# The Importance of Modeling Path Choice Behavior in the Vehicle Routing Problem
## 1 Introduction  
The transport system has an important role within smart cities (SC) and intelligent transport systems (ITS). Further details are given in Section 2. The introduction of increasingly advanced technologies and the development of smart cities has made the transport system increasingly important, and it is one of the three pillars of the smart city. Of fundamental importance is the construction of adequate reliable mathematical models to be used for the design and management of transport systems. In this context, the problem of the path choice between a pair of points and the influence of the best sequence of paths between serial pairs of points in a transport system are studied. It can support the decision-takers and the analysts for the transport system design and management. In this paper, considering a transport network model, the following definitions are assumed:  
1. A path is a set of consecutive and loopless transport network links connecting two consecutive nodes of the route; between two nodes, one or more paths can be perceived; the path choice is studied with the path choice problem (PCP);
2. A route is an ordered list of network nodes (in the case of freight for pick-up and delivery activities) to be visited following paths; the optimal routes are studied with the vehicle routing problem (VRP). The general objective of this work is to investigate the role of PCP in the VRP, in the transport network, in order to identify the influence of user choice. The specific objective concerns the study of the importance of the correct path choice modeling for the evaluationof the optimal routing configuration. Proper modeling and optimization can help the system manager in the context of transport system planning activities for the optimization of the entire transport system from the point of view of users, citizens and companies (i.e., freight distribution for the delivery company, dial and ride for the passenger companies), for the optimization of the routes and paths. The method proposed can support decision-takers and for this reason, this paper proposes the problem at a methodological level. This paper does not have a specific objective concerning the study of the algorithms to be adopted for the solution of the PCP and the VRP. The literature on PCP and VRP solution models and algorithms is very extensive. Considering the objectives of this paper, some references are reported in each research area and the references reported are not exhaustive. A classification of the reported state of the art is reported in Figure 1. For further details, the references reported in the cited papers can be considered. 1. The PCP has the main objective of finding the paths taken by users, given two consecutive pick-up and delivery nodes. In the transport demand assignment to the network, the path choice follows [1] a user equilibrium approach ([PERSON]'s first principle [1]) and does not follow an optimal system approach ([PERSON]'s second principle [1]). The Wardrop's first principle provides that each user moves by maximizing his utility function; the system reaches a user equilibrium condition in which no user can improve its utility by a unilateral decision. [PERSON]'s second principle considers that users choose cooperatively by maximizing the sum of all users' utilities; the system reaches the optimum system configuration. In uncongested networks, the two configurations give the same overall utility; in congested networks the optimum configuration of the system generates an overall utility for users greater than or equal to that obtained with the user equilibrium configuration. [PERSON]'s first principle is applied at the network level within the assignment problem [1; 2; 3]. The PCP can be modeled with [PERSON]'s [4] formulation and two submodels are included: the perception model related to the choice set of alternatives [5] and the choice model related to each alternative [6].