Modeling two-stage UHL problem with uncertain demands

In hub location problems, a decision-maker may encounter hybrid uncertain environments where randomness and fuzziness are in the state of affairs. The purpose of this paper is to develop a new two-stage uncapacitated hub location (UHL) problem with recourse, in which uncertain parameters are characterized by both probability and possibility distributions. When demands are the only uncertain parameters, we show that the proposed two-stage UHL model is equivalent to a static optimization problem subject to equilibrium constraint. In the case that the randomness of uncertain demands follows normal distributions, we reduce the equilibrium constraint to its equivalent credibility constraint. Furthermore, when the fuzziness of uncertain demands follows triangular distributions, we discuss the convexity of equilibrium objective function, and establish the equivalent deterministic programming model of the original UHL problem. In general case, we adopt fuzzy simulation (FS) method to approximate uncertain parameters. To solve the proposed hub location problem, we design a hybrid heuristic algorithm by integrating genetic algorithm (GA), variable neighborhood search (VNS) and FS. We conduct some numerical experiments and compare the computational results obtained by the VNS-based GA and standard GA. The computational results together with convergence analysis demonstrate that the VNS-based GA achieves the better performance than standard GA. Finally, we carry out the sensitivity analysis to recognize the most significant parameter of the proposed optimization model.