Recursive delivery multiple flying sidekicks traveling salesman problem: An enlightenment of the Covid-19 pandemic

The last-mile delivery of COVID-19 vaccines required a structured timeframe, particularly for double-dose vaccines, where follow-up deliveries had to occur after vaccine type-specific intervals. This recurring service pattern inspired the development of a novel recursive delivery concept—where services must be repeated based on prior deliveries and elapsed time. We define recursive delivery as a multi-period mechanism that dynamically establishes whether, when, and how each customer should be revisited, built upon both service history and time-based constraints. To capture such recursive service patterns, this paper introduces a coordinated truck-drone delivery system, where either vehicle may recursively revisit previously serviced locations depending on the type of service—single or recursive— provided in earlier periods. To formalize this concept, we present a new variant of the Traveling Salesman Problem, termed the Recursive delivery multiple Flying Sidekicks Traveling Salesman Problem (R−mFSTSP). This model extends traditional TSP by incorporating dynamic, service-type-dependent revisit scheduling. The R−mFSTSP has wide applicability in various domains requiring structured, time-sensitive service repetition, such as maintenance, health services, and supply replenishment. We formulate the R−mFSTSPas a mixed-integer linear programming model aimed at minimizing total transportation costs. Given the computational limitations of exact solvers for large-scale instances, a tailored metaheuristic algorithm has been developed to address the structural characteristics of the proposed R−mFSTSP. Its performance has been benchmarked against results from a relevant study, demonstrating competitive outcomes. Furthermore, a lower bound is provided to evaluate the quality of the obtained solutions.