Optimizing Multi-Commodity Fixed-Charge Solid Transportation Under Spherical Fuzzy Environment: A Multi-Objective Perspective

Abstract

In today's dynamic and complex logistics networks, decision makers must simultaneously balance transporta-tion cost, delivery time, and service quality in multi-commodity systems involving fixed charges and uncertainty. This study proposes a novel multi-objective fixed-cost solid transportation problem formulated under the framework of spherical fuzzy sets, which effectively capture uncertainty in transportation costs, supplies, demands, and conveyance capacities. Although fixed costs are typically assumed to be constant in classical models, real-world factors such as fluctuating contractual conditions and administrative constraints often introduce indirect uncertainty, which is explicitly addressed in the proposed formulation. The model optimizes multiple conflicting objectives, including the minimization of transportation and fixed costs and delivery time, while simultaneously maximizing service reliability and customer satisfaction. An efficient solution methodology is developed to manage the resulting complexity. Numerical experiments and real-world case studies demonstrate that the proposed approach achieves an average reduction of approximately 15% in total transportation cost and an improvement of about 25% in computational efficiency compared with existing methods. These results highlight the robustness and effectiveness of the proposed framework in supporting reliable and informed decision-making. The methodology provides valuable insights for logistics, urban distribution, and supply chain systems seeking cost-effective and sustainable transportation solutions under uncertainty.