# The Importance of Modeling Path Choice Behavior in the Vehicle Routing Problem
## 3 Path Choice Problem and Assignment (Behavioral Constraint)
### Models
#### 3.2.2 Behavioral Constraint  
The behavioral constraint considers the interaction between supply and demand and starts from supply and demand models. The demand model refers to the dependence of a user's choice on service transport level indicators and socioeconomic data. The supply model relates to the dependence of the service transport level on the traffic flow. For simplicity's sake, before introducing the behavioral constraint, the single dependency is reported considering the dependency between link flow, link cost, path cost, path choice, demand and flow. * Link flow-link cost Considering the small number of VRP vehicles designed in a real problem compared to the total number of vehicles in the transport system, it can be assumed that the output of the VRP (best route) does not affect the state of the transport system in terms of the cost of the link and flow. With this assumption, the VRP cost can be obtained from the flow on the  
Figure 3: Path choice model: variables and parameters. transport system using the link cost function. In a link 1, the link cost function gives the link cost from the link flow:  
\[c_{l}=\gamma_{l}(\mathbf{f})\] in static a context (7) \[c_{l}(t)=\gamma_{l}(\mathbf{f}(t))\] in dynamic a context (8)  
The time-dependent link cost functions is generally assumed as a weighted sum of the cost at the same instant (t) and the cost obtained in a previous interval (t-\(\tau\)). In this specific case, a dependence on multiple time intervals should be adopted: \(\gamma_{l}(\mathbf{f}(t),\mathbf{f}(t\text{-}\tau))\). Considering the scope of this paper, the generic time-dependent function \(\gamma_{l}(0)\) is reported; in this paper, the dynamic assignment is not explored and reference is made to the specific literature for further information [36]. The static cost can be assumed as a particular case of the dynamic cost considering that it is assumed to depend on the flow, but it is constant over time. From the link cost functions, the vector of link cost functions are obtained:  
\[c(t)=\gamma(\mathbf{f}(t))=[\ \ldots\ \gamma_{l}(\mathbf{f}(t)),\ \ldots\ ]^{\prime} \tag{9}\]  
* Link cost-path cost The path cost \(\mathbf{g}\) is obtained as the sum of the additive (with respect to the link) path cost (\(\mathbf{g}^{\text{A}}\) with entry \(g_{k}{}^{\text{A}}\)) and the nonadditive path cost (\(\mathbf{g}^{\text{NA}}\) with entry \(g_{k}{}^{\text{NA}}\)): \[g_{k}(t)=g_{k}{}^{\text{A}}(t)+g_{k}{}^{\text{NA}}=\Sigma_{l}\in\kappa\ \gamma_{l}(\mathbf{f}(t))+g_{k}{}^{\text{NA}}\] (10)  
Considering the whole path \(k\) between all nodes \(u\) and \(v\), the path choice vector is defined as (\(\mathbf{g}\)), dependent on the link flow vector:  
\[\mathbf{g}(t)=\bar{\zeta}(\mathbf{f}(t)) \tag{11}\]  
* Path cost-path choice In the DUM, it is assumed that each user, for each alternative \(k\), from \(u\) to \(v\), perceives a deterministic utility given by the linear combination (\(v_{k}=\theta_{k}{}^{\prime}\ \mathbf{x}_{k}\)) of the attributes (\(\mathbf{x}_{k}\)) with respect to the parameter (\(\theta_{k}\)). Users choose the alternative of greatest perceived utility. In nondeterministic models it is assumed that each user:
* Perceives a nondeterministic utility (\(u_{k}\)) with expected (or barycenter) value (\(v_{k}\)=E(\(u_{k}\))) given by the linear combination of the attributes with respect to the parameter (\(v_{k}=\theta_{k}{}^{\prime}\ \mathbf{x}_{k}\));
* Chooses the alternative of maximum perceived utility, and the analyst can evaluate the choice rate (\(p_{k}\)) for each alternative (\(k\)) belonging to the perceived choice set (\(I_{\text{rs}}\)). Note that the quantitative attributes of the alternative \(k\) include all the variables that influence the path choice. In vector \(\mathbf{x}_{k}\), one element (it could also be the only one) is the cost \(g_{k}\) of the path \(k\). Considering all the alternative, the choice rate vector \(\mathbf{p}\) depends on the path cost vector (\(\mathbf{g}\)), and the path cost depends on the link flow vector (\(\mathbf{f}\)).