Integrating battery-related decisions into truck-drone tandem delivery problem with limited battery resources

The truck-drone tandem delivery mode provides a promising application in last-mile package delivery but is limited by the duration of drones. The battery swap strategy is a widely adopted approach to extend the cruising ranges of drones, ensuring that depleted batteries can be swapped for fully charged ones in a matter of minutes. However, most existing studies assume that there are sufficient batteries available at the depot, which is impractical as storing a large number of batteries is expensive. To bridge this gap, this paper considers the truck-drone tandem delivery problem with a battery swap strategy, under the condition of a limited number of batteries. To address the challenges posed by the practical limitation of the number of batteries, we propose a joint optimization problem integrating two types of interdependent decisions, i.e., battery-related decisions and route-related decisions. The battery-related decisions identify which batteries to be installed on drones and establish optimal battery charging schedules at the depot. And the route-related decisions determine the truck-drone tandem delivery routes. The studied joint optimization problem is formulated as a mixed integer linear programming model, and this model is integrated into a well-designed adaptive large neighborhood search algorithm to determine the two types of decisions. Specifically, on the basis of traditional operators, we design a series of depot-related operators tailored to the feature of route-related decisions. Furthermore, regarding the features of battery swapping and charging schedules, we introduce a novel battery operator to determine optimal battery-related decisions. The numerical experiments show that introducing the battery-related decisions can bring flexible battery schedules when the total number of batteries is limited. The effects of battery capacity, charging rate, charging cost, drone speed, and the number of batteries and drones are analyzed to provide practical suggestions for companies.