Efficiently solving open capacitated location-routing problems through a discrete fireworks algorithm

In the open capacitated location-routing problem (OCLRP), a fleet of distribution vehicles departs from selected depots to satisfy customers’ demands, but they do not need to return to their starting depots after serving all customers. Comparing the solutions of the OCLRP and its corresponding capacitated location-routing problem (CLRP) can provide valuable insights for companies considering whether to outsource their delivery activities. To effectively solve the OCLRP, this study proposes a novel discrete fireworks algorithm (DFWA) with two key innovations. (1) An adaptive search mechanism: embedding swap, insertion, and reverse operations into explosion/mutation breaks the limitations of traditional fireworks algorithms in discrete optimization by expanding the search space while enhancing local tuning, thus boosting global optimal solution discovery. (2) A diversified selection strategy: integrating fitness value and Hamming distance improves the “premature convergence from single fitness selection” defect in existing algorithms. It retains high-performance solutions while maintaining population diversity. Evaluated on OCLRP instances adapted from standard CLRP benchmarks, the DFWA shows strong competitiveness, consistently generating high-quality solutions within reasonable computation time. A real-world OCLRP case further verifies its practical applicability in complex industrial scenarios.