Hub networks play a crucial role in optimizing transportation costs in air and road systems. Their main objective is to strategically locate hubs and allocate non-hub nodes within the network. The modular hub location problem is a specific area of hub network design that focuses on accurately calculating transportation costs, considering factors like trip numbers and capacity constraints in network routes. This study proposes a mixed-integer programming model to address the modular hub location problem with multiple allocations. It considers dependent and independent costs associated with vehicles per trip between hub network routes, considering specific vehicle capacities. Two datasets are utilized for validation: the CAB dataset representing 25 nodes of US airports and the TR dataset representing the Turkish transportation system with 81 nodes. To tackle the NP-hard nature of hub location models and the computational complexity of the proposed model, two solutions are developed. Firstly, a novel LP relaxation-based method using GAMS software provides near-optimal solutions for medium-sized instances. Additionally, a Genetic Algorithm (GA) implemented in MATLAB handles larger instances. The GA's efficiency is enhanced by tuning its parameters using the Taguchi method. Results analysis shows that both proposed algorithms yield high-quality solutions within significantly reduced timeframes compared to the CPLEX solver in GAMS software. The LP relaxation-based method performs well for medium-sized instances, while the GA approach is efficient for larger instances after parameter tuning with the Taguchi method.

Recovery of used products has steadily become interesting issue for research due to economic reasons and growing environmental or legislative concern. This paper presents a closed-loop logistics network design based on reverse logistics models. A mixed integer linear programming model is implemented to integrate logistics network design in order to prevent the sub-optimality caused by the separate design of the forward and reverse networks. The study presents a single product and multi-stage logistics network problem for the new and return products not only to determine subsets of logistics centers to be opened, but also to determine transportation strategy, which satisfies demand imposed by facilities and minimizes fixed opening and total shipping costs. Since the deterministic estimation of some parameters such as demand and rate of return of used products in closed loop logistics models is impractical, an uncertain programming is proposed. In this case, we assume there are several economic conditions with predefined probabilities calculated from historical data. Then by means of expert & apos; s opinion, a fuzzy variable is offered as customer & apos; s demand under each economic condition. In addition, demand and rate of return of products for each customer zone is presented by fuzzy-random variables, similarly. Therefore, a fuzzy-random programming is used and a priority-based genetic algorithm is proposed to solve large-scale problems.