Solving the fuzzy p-hub center problem using imperialist competitive algorithm

The p-hub center location tries to minimize the maximum travel times by locating p hubs and allocating other nodes to these hub nodes. There is a wide application in various fields of study for the p-hub center problems, such as telecommunication and transportation systems. In the proposed fuzzy p-hub center problem (FPHCP) by Yang et al. (Yang et al. in Appl Soft Comput 13:2624–2632, 2013), a risk aversion PHCP is considered where travel times are trapezoidal fuzzy variables and value-at-risk criterion is the objective function. In the mentioned research, the original value-at-risk PHCP has turned into two equivalent mixed-integer linear programming models, and a genetic algorithm incorporating local search (GALS) has been applied to solve the equivalent mixed-integer linear programming models. Considering the properties of FPHCPs, the Imperialist competitive algorithm (ICA) incorporating edge recombination operator (ERO) and Elzinga-Hearn algorithm (EHA) is a more suitable method to solve FPHCPs proposed in the current study. The ICA, which uses assimilation and revolution operators, is an effective tool for solving hard combinational optimization problems. The ERO is applied for assimilation in the proposed ICA, an operator that creates a parent like a set of existing parents by looking at the edges. In addition, the EHA is applied for revolution in the proposed ICA, which maintains a covering circle for a subset of the points. We evaluated the performance of the exact method (EM), GALS, and ICA incorporating ERO and EHA by solving and comparing their accuracy and speed on 81 small-scale and large-scale FPHCP samples. Our experiments demonstrate that the proposed method achieves optimal solutions in solving all small-scale sample tests of FPHCPs. Also, it was faster than EM and GALS for solving these sample tests. Finding optimum solutions for large-scale FPHCP sample tests by EM was impossible due to the limitation of computational CPU time. In solving large-scale sample tests, ICA achieved significantly higher accuracy in a shorter time than GALS. So, results show that ICA outperformed EM and GALS in terms of accuracy and speed for solving all FPHCP sample tests.