# Route Optimization for Open Vehicle Routing Problem (OVRP): A Mathematical and Solution Approach
## 2 Related Work  
The literature frames a series of works related to the vehicle routing problem from the perspective of the number of distribution centers. From the context of single-depot route planning, recent research contributions included those developed by [PERSON] et al. Figure 1: A sample of vehicle routing networks. (2024) [8]; [PERSON] et al. (2024) [9]; [PERSON] and [PERSON] (2023a) [10]; [PERSON] and [PERSON] (2023) [11]; [PERSON] and [PERSON] (2023b) [12]; [PERSON] et al. (2023) [13]; [PERSON] et al. (2022) [14]; [PERSON] et al. (2022) [15]; [PERSON] et al. (2022) [16]; [PERSON] et al. (2022) [17]; and [PERSON] (2022) [18]. From another approach, with multiple warehouses or distribution centers, planning becomes more complex but also more flexible. In this scenario, the works of [PERSON], [PERSON] et al. (2024) [19]; [PERSON] et al. (2024) [20]; [PERSON], [PERSON] et al. (2023) [21]; [PERSON] et al. (2023) [22]; [PERSON] and [PERSON] (2023) [23]; and [PERSON] (2023) [24] stand out. The optimization challenges is to find the best routes for a set of vehicles to satisfy the demands of a set of customers or locations. Some common problems appear in works developed by [PERSON], [PERSON] et al. (2024) [8]; [PERSON], [PERSON] et al. (2023) [21]; [PERSON] and [PERSON] (2023b) [12]; [PERSON], [PERSON] and [PERSON] (2023) [23]; and [PERSON] (2023) [24] with the Time Windowed Freight Vehicle Problem. Likewise, problems appear in the studies of [PERSON] et al. (2024) [9]; [PERSON] and [PERSON] (2023a) [10]; [PERSON] and [PERSON] (2023b) [12]; and [PERSON] (2022) [18] with the heterogeneity fixed fleet of vehicles problem. The works of [PERSON] and [PERSON] (2023) [11] and [PERSON] et al. (2022) [14] exhibit common problems with the third-party logistics services problem; the work of [PERSON] et al. (2024) [20] with the resource sharing problem; the work of [PERSON], [PERSON] et al. (2023) [13] with the supply sharing problem, considering demand urgency; the work of [PERSON], [PERSON] et al. (2022) [15] with the problem of excess reserves with limited vehicle capacity; and the works of [PERSON] et al. (2022) [16] and [PERSON] et al. (2022) [17] with the crowd-shipping problem and split deliveries, respectively. In synthesis, these studies have improved computational efficiency and flexibility in route planning. Implementing these approaches in open vehicle routing optimization (OVRP) provides efficient and adaptable solutions to complex logistics challenges. Considering the type of techniques or tools to address the solution of optimization problems in VRP and OVRP environments, which allow for finding high-quality solutions in a reasonable time, heuristic approaches are among them and have been applied in the works of [PERSON], [PERSON] et al. (2024) [19]; [PERSON] et al. (2022) [16]; and [PERSON] (2022) [16]; and those of a metaheuristic nature, have been addressed by the following researchers: [PERSON] et al. (2024) [8]; [PERSON] et al. (2024) [9]; [PERSON] et al. (2024) [20]; [PERSON] et al. (2023) [21]; [PERSON] and [PERSON] (2023a) [10]; [PERSON] and [PERSON] (2023) [11]; [PERSON] and [PERSON] (2023b) [12]; [PERSON] et al. (2023) [13]; [PERSON] et al. (2023) [22]; [PERSON] and [PERSON] (2023) [23]; [PERSON] (2023) [24]; [PERSON] et al. (2022) [14]; [PERSON] et al. (2022) [15]; and [PERSON] et al. (2022) [17]. In search of developing tasks quickly and efficiently and given the complexity of the problems addressed, in recent years, a series of fundamental algorithms have been proposed for decision making: Authors such as [PERSON] et al. (2024) [19] with the Harris Hawks Optimization and Particle Swarm Optimization algorithm. [PERSON] et al. (2024) [8] with the multi-objective whale learning optimization algorithm. [PERSON] et al. (2024) [9]; [PERSON] and [PERSON] (2023b) [12]; and [PERSON] (2023) [24] with the modified ant colony algorithm. [PERSON] et al. (2024) [20] with the adaptive hybrid simulated tempering and tempering algorithm. [PERSON] et al.