with delivery options, Attended home delivery, Shared delivery locations.

The explosive growth of e-commerce and the rapidly changing customer expectations have driven a significant transformation in last-mile logistics in recent years. This shift in consumer behavior was further accelerated by the COVID-19 pandemic, which led to a strong preference for flexible, contactless delivery methods [1], [2]. Today’s customers demand a range of delivery options, including home delivery, parcel lockers, and shared collection points, alongside prompt and dependable service. Consequently, logistics providers now face additional operational challenges, underscoring the need for more flexible, customer-focused distribution strategies.

The Vehicle Routing Problem (VRP) is a well-known and extensively studied logistics optimization problem that seeks to determine the most effective routes for a fleet of vehicles to serve a set of customers under various constraints [3]. Recent extensions of the VRP have incorporated advanced features such as time windows, heterogeneous fleets, and flexible delivery options. Among these, the concept of roaming delivery, where a customer can be served at multiple potential locations, has gained increasing attention [4]. However, much of the existing research focuses on dynamic routing to private addresses. It often overlooks Shared Delivery Locations (SDLs), such as parcel lockers or common pickup points, when compiling the customer’s candidate delivery list. Moreover, these shared facilities are frequently modeled with unlimited capacity, disregarding practical limitations such as congestion, service fairness, and operational constraints. As a result, current approaches may fail to capture both the logistical benefits and the real-world challenges associated with shared delivery infrastructure in last-mile distribution systems.

This study explores a single-depot VRP that permits SDLs and accounts for delivery options at a more detailed level, thereby addressing this gap. As illustrated in Fig. 1, the delivery network includes both private home delivery points (circles) and SDLs (pentagons). Each dashed oval region (L1, L2, L3) represents an SDL cluster where customers can choose to receive their packages instead of home delivery. A single central depot (square) dispatches vehicles that visit a mix of home addresses and shared centers depending on the routing decisions. Additionally, each color in the figure corresponds to a unique customer, meaning that a customer may have multiple delivery options (private or shared), all represented with the same color across different nodes. This visual layout illustrates how incorporating roaming and centralized pickup options into routing plans increases delivery flexibility while maintaining service efficiency.

In contrast to conventional models, this study assesses two frameworks: one with fixed delivery locations and another with centralized pickup locations and roaming delivery options. The comparative analysis demonstrates significant efficiency gains enabled by flexible delivery structures, including reduced fleet size and operating costs. This research advances the theoretical development and practical implementation of sophisticated VRP models, providing insights for optimizing urban logistics in increasingly dynamic and

The remainder of this paper is organized as follows. Section 2 reviews the relevant literature on VRPs. Section 3 presents the mathematical formulation and methodology. Section 4 discusses the numerical example and computational results, and finally, Section 5 concludes the study with managerial implications and future research directions.

To evaluate the operational impact of delivery option flexibility in single-depot VRPs, two scenarios were formulated and solved using a mixed-integer programming model implemented in Python with PuLP. Both scenarios utilized a heterogeneous fleet of five vehicles with capacities fixed at 100, 120, 90, 110, and 95 units for vehicles K0 through K4, respectively. The depot was located at (50, 50), and the vehicle speed was assumed to be 60 km/h. A service time of 10 minutes was imposed at each customer location to account for loading and unloading. Additionally, cost parameters were set at 30 currency units per kilometer traveled, 100 units per hour of waiting time, and a penalty of 50 units for using shared delivery options to capture potential coordination costs. The maximum allowed route duration was set at 1000 minutes.

In the first scenario, each customer was allowed to have multiple delivery options, enabling the model to select the optimal delivery node to minimize operational costs. The results showed an objective value of 5444.69 units, with only 3 vehicles serving all 6 customers. Additionally, one SDL was used, and no waiting costs were incurred. This outcome is illustrated in Fig. 3, which depicts the optimized routes and selected delivery points under flexible options. The results indicate that offering multiple delivery options enables effective route consolidation, thereby reducing both total travel costs and the number of vehicles needed. In contrast, the second scenario considered a system where each customer was assigned only a single delivery location. Although the model allowed SDCs, in this dataset, only the first customer’s delivery location was designated as a shared center, while the others had no alternative. The results showed a substantially higher objective value of 9183.21 units, with all five vehicles required to serve the same six customers. No shared deliveries were used beyond the predefined shared node, and no waiting costs were incurred. This outcome is shown in Fig. 4, where restricted delivery options led to longer total routes and higher fleet utilization. The significant increase in operational cost demonstrates the inefficiency that arises when customers are restricted to a single delivery option, even if shared centers are permitted in the system.

Overall, these findings demonstrate the critical impact of delivery flexibility on vehicle routing performance. For instance, allowing multiple delivery options per customer reduced total operational costs by over 40% and led to a marked decrease in fleet utilization compared to the single-option scenario. These results highlight the managerial importance of integrating customer flexibility into last-mile delivery planning to achieve substantial efficiency gains in urban logistics networks.

The VRP with private Roaming and SDLs (VRP-RSD) was examined in this study, accounting for SDCs, customer delivery choices, and heterogeneous fleet capacities. The model incorporated realistic features such as alternative customer delivery locations, time-window restrictions, service times, and vehicle and shared-facility capacity limitations by developing a mixed-integer linear programming formulation. By optimizing vehicle routing, cutting operational costs, and minimizing fleet requirements, the results showed that adding flexibility through shared delivery points and multiple delivery options can improve overall system efficiency. To achieve sustainable, cost-effective last-mile operations, the proposed model emphasizes the importance of combining customer choice with cooperative delivery infrastructure. It offers useful insights for urban logistics planning. However, advanced heuristic and metaheuristic solution approaches are advised for practical implementation due to the computational complexity of solving such detailed models, particularly in large-scale urban contexts. Methods such as large neighborhood search, in conjunction with hybrid evolutionary algorithms, can effectively address scalability issues without sacrificing solution quality.

Overall, by demonstrating how delivery flexibility and shared resource utilization can significantly impact performance, this study advances the field of last-mile logistics optimization. It illustrates how operational guidelines that allow multiple delivery options for each customer can significantly reduce fleet size and routing costs while simultaneously enhancing customer satisfaction and delivery reliability. To increase the work's practical applicability in rapidly changing urban environments, future research should account for stochastic travel times, dynamic demand, and integration with real-time vehicle routing systems. Furthermore, integrating environmental goals, such as reducing carbon emissions, and considering the use of electric vehicles may offer a more comprehensive strategy for sustainable urban distribution planning.

The authors would like to express their sincere appreciation to all individuals who contributed to this research through their valuable comments, suggestions, and support.

This research received no external funding.

The data used in this study were generated and processed by the authors for numerical analyses and model validation. The datasets supporting the findings of this study are available from the corresponding author upon reasonable request.