# The Importance of Modeling Path Choice Behavior in the Vehicle Routing Problem
## 3 Path Choice Problem and Assignment (Behavioral Constraint)
### Discussion  
In the DUM, the deterministic perceived utility function can be viewed mathematically as a random perceived utility function with zero limiting variance. The sum of all the percentages on all the alternatives is equal to one. The assumption of a deterministic choice of the path is not possible for several reasons: the analyst does not know exactly the behavior of the system manager and the drivers; the analyst cannot accurately predict the state of the transport system; and the state of the transport system is congested and is time-dependent.  
In the RUM, different distributions can be adopted for the continuous random utility function (i.e., Gumbel for the Logit class; Weibull for the Weibit class; multivariate Gauss for the Probit class; multivariate gamma for the Gammit class [37]; more than one random utility functions for the mixed class). Considering all the alternatives, the random utility function can be: identical (homoscedastic) or nonidentical (heteroscedastic) between users, alternative or time; independent or joint distributions; with deterministic \(\theta\) parameters or with parameters distributed with a random function; in this case, the utility has two random effects, one relating to the entire utility and one for each random parameter; for more details see [38]. The sum of all probabilities on all alternatives is equal to one.  
In the FUM, the sum of all possibilities over all alternatives can be greater than one. For this reason, the choice rate (\(p_{k}\)) is evaluated by the possibility \(q_{k}\) with a transformation function; one of these functions was proposed by [PERSON] [39] to be:  
\[p_{k}=(p^{f}_{k})^{\gamma}/\Sigma_{h\in I}(p^{f}_{h})^{\gamma} \tag{16}\]  
with \(\gamma\) a parameter greater than zero to be calibrated.  
In the QUM, the interference term decreases or increases the probability evaluated with a random utility function in relation to the level of interference (unclear perception) between the alternatives. It can be demonstrated that the interference term \(\mathrm{p}^{\mathrm{q}}_{\mathrm{k}}\) assumes values in relation to the level of interference and guarantees that the sum of all rate probability on all alternatives is one (\(\mathrm{p}^{\mathrm{q}}_{\mathrm{k}}\in[-\mathrm{p}^{r}_{\mathrm{k}};1-\mathrm{ p}^{r}_{\mathrm{k}}]\) and \(\Sigma_{\mathrm{k}}\)\(\mathrm{p}^{\mathrm{q}}_{\mathrm{k}}=0\)) and the rate (\(\mathrm{p}_{\mathrm{k}}\)) is evaluated with:  
\[\mathrm{P}_{\mathrm{k}}=\mathrm{p}^{r}_{\mathrm{k}}+\mathrm{p}^{\mathrm{q}}_{ \mathrm{k}} \tag{17}\]  
For the FUM and the QUM, in relation to the specification of the utility function, the same model described in the case of the random perceived utility can be obtained (identical or not identical distribution; homoscedastic or heteroscedastic; not jointly or jointly; deterministic or random parameters).  
Note that the FUM and the QUM cannot be considered as a particular or a general case of the RUM: they are different from the RUM. The main reasons are as follows: they consider a different user's behavior not representable by the analyst with random utility models; the fuzzy model does not evaluate a probability, it evaluates a possibility, and the sum on all the alternatives is not equal to one; the quantum model evaluates the interference deriving from considering the interference between the alternatives unobservable by the analyst and by the users; the quantum model modifies with an additive term the random probability.  
A summary of the reported path choice models with a comparison of the main characteristics is reported in Table 1.