Marzieh Zaker - M.Sc. Grad., Department of Industrial Engineering, Bu-Ali Sina University, Hamedan, Iran
Amir Saman Kheirkhah - Associate Professor, Department of Industrial Engineering, Bu-Ali Sina University, Hamedan, Iran
Reza Tavakkoli-Moghaddam - Professor, School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran

A capacitated location-routing problem (CLRP) is one of the new areas of research in distribution management. It consists of two problems; locating of facilities and routing of the vehicle with a specific capacity. The purpose of the CLRP is to open a set of stores, allocate customers to established deposits, and then design vehicle tours in order to minimize the total cost. In this paper, a new mathematical programming model for multi-depot location-routing problems is considered. This model considers heterogeneous vehicles and fuzzy travel times, which are innovative and practical limitations compared to the previous studies (e.g., simultaneous pickup and delivery). This makes the model close to real-world situations. After modeling, the fuzzy model is changed to a deterministic model by credibility theory. Since this problem belongs to a class of NP-hard ones because of its computational complexity, it is impossible to find the optimal solution in reasonable time. Therefore, a particle swarm optimization algorithm is proposed and designed to solve the presented model. To show the efficiency of the proposed PSO, its results are compared with the optimal solutions obtained by an exact method embedded in the optimization software. Furthermore, the proposed PSO is able to solve medium- and large-sized problem efficiently.

Multi-depot location-routing problem, Simultaneous pickup, and delivery, heterogeneous vehicles, Fuzzy Travel Time, meta-heuristic algorithm