# The Importance of Modeling Path Choice Behavior in the Vehicle Routing Problem
## 3 Path Choice Problem and Assignment (Behavioral Constraint)
### Models
#### 3.2.1 Path Choice  
The analyst hypothesizes a functional form for the perceived utility function (\(v_{\text{k}}\)(x\({}_{\text{k}}\),\(\theta_{\text{k}}\))). In relation to the hypothesis, the choice rate (p\({}_{\text{k}}\)) of the users who choose each alternative (k) belonging to the set of perceived alternatives is obtained from:  
* [... ] A deterministic utility model (DUM), assuming a perceived deterministic utility function; a choice percentage (p\({}^{\text{d}}\)\({}_{\text{k}}\)) is obtained;
* [... ] A random utility model (RUM) assuming a perceived random utility function; a choice probability (p\({}^{\text{r}}\)\({}_{\text{k}}\)) is obtained;
* [... ] A fuzzy utility model (FUM) assuming a perceived fuzzy utility function; a fuzzy possibility (p\({}^{\text{f}}\)\({}_{\text{k}}\)) is obtained;
* [... ] A quantum utility model (QUM) assuming a perceived quantum utility function; a random choice probability (p\({}^{\text{r}}\)\({}_{\text{k}}\)) and an interference term (p\({}^{\text{q}}\)\({}_{\text{k}}\)) are obtained.  
It should be noted that in this paper the path rate was evaluated using discrete choice models and random utility theory. The same considerations can be extended by introducing models that do not belong to the random utility class for evaluating the path rate.  
Given a user n and two route nodes, the procedure for evaluating the choice rate is shown in Figure 3.  
In the DUM [27], the choice rate (p\({}_{\text{k}}\)) is evaluated with the percentage p\({}^{\text{d}}\)\({}_{\text{k}}\) assigned to the alternatives with the maximum utility value. An alternative has p\({}_{\text{k}}\) = p\({}^{\text{d}}\)\({}_{\text{k}}\) = 1 (or 100%) if it is single with maximum value; the solution is not unique in the case of several alternatives with the maximum perceived utility (any division of 1 between these alternatives is allowed):  
\[\text{p}^{\text{d}}\)\({}_{\text{k}}\)\(\geq\) 0 for the alternative of maximum utility with \(\Sigma_{\text{k}}\) p\({}^{\text{d}}\)\({}_{\text{k}}\) = 1  
\[\text{p}_{\text{k}}\) = p\({}^{\text{d}}\)\({}_{\text{k}}\) (2)  
In the RUM [27], the perceived utility is modelled as a continuous random utility function. The choice rate (p\({}_{\text{k}}\)) is evaluated with the probability p\({}^{\text{r}}\)\({}_{\text{k}}\) that the perceived utility of the alternative is the greatest with respect to all the perceived alternatives:  
\[\text{p}^{\text{r}}\)\({}_{\text{k}}\) = probability(\(v_{\text{k}}\)(\(\mathbf{x}_{\text{k}}\), \(\theta\)) \(>\)\(v_{\text{h}}\)(\(\mathbf{x}_{\text{h}}\), \(\theta\)), \(\forall\) k, h \(\in\) I\({}_{\text{rs}}\))  
\[\text{p}_{\text{k}}\) = p\({}^{\text{r}}\)\({}_{\text{k}}\) (4)  
In the FUM [30; 31], the perceived utility is modelled as a continuous fuzzy utility number; the possibility q\({}_{\text{k}}\) that the perceived utility of the alternative is greatest is:  
\[\text{p}^{\text{f}}\)\({}_{\text{k}}\) = possibility(\(v_{\text{k}}\)(\(\mathbf{x}_{\text{k}}\), \(\theta\)\({}_{\text{k}}\))\(>\)\(v_{\text{h}}\)(\(\mathbf{x}_{\text{h}}\), \(\theta\)\({}_{\text{h}}\)), \(\forall\) k, h \(\in\) I\({}_{\text{rs}}\))  
In the QUM (the theory and some applications are reported in [32; 33; 34]; an aggregate calibration is reported in [35]), the perceived utility is modelled as a quantum utility function with interference between alternatives. The analyst assumes an intermediate decision level with alternatives not clearly perceived by users and not directly observable by the modeler (interference). From the quantum theory, the interference term (p\({}^{\text{d}}\)\({}_{\text{k}}\)) is a function of the probability vector p\({}^{\text{r}}\) with entry p\({}^{\text{r}}\)\({}_{\text{k}}\) and the vector \(\alpha\) containing the angle in the Hilbert space relative to the interference of each pair of alternatives:  
\[\text{p}^{\text{q}}\)\({}_{\text{k}}\) = interference(p\({}^{\text{r}}\), \(\alpha\))