (FUM) & & Continuous \\ \cline{1-1} \cline{3-5}  & & & \(\forall\) k, h\(\in\)l \\ \cline{1-1} \cline{3-5} Quantum & & & (\(\forall\) k  
* \(\mathbf{g}^{\text{(VRP)}}=[\ \ldots\,\mathbf{g}^{\text{(VRP)}}{}_{\text{rs}}\,\ldots\ ]^{\prime}\) with \(\mathbf{g}^{\text{(VRP)}}{}_{\text{rs}}=[\ \ldots\,\mathbf{g}^{\text{(VRP)}}{}_{\text{k}},\ldots\ ]^{\prime}\) the generalized path cost vector perceived by the decision-maker (system manager or driver) connecting the pairs of nodes r and s to be visited for pick-up and delivery;
* \(\mathbf{g}^{\text{(VRP)}}(t)=\xi(\mathbf{f}(t))\) with \(\xi(t)\) the vector of path cost functions defined in the Section 3;
* \(\mathbf{p}^{\text{(VRP)}}=[\ \ldots\,\mathbf{p}^{\text{(VRP)}}{}_{\text{rs}} \,\ldots\ ]^{\prime}\) with \(\mathbf{p}^{\text{(VRP)}}{}_{\text{rs}}=[\ \ldots\,\mathbf{p}^{\text{(VRP)}}{}_{\text{k}},\ldots\ ]^{\prime}\) the vector of path rate connecting the pairs of nodes r and s to be visited for pick-up and delivery;
* \(\mathbf{p}^{\text{(VRP)}}(t)=\boldsymbol{v}(\mathbf{f}(t))\) with \(\boldsymbol{v}(\mathbf{f}(t))\) the vector of path rate functions defined in Section 3;
* \(z(\mathbf{f},\ \mathbf{\Omega},\ \mathbf{g}^{\text{(VRP)}},\ \mathbf{p}^{\text{(VRP)}},\ t)=[\ \ldots\,z_{b}( \mathbf{f},\ \mathbf{\Omega},\ \mathbf{g}^{\text{(VRP)}},\ \mathbf{p}^{\text{(VRP)}}),\ \ldots\ ]^{\prime}\) is the multicriteria objective function with entry \(z_{b}()\) relative to the generic criteria adopted in the multicriteria optimization;
* \(\mathbf{S}_{\text{TE}}\) is the set of feasible technical and external constraints;
* \(\mathbf{S}_{\text{f}}\) is the set of admissible flow (i.e., non-negative, congruent with the passenger and freight demand on the routes, paths and nodes).