Exact algorithms and heuristics for capacitated covering salesman problems (2403.06995v1)

Abstract: This paper introduces the Capacitated Covering Salesman Problem (CCSP), approaching the notion of service by coverage in capacitated vehicle routing problems. In CCSP, locations where vehicles can transit are provided, some of which have customers with demands. The objective is to service customers through a fleet of vehicles based in a depot, minimizing the total distance traversed by the vehicles. CCSP is unique in the sense that customers, to be serviced, do not need to be visited by a vehicle. Instead, they can be serviced if they are within a coverage area of the vehicle. This assumption is motivated by applications in which some customers are unreachable (e.g., forbidden access to vehicles) or visiting every customer is impractical. In this work, optimization methodologies are proposed for the CCSP based on ILP (Integer Linear Programming) and BRKGA (Biased Random-Key Genetic Algorithm) metaheuristic. Computational experiments conducted on a benchmark of instances for the CCSP evaluate the performance of the methodologies with respect to primal bounds. Furthermore, our ILP formulation is extended in order to create a novel MILP (Mixed Integer Linear Programming) for the Multi-Depot Covering Tour Vehicle Routing Problem (MDCTVRP). Computational experiments show that the extended MILP formulation outperformed the previous state-of-the-art exact approach with respect to optimality gaps. In particular, optimal solutions were obtained for several previously unsolved instances.

• The paper introduces novel ILP and MILP formulations for CCSP and MDCTVRP that minimize travel distances while satisfying coverage constraints.
• The paper demonstrates that a Biased Random-Key Genetic Algorithm and a matheuristic approach yield superior near-optimal solutions for challenging large-scale instances.
• The study validates the integration of exact and heuristic methods through extensive computational experiments, setting new benchmarks in multi-depot vehicle routing performance.

Exploring the Capacitated Covering Salesman Problem (CCSP) and Multi-Depot Covering Tour Vehicle Routing Problem (MDCTVRP): An Insightful Synthesis and Computational Study

Introduction to CCSP and MDCTVRP Formulations and Solutions

The paper introduces novel frameworks for addressing two complex routing problems within logistics and distribution networks – the Capacitated Covering Salesman Problem (CCSP) and its extension, the Multi-Depot Covering Tour Vehicle Routing Problem (MDCTVRP). These problems are significant in scenarios where direct access to customers is not feasible, necessitating solutions that cater to coverage-based servicing. Introduced are intricate Integer Linear Programming (ILP) models for the CCSP and a Mixed Integer Linear Programming (MILP) for the MDCTVRP, presenting a fresh perspective on optimizing routing by incorporating coverage concepts.

CCSP Formulation and Insights

The CCSP is formulated as a vehicle routing problem where the objective is to minimize the total distance traveled while servicing customer demands within a specified coverage range. The proposed ILP model accounts for vehicle capacities and enforces constraints ensuring every demand is met either through direct visitation or coverage. Computational experiments demonstrate the model's ability to derive optimal solutions for smaller instances and competitive bounds for larger scenarios when paired with heuristic methods.

MDCTVRP Formulation and Advances

The paper extends the paper into MDCTVRP, which combines aspects of depot-based vehicle routing with the coverage servicing mechanism. The novel MILP formulation outperforms existing exact algorithms in literature, offering substantial improvements in terms of optimality gaps and solving previously unsolved instances. This formulation demonstrates strategic advancements in handling multi-depot scenarios with coverage considerations, marking a significant edge over previous state-of-the-art methodologies.

Heuristic and Metaheuristic Solutions for CCSP

Acknowledging the computational complexity of exact solutions for larger instances, the paper proposes a Biased Random-Key Genetic Algorithm (BRKGA) tailored for the CCSP. This metaheuristic approach yields superior upper bounds across a broad set of instances, showcasing its effectiveness in finding near-optimal solutions efficiently. Further, an innovative matheuristic integrates this BRKGA with a covering and packing model, enhancing solution quality through focused intensification strategies. The comparative analysis highlights the substantial benefits of combining heuristic strategies with exact formulations, particularly in tackling large-scale instances where direct ILP solutions may not be tractable.

Computational Study Results and Implications

Through extensive computational experiments, the paper evaluates the performance of proposed models and methodologies across a diverse set of instances. The findings reveal the potential of integrating exact and heuristic approaches, where the proposed BRKGA and matheuristic significantly improve solution bounds derived from ILP formulations. In the context of MDCTVRP, the new MILP formulation sets new benchmarks by delivering improved lower bounds and resolving instances that have remained unsolved, affirming its methodological superiority and practical relevance.

Future Directions and Concluding Remarks

The paper closes with suggestions for future research, emphasizing the exploration of valid inequalities and branch-and-cut strategies to further enhance the exact solution frameworks for CCSP and MDCTVRP. The possibility of extending these problems into multi-objective formulations, incorporating covering range as an additional optimization objective, presents an exciting avenue for expanding the scope of vehicle routing problems.

In summary, this work lays a robust foundation for tackling the CCSP and MDCTVRP, incorporating coverage principles into vehicle routing optimizations. Through methodological innovations and extensive computational validations, the paper contributes significantly to the field, offering new perspectives and tools for addressing complex distribution and servicing scenarios encountered in logistics and supply chain management.