A Non-Linear Optimization Model for the Multi-Depot Multi-Supplier Vehicle Routing Problem with Relaxed Time Windows

In the realm of supply chain logistics, the Multi-Depot Multi-Supplier Vehicle Routing Problem (MDMSVRP) poses a significant challenge in optimizing the transportation process to minimize costs and enhance operational efficiency. This problem involves determining the most cost-effective routes for a fleet of vehicles to deliver goods from multiple suppliers to multiple depots, considering various constraints and non-linear relationships. The routing problem (RP) is a critical element of many logistics systems that involve the routing and scheduling of vehicles from a depot to a set of customer nodes. One of the most studied versions of the RP is the Vehicle Routing Problem with Time Windows (VRPTW), in which each customer must be visited at certain time intervals, called time windows. In this paper, it is considered that there are multiple depots (supply centers) and multiple suppliers, along with a fleet of vehicles. The goal is to efficiently plan routes for these vehicles to deliver goods from the suppliers to various customers while considering relaxed time windows. This research is intended to establish a new relaxation scheme that relaxes the time window constraints in order to lead to feasible and good solutions. In addition, this study develops a discrete optimization model as an alternative model for the time-dependent VRPTW involving multi-suppliers. This research also develops a metaheuristic algorithm with an initial solution that is determined through time window relaxation.