# The Importance of Modeling Path Choice Behavior in the Vehicle Routing Problem
## 5 Conclusions  
In this paper, the influence of the PCP in the VRP was investigated. Initially, the transport system analysis and the main actors involved were described, as well as the objectives considering different classifications and constraints. From this analysis emerged the need to study the problem through a multicriteria approach that considered the choice of the path within a behavioral constraint.  
The behavioral nonlinear constraint can be represented by studying the state of equilibrium and the dynamic evolution of the transport system using path choice models. For path choice models, deterministic models cannot be used considering that a future state of the transport system is influenced by various factors such as congestion, incomplete information of the system, etc.  
The main discrete choice models of random utility were reported together with the specifications for calculating the path rate. The expected value of the perceived cost between each pair of nodes was calculated and inserted into the general optimal model of the VRP. In real problems, the number of vehicles considered in the VRP is lower compared to the total number of vehicles in circulation. Therefore, it is possible to assume that the path choice probability of the vehicles of the VRP does not affect the state of the transport system.  
The model was applied to a test network with assumptions on the cost values and choice rate. It was seen that, in the adopted test network, the best route changed by establishing a deterministic or nondeterministic path choice model.  
The results obtained provide policy recommendations for analysts regarding the path choice models to be adopted for the VRP solution. Using costs for a single path could provide different VRP solutions than using all perceived paths. However, considering that the VRP refers to a future state of the system, it is not possible to solve it through deterministic and hypothetical costs, even if historical. The solution of the PCP within the VRP is recommended. Decision-makers also benefit from this model specification because it better represents user behavior.  
In term of applicability, the model has limits regarding the optimization as the nonlinear behavioral constraint requires adequate procedures to find a solution in acceptable processing times. In the proposed methodology, the consolidated algorithms used for the solution of the assignment models, including the solution of the PCP, and for the solution of the VRP can be adopted. In practice, in real-size problems and for real-time solutions, it requires suitable activities for the integration of the two algorithms. The model presented requires the development of optimal procedures for the solution of real-scale problems in the future. It could start from excellent metaheuristic procedures adopted for the solution of the VRP, integrating them with the consolidated procedures that solve the behavioral constraint.  
This research is partially supported by DIIES--Universita di Reggio Calabria and by the project \"La Mobilita per i passeggeri come Servizio-MyPass\", Fondi PON R&I 2014-2020 e FSC, Progetti di Ricerca PNR 2015-2020, codice identificativo ARS01_01100.  
Not applicable.  
The author thanks the anonymous reviewers for their useful comments and interesting suggestions.  
The author declares no conflict of interest.  
DSS, decision support system; DUM, deterministic utility model; ICT, intelligent and communication technology; FUM, fuzzy utility model; PCP, path choice problem; QUM, quantum utility model; RUM, random utility model; SC, smart city; VRP, vehicle routing problem.