# The Importance of Modeling Path Choice Behavior in the Vehicle Routing Problem
## 4 Vehicle Routing Problem
### Discussion  
The first consideration concerns the multicriteria objective function. As reported in Section 2, different actors interact with each other (decision-takers, analysts, users) and objectives of different classes can be considered (e.g., transport, information and communication, energy considering the smart city; economic, environmental, social criteria considering sustainability). The objectives can also be conflicting with each other (e.g., a high speed and short travel time could conflict with the pedestrian area for citizens). For this reason, it is necessary to adopt a multicriteria optimization. In the real case, it can be observed that the drivers do not follow the best route and do not consider the same cost estimation criterion. The results are also confirmed in [40]. The paper [40] analyzed 12 routes with more than four visited nodes in each route. In the 12 routes, 73 paths were chosen. Three criteria were adopted for the VRP and the PCP for route optimization and path generation: minimum time, minimum distance and minimum energy consumption. The paper [40] showed that there was not a single criterion adopted in the choice of routes and paths. Minimum times were frequently adopted in the VRP (about 70% of similarity); the three criteria were used in the PCP (approximately 75% of similarity). It confirmed the necessity to use a multicriteria approach in the objective function for the VRP and the need to evaluate the rate choice. The importance of path planning was also simulated in [41]. The second consideration relates to the evaluation of the path rate. Given a pick-up and delivery pair of nodes r and s (the time dependence is omitted for simplicity's sake):  
* In the DUM, for each pair of pick-up and delivery nodes, the generalized minimum cost is: \[\mathrm{g}^{\mathrm{(VRP)min}}{}_{\mathrm{rs}}=\mathrm{min}{}_{\mathrm{rs}} \mathrm{g}^{\mathrm{(VRP)}}{}_{\mathrm{k}}\ \forall\ \mathrm{g}^{\mathrm{(VRP)}}{}_{\mathrm{k}}\in\mathrm{g}^{\mathrm{(VRP)}}{}_{ \mathrm{rs}}\] (23)
* In the RUM, the FUM and the QUM, which best represent reality, a rate choice is evaluated for each perceived path and some indicators must be evaluated (i.e., the expected perceived cost for all paths or for the path with a rate choice higher than a threshold or a path that satisfies a decision criterion); if the expected cost values for all the perceived paths are considered, the generalized cost is: \[\mathrm{g}^{\mathrm{(VRP)E}}{}_{\mathrm{rs}}=\Sigma_{\mathrm{k}}\in{}_{ \mathrm{rs}}\mathrm{P}^{\mathrm{(VRP)}}{}_{\mathrm{k}}\bullet\mathrm{g}^{ \mathrm{(VRP)}}{}_{\mathrm{k}}\ \forall\ \mathrm{k}\in\mathbf{I}_{\mathrm{rs}}\] (24)  
Note \(\mathrm{g}^{\mathrm{(VRP)E}}{}_{\mathrm{rs}}\) is obtained with a convex combination of path cost \(\mathrm{g}^{\mathrm{(VRP)}}{}_{\mathrm{k}}\geq\mathrm{g}^{\mathrm{(VRP)min}}{}_ {\mathrm{rs}}\) and \(\mathrm{g}^{\mathrm{(VRP)E}}{}_{\mathrm{rs}}\geq\mathrm{g}^{\mathrm{(VRP)min}}{}_ {\mathrm{rs}}\). It means that the VRP solved with the minimum cost for each pair of nodes is a lower bound of the real perceived cost and for this reason, the VRP solution cannot be obtained with the DUM. The DUM cannot guarantee that the best solution is the same as that obtained with the RUM, the FUM or the QUM. These considerations were verified in a numerical application. A small network size was considered (Figure 4) to allow the reader to follow all the steps of the procedure. In the test network, one depot (number 1) and four pick-up and delivery nodes (2, 3, 4, 5) were considered. The pick-up and delivery times at each node were fixed and were not reported into the routing time because it did not affect the best sequence of nodes to visit. The path cost was obtained with a fixed-point model in a static approach assuming a congested network or with a day-to-day and/or within-day model in a dynamic approach, assuming a time-dependent network.