We study the vehicle routing problem with drones (VRPD), a variant of the classical vehicle routing problem, in which both drones and trucks serve customers with the objective of minimizing transportation cost. Drones can reduce operating expenses because their per-unit travel cost is lower than that of trucks. However, due to limited carrying capacity and flying range, a fleet of drones must rely on trucks to serve long-distance or heavy-demand customers. Drones can be dispatched from and picked up by the trucks at the depot or any of the customer locations. VRPD is inspired by the increasing interest in commercial drone delivery by companies such as Amazon, Federal Express, DHL, and Walmart. Utilization of drones helps reduce energy usage and carbon emissions, representing a step toward fostering a sustainable and environmentally friendly supply chain.

We develop a mixed-integer linear programming model for VRPD. A number of valid inequalities are proposed to strengthen the MILP formulation. The effectiveness of the valid inequalities is verified using a computational study. The computational results demonstrate a significant reduction in the CPU time required to solve these problems to optimality.

In modern supply chains, improved coordination among different parties is sought-after as it can enhance overall efficiency and lower costs through integration. Therefore, the production routing problem (PRP) has emerged to tackle this integrated problem, combining production, inventory, and routing decisions into a unified framework to improve coordination throughout the supply chain. However, neglecting demand uncertainty can lead to profit losses and poor customer satisfaction.

To mitigate this risk, companies often adopt service level targets to meet a certain portion of demand while considering operational constraints and minimizing costs. In our study, we address the stochastic PRP with adaptive routing (i.e., second-stage) and service level constraints. We examine four distinct service levels, each tailored to address specific metrics aligned with different real-world requirements, such as managing stockout occurrences and backorder or backlog ratios.

We propose a two-stage stochastic formulation where the setup decisions are made in the first-stage while the production, inventory, and routing decisions are made after demand realization, facilitating rapid responses to demand fluctuations. However, this approach brings computational challenges, which we overcome through an iterative matheuristic algorithm designed to deliver high-quality solutions within reasonable computation times.

This algorithm decomposes the problem into three subproblems. Initially, we tackle a two-level lot sizing problem, focusing on a single aggregate customer and direct shipments to rapidly generate setup decisions. Subsequently, we handle a restricted PRP with predefined setup plans and a vehicle with aggregate capacity, integrating service level constraints. Lastly, we focus on deriving feasible routing decisions, allowing for some flexibility in production and inventory choices while ensuring adherence to service level criteria. We then iterate over these phases to enhance the quality of solutions. By applying our algorithm to various benchmark scenarios, we compare results across different service levels and offer insights based on our findings.