A new formulation for multi-modal shortest path problem with fuzzy arc travel times

This paper mainly investigates multi-modal shortest path problems in fuzzy environment, in which the arc travel times are represented by fuzzy variables. Within the framework of fuzzy programming, a chance-constrained programming model is formulated, and then its corresponding crisp equivalent is given. It is noted that we introduce transfer arcs to represent the travel mode change for the convenience of the model´s establishment. Finally, some numerical examples are given to show the applications of the proposed models.