1 Introduction

Same-day delivery is a growing service due to the increased customers’ expectations for faster delivery, the continued boom in e-commerce, and advances in e-commerce technologies. Target, for example, announced that the number of fulfilled same-day delivery orders increased from 30% to 75% in 2017 (Gartner Inc. 2022). This service is among the major sources of revenues (Collins 2022), providing companies with customer satisfaction and trust, and reduced logistics costs (Yahoo!Finance 2022). Same-day delivery service has considerably increased during the pandemic by 41% (Chandra 2002), and it is expected to grow from $5.14 billion in 2021 to $6.43 billion and $24.9 in 2022 and 2028, respectively, at a compound annual growth rate of 25.1% (Yahoo!Finance 2022; Newsmantraa 2022).

According to (Heitz-Spahn et al 2018; Nguyen et al 2019), the main drivers of customer preferences for delivery choice are user experience, price, and delivery time. To satisfy these drivers and achieve an efficient same-day delivery service for customers, companies need to improve their logistics operations, primarily the last-mile delivery operations. Last-mile delivery is known as the costliest part of order fulfillment operations, accounting for more than 40% of the total costs of transportation (Goodman 2005). Some of the underlying decisions for optimizing last-mile delivery include determining the number of vehicles, assignments of packages to trucks, and sequence of deliveries to customers by each vehicle.

Companies seek to improve last-mile delivery operations using different strategies and technologies. Some companies optimize conventional truck-based delivery models by leveraging data and capturing additional real-world constraints and operational conditions into their route optimization tools. For instance, UPS considers the impact of left turns on the route performance of their delivery trucks, to reduce accidents, wait times at intersections, fuel consumption, and pollutant emissions (Prisco 2017). Other companies integrate alternative delivery vehicles, such as bikes, tricycles, or electric bikes with trailers, in their last-mile operations to minimize the effects of traffic conditions, and therefore, reduce the wasted time of vehicles in traffic (Meza 2017; Barber 2018; McCurdy 2022). Given several advantages of drones over conventional vehicles, they recently attract attention to be used in last-mile logistics. Since drones are not limited to the road network infrastructure, they can fly in a straight line from the origin to the destination. In addition, considerable time-saving can be obtained as drones are not limited to road traffic. Despite these advantages, drones have a limited potential capacity concerning both payload weight and size and battery endurance time. Figure 1 shows three examples of drone deliveries.

Fig. 1
figure 1

Drone deliveries in practice: a drone delivery by Walmart (Guggina 2022), b drone delivery by Google Wing (Straight 2022), c drone delivery by Amazon (Reed 2022)

Full size image

To enhance the efficiency of the last-mile delivery operations by drones towards same-day delivery service, two categories of models exist. While the first category of models allows drones to deliver packages to customers, the second category allows only trucks to deliver packages to customers, and drones are primarily used to resupply trucks with packages (see Fig. 2). Each of these categories consists of two types of models. The first model in category one assumes drones are loaded with packages, and they are dispatched to customers to deliver the packages. In this model, drones can pick up one or multiple packages depending on the drone capacity, at the DC, and deliver them to customers (see Fig. 2a). Other vehicle types, such as trucks can perform deliveries to customers independently of drones. Using this model, Amazon, UPS, Zipline, and Google Wing conduct several pilot experiments in different countries (Ackerman and Strickland 2018; Webb 2019; UPS 2019; Vincent 2021). Due to the dense population concentration in urban areas, this model likely leads to over-crowded airspace around the apartment buildings and also the DC, increasing the chance of accident (Rojas Viloria et al 2020). To deal with this problem, a second model can be used which is a hybrid synchronized truck-and-drone delivery model. In this model, drones are launched from the truck to deliver packages to customers (see Fig. 2b). UPS and Mercedes Benz conduct some pilot studies using this model (Grossman 2017; Yvkoff 2017; Trop 2016). This model can effectively be used to deliver bulky packages using the truck, as well as light packages using drones. However, the truck has to return to the DC to pick up the next packages for same-day delivery. This reduces the efficiency of this model since the vehicles have to spend a considerable amount of time and energy on back-haul trips to the DC (Murray and Chu 2015; Moshref-Javadi et al 2020b).

The second category of models assumes that drones are used to support deliveries by trucks. This is achieved by using drones to resupply trucks with packages throughout the day while customers place their orders. The first model in this category -which is the focus of this paper- allows drones to resupply Transshipment Points (TPs) located in the delivery region. The trucks pick up packages at the TPs and deliver them to the customers, enabling same-day delivery without the need to return to the DC. The truck route is dynamically optimized and updated based on the packages resupplied at the TPs. A representation of the drone resupply model with one TP is shown in Fig. 2c. In the second model in this category, drones can deliver packages directly to the trucks rather than to the transshipment points. This model assumes that drones have to rendezvous with the truck at a location to transfer packages to the truck. While drones continuously resupply the truck with packages based on the arriving customer orders throughout the day, the trucks perform package deliveries to customers. This model is shown in Fig. 2d. The resupply models can effectively save the truck time by eliminating the truck trips from and to the depot during the day (Otto et al 2018; Moshref-Javadi and Winkenbach 2021). Mercedes-Benz and Siroop conduct several experiments using this model to provide on-demand delivery service to reduce delivery time (Mercedes-Benz 2021).

Fig. 2
figure 2

Categories of drone delivery models: a drones deliver directly to customers from the DC while a truck independently performs deliveries to customers, b Drones are dispatched from the truck to deliver packages to customers, c Drones deliver packages from the DC to TPs, and trucks pick them up to perform deliveries to customers, d Drones meet the truck to resupply it with packages, and trucks deliver the packages to customers

Full size image

In this paper, we study a drone resupply model for same-day deliveries, named the Drone Resupply Model with Transshipment Points (DRMTP). Drones are used to pick up packages from the DC to deliver them to a pre-specified number of transshipment points located in the region. Trucks pick up packages from the transshipment points and deliver them to customers. We develop a simulation model that simulates the same-day delivery operations throughout the day. The model is used to evaluate the effectiveness of the DRMTP model in comparison to the truck-only delivery model, followed by several sensitivity analyses on the underlying parameters of the model. The simulation model is used to evaluate the DRMTP in two case studies in Boston, MA and Pittsfield, MA in the USA. The contributions of this paper are as follows:

  • Develop a simulation model for the DRMTP

  • Quantify the effectiveness of the DRMTP in comparison to the truck-only model

  • Analyze the performance of the model based on the underlying parameters of the model through sensitivity analysis

The remainder of the paper is organized as follows. A review of the literature on same-day delivery logistics and drone logistics is presented in Sect. 2. Section 3 describes the DRMTP and Sect. 4 presents the details of the simulation model. The assessment of the DRMTP in two case studies along with the sensitivity analyses is presented in Sect. 5. Finally, Sect. 6 presents the conclusions and some directions for future research.

2 Literature review

In this section, we review the research literature on the same-day delivery logistics, and hybrid delivery models depicted in Fig. 2.

Same-day delivery recently received considerable attention in research. Klapp et al (2018) formulate the Dynamic Dispatch Waves Problem (DDWP) which determines the set of packages to load on a vehicle to dispatch for delivery to customers. The problem is formulated as an arc-based integer programming model and solved using heuristics. The results show more than a 9% increase in the number of served orders. A multi-vehicle dynamic pick-up and delivery problem with time constraints are proposed in (Voccia et al 2019). To achieve more informed delivery, authors optimize vehicle dispatch time to minimize delivery time and maximize the number of served orders per day. A heuristic method based on sample-scenario planning is also proposed to optimize vehicle departure time at the depot. Later, Ulmer and Streng (2019) improve this model by incorporating pick-up lockers in the delivery network. The results show significant improvement in the number of deliveries per day. Ulmer and Thomas (2018) develop a same-day delivery model using trucks and drones which depart from the depot to serve customers. Trucks serve customers that are located closer to the depot, while drones deliver packages to farther-located customers. An approximate dynamic programming method is developed to maximize the number of served customers each day.

Several articles consider the hybrid truck-and-drone models in which drones are launched from the truck to perform package deliveries. Murray and Chu (2015) propose the Flying Sidekick Traveling Salesman Problem (FSTSP), where a drone is launched from a truck to deliver packages, considering a travel time constraint. The goal is to find the sequence of deliveries by truck and drone to minimize the return time of the vehicles to the depot. Dell’Amico et al (2019) develop a new mathematical model of this problem, strengthened by valid inequalities. This model is extended by Yoon (2018) and Kitjacharoenchai et al (2019) to the problem with multiple trucks and drones. An extension to the FSTSP is developed by (Jeong et al 2019) in which no-fly zones are considered for the drone travel space.

Due to the computational complexity of the truck-and-drone routing problems, several articles focus on developing efficient algorithms to solve these problems. Ponza (2016) propose a Simulated Annealing algorithm to solve the FSTPS on problems with up to 200 customers. Bouman et al (2018) and Agatz et al (2018) develop dynamic programming methods to solve the FSTSP and its variants. Yurek and Ozmutlu (2018) propose a two-step iterative optimization method that determines the assignments of customers to trucks and drones. Thereafter, it optimizes the launches and retrievals of drones at the truck. Dell’Amico et al (2021) develop four formulations of the FSTSP with multiple drones, and a branch and bound algorithm to solve the problem. The method can be used to solve problems with up to 10 customers. Moshref-Javadi et al (2020b) present a new variant of the problem, named the Traveling Repairman Problem with Drones, aiming at minimization of customer waiting times. It is assumed that the truck has to stop at the launch location for all the drones to return to the truck before the truck can continue to the next customer location on its route. Later, Moshref-Javadi et al (2020a, 2021) extend the model such that the truck is allowed to retrieve drones at later locations on its route. A metaheuristic algorithm based on the Adaptive Large Neighborhood Search (ALNS) is developed to solve three variants of the problem. Similarly, an Adaptive Large Neighborhood Search algorithm is proposed in Sacramento et al (2019) to solve the extension to the FSTSP, so-called the Vehicle Routing Problem with Drones (VRPD), assuming multiple trucks and one drone per truck. Considering a problem with multiple trucks and drones, and time windows at customer locations, Di Puglia Pugliese et al (2021) propose a two-phased heuristic with a multi-start framework to solve problem instances with up to 15 customers.

There exist a few articles that consider the hybrid truck-and-drone models in which drones are used to resupply trucks with packages for delivery by trucks. Dayarian et al (2020) propose a model in which drones can resupply trucks at any customer location on the truck’s route. This model aims to maximize the number of packages that are delivered within a given time window. Due to the complexity of the model, the model is simplified by considering one DC, one truck, and one drone. Pina-Pardo et al (2021) formulate a drone resupply model, so-called the Traveling Salesman Problem with release dates. Because the order release dates are known in advance, the drones can resupply the trucks with packages to fulfill demands before their due dates. Using a decomposition method for solving the underlying scheduling problem, the model can obtain up to 20% savings in delivery time compared to the truck-only model.

The literature review indicates that there exist a limited number of studies on modeling and evaluations of the resupply truck-and-drone delivery models. All the existing studies consider that drones deliver supplies to trucks rather than to TPs. The models are also simplified concerning the number of trucks and drones. To the best of our knowledge, no extant research models the resupply truck-and-drone delivery model with transshipment points.

In this article, we propose the Drone Resupply Model with Transshipment Points. The model is a two-stage delivery model. The first stage utilizes drones to resupply the transshipment points with packages. In the second stage, the trucks pick up packages from the TPs and deliver them to customers. We develop a simulation model and evaluate the effectiveness of the DRMTP in two case studies in Massachusetts, USA. The performance of this model is evaluated and compared with the conventional truck-only model under several scenarios.

3 The drone resupply model with transshipment points (DRMTP)

The DRMTP is a hybrid truck-and-drone delivery model aiming to enhance delivery operations by incorporating a multi-modal fleet of vehicles. DRMTP is a two-echelon location-routing problem (2E-LRP), where vehicles deliver packages in the first echelon from the DC to TPs, and in the second echelon from the TPs to customers (Cuda et al 2015; Moshref-Javadi and Lee 2016; Darvish et al 2019). The routes of the vehicles in the first and second echelons, as well as the locations of the TPs, are determined in this problem. In this model, customer orders are received by the company throughout the day and prepared for shipment to customers at the DC. Once a drone picks up the packages at the company’s DC, it travels to a specified TP to deliver them to the TP. TPs can be (temporary) manual trailers or automated facilities located in the region (Fig. 3). Drones do not have to stay at the TP for a truck to arrive at the TP after they unload the packages at the TP. Rather, they return to the DC to load and deliver the next packages to the TPs. Each TP is used to serve a specified set of customers. The number and locations of TPs and allocation of customers to TPs are determined using the optimization model described in Sect. 4.2.4. In the second echelon, trucks pick up packages from the TPs and deliver them to customers. Once a truck picks up the packages at a TP, the sequence of deliveries is determined to minimize the travel time of the truck. Trucks are not dedicated to specific TPs, that is, trucks can pick up packages from any TPs.

The model uses one DC with multiple trucks, multiple drones, and multiple TPs. Considering that the current drone technology allows drones to carry multiple packages with a total payload of more than 200 kg (Brown 2021; Dahl 2021), the proposed delivery model can effectively eliminate the trips of trucks from and to the DC. It should be noted that the defined model can be used in several industries, such as food and beverage, healthcare, retailers, postal services, and grocery stores.

Fig. 3
figure 3

Illustration of Transshipment Points: a a fully loading and offloading station by DHL in China (DHL 2019), b a UPS transshipment point in downtown Hamburg, Germany (Dobos 2018), c a modern pickup and delivery station for drones (Pixta 2021), d a Caltex gas station served as a drone delivery station in South Korea (Byung-Wook 2020)

Full size image

4 The simulation model of the DRMTP

In this section, we describe the details of the simulation model based on the Strengthening the Reporting of Empirical Simulation Studies (STRESS) guideline proposed by (Monks et al 2019). The summary of the simulation model is given in Table 1.

Table 1 Details of the simulation study based on the STRESS guideline Monks et al. (2019)
Full size table

4.1 Objectives

This model aims is to evaluate the effectiveness of the DRMTP towards enabling same-day delivery by a two-echelon truck-and-drone model with transshipment points. The effectiveness of the model is evaluated in two sets of analyses. First, we compare this model in its baseline scenario with a truck-only model which is a single-echelon delivery model. This model uses only trucks to pick up packages from the DC and deliver them to customer locations without using drones and TPs. The model is used to set a benchmark for comparison with the DRMTP. The second set of analysis aims to evaluate the performance of the DRMTP in several scenarios defined based on the values of the underlying parameters of the problem, such as order inter-arrival time, and parameters of the model, including the number of TPs, number of trucks, and number of drones, drone capacity, and drone speed. The goal of this analysis is to understand how the model behaves in response to these parameters and to understand the sensitivity of the model to the underlying parameters of the model.

Two key performance metrics are used to evaluate the effectiveness of the DRMTP in comparison to the truck-only model, and also assess the effects of the parameters of the model on the performance of the model. The first metric is the truck travel distance which can be used for the calculations of the transportation cost. We also use this measure to compare the performance of the DRMTP with the truck-only model. Once the truck travel distance is calculated at the end of the simulation run, we divide it by the number of delivery days to report the average truck travel distance per day (km/day). The second metric is the average package delivery time to customers throughout the day. Package delivery time measures the time between when the order is placed by a customer, i.e. package entity is created until it is delivered to the customer location. Once the package delivery time is calculated for all the packages, the sum of them is divided by the number of packages to report the average package delivery time.

Fig. 4
figure 4

Process flow diagram, showing the flow of activities for a package, a drone, and a truck

Full size image

4.2 Logic

In this section, we explain the details of the components of the model. For each component, all the related algorithms and activities are described. Figure 4 illustrates the process flow of a package along with a truck and a drone.

4.2.1 Packages

Customer orders are generated at the DC throughout the day. Each customer order may contain one or multiple packages. These packages are defined as the entities of the model which are created when a customer order is placed and terminated once they are delivered to their customers. The creation time of each order is based on the Inter-Arrival Time (IAT) based on the given data by the company. IAT represents the average time between the arrival of orders to the DC information system. Once a package is created, it enters a queue at the DC, waiting for a drone to pick up the package. Packages are loaded on drones based on the assorted First-In-First-Out (FIFO) rule. That is, the drone loads the first package in the queue, and it continues to load more packages from the queue if there is any package whose destination is the same TP as the TP of the first loaded package until the drone does not have any more free space. Each package has one intermediate destination and one final destination. The intermediate destination is the transshipment point, while the final destination is the customer location. The intermediate destination, i.e. TP, is determined based on the location-allocation model described in Sect. 4.2.4. Table 2 shows the parameters of the problem based on deterministic/stochastic and static/dynamic classification.

Table 2 Parameters of the problem based on Stochastic/Deterministic and Static/Dynamic classification
Full size table

4.2.2 Customer locations

Customer locations represent the final destinations of packages. When a customer order is generated during the simulation run, a number of packages and a delivery address are assigned to it. For generating orders during simulation runs, we consider the frequency of orders and the number of requested packages at each delivery address in the historical data. That is, the higher the frequency of deliveries to a customer location, the more the created packages are assigned to it during the simulation run. The delivery address assigned to a package is selected from a set of customer locations in the given historical data by the company.

4.2.3 Distribution center

All packages originate from a single distribution center in the region. The location of the DC in our experiments is pre-specified based on the given data by the company.

4.2.4 Transshipment points

The Transshipment Points are resupplied by drones with packages throughout the day, and trucks pick up packages from the TPs to deliver to customer locations. Once a package is transported by a drone to a TP and unloaded, it enters a queue, waiting for a truck to pick it up. Each TP can serve a pre-specified set of customer locations. We assume that the number of TPs in the region is given while the locations of the TPs and allocations of customers to TPs are determined using an Integer Programming model. This model is a location-allocation problem that seeks to minimize both the inbound and outbound travel time to and from the TPs. This model is flexible to be used to minimize the total cost of delivery if the delivery cost per unit of time/distance is provided. Table 3 shows the notations used in the mathematical model. The Transshipment Points Location-Allocation problem is formulated as follows:

Table 3 Notation used in the mathematical model
Full size table
$$\begin{aligned} \text {Minimize} \quad \sum _{s\in {S}}\sum _{i\in {V}}x_{is}t_{si} + \alpha \sum _{s\in {S}}y_{s}\hat{t}_{0s}{} & {} \end{aligned}$$
(1)

subject to

$$\begin{aligned} \sum _{j\in {S}}x_{is}&=1,&\forall i \in V, \end{aligned}$$
(2)
$$\begin{aligned} \sum _{s\in {S}}y_{s}&=N_T,&\end{aligned}$$
(3)
$$\begin{aligned} \sum _{\begin{array}{c} i\in {V} \end{array}}x_{is}&\le Cy_s,&\forall s \in S, \end{aligned}$$
(4)
$$\begin{aligned} {x}_{is},{y}_{s}&\in \{0,1\}&\forall i \in V, \forall s \in S. \end{aligned}$$
(5)

The Objective function (1) minimizes the sum of the total time needed for the drones to travel from the DC to TPs and the total travel time for the trucks to drive from TPs to customer locations. The drone travel time is adjusted using the coefficient factor \(\alpha\) since we assume that the speed of the drones is higher than the speed of the trucks. Note that it is necessary to consider the distances between the DC and TPs and between the TPs and customers in the objective function to ensure minimizing the total travel distance. Equation (2) guarantees that each customer must be allocated to exactly one TP. The number of TPs in the network must be equal to \(N_T\), which is represented by Equation (3). Equation (4) ensures that TP location s must be opened if any customer is allocated to it. Finally, Equation (5) determines the types of variables.

4.2.5 Drones

Drones are dedicated to transportation of packages from the DC to TPs. After the drone unloads the packages at the TP, it returns to the DC without having to stay at the TP. This level of synchronization enhances the flexibility and viability of the model as the drone does not have to stay at the TP for a truck to arrive for pickup. They have a fixed carrying capacity specified by the number of packages that can be carried. The loading or unloading of each package requires 20 s per package. Drones can fly at a pre-specified speed and they do not follow the road network, that is, they can fly in a straight line. Once a drone arrives at the DC, it is loaded with as many packages available at the DC up to its carrying capacity. The drone loads packages with the same TP destination as it can visit only one TP in each trip from the DC.

4.2.6 Trucks

Trucks are dedicated to the transportation of packages from the TPs to customer locations. Trucks are not capacitated as the space in a truck is considerably higher than the carrying space of the drones. Once a truck arrives at a TP, it loads all the packages at the TP based on the FIFO rule. The package load time for a truck is proportional to the number of packages that are loaded, requiring 20 s per package. We also assume a park time of 90 s at each TP and a service time of 90 s at each customer location. Once the truck picks up the packages at a TP, the sequence of deliveries by the truck is determined to minimize the total travel time of the truck using the greedy nearest neighbor method. All travel distances are real since trucks follow the real-road road network obtained from the Google Maps API. Once a truck performs all the deliveries, it waits at its latest location until a TP requests a truck to pick up packages. Therefore, trucks are not dedicated to the TPs, that is, they can perform deliveries for different TPs. If multiple trucks are available to pick up packages at a TP, the closer truck is chosen. Since the truck-only model does not use any drones, trucks pick up packages from the DC based on the FIFO rule and deliver them to customer locations using the greedy nearest neighbor method.

4.3 Data

The simulation model is examined in two case studies in the cities of Boston, MA and Pittsfield, MA in the USA to evaluate the DRMTP. The city of Boston represents an urban region, while Pittsfield represents a suburban area. All the packages are distributed from one distribution center located in the region based on the given data. The distance metric is real and all the distances are collected using the Google Maps API.

Since Boston and Pittsfield have different package delivery demand volumes, the package arrival rates are different. In Boston, the region has 5,020 packages delivered in 8 h, with 10% requested for same-day deliveries. Hence, Boston has an average package arrival rate of 1.0 packages per minute (i.e., IAT is 1.0 min). In Pittsfield, an average of 260 packages are delivered in 8 h, with 10% same-day deliveries. Pittsfield, therefore, has an average package arrival rate of 0.055 packages per minute (i.e., IAT is 18 min). The IATs are used to generate orders based on the Exponential distribution. We also conduct a sensitivity analysis for both case studies with an inter-arrival time of 0.5 and 9 min for Boston and Pittsfield, respectively, to assess the potential impacts of growing demand. Although we use the model for these two case studies, the model is flexible and it can be used to simulate delivery operations in any city.

4.3.1 Parameter values for scenarios

The parameter values for the baseline scenarios for both case studies are given in Table 4. The parameter values are selected such that a 3-hour average delivery time is obtained in the steady state after the warm-up period. Note that since the number of resources used in the two case studies is different, a comparison of the results of case studies together may not be fair.

Table 4 Baseline parameter values for the two case studies
Full size table

To evaluate the performance of the DRMTP, we simulate it in several scenarios, defined based on different parameter values. Note that in each experiment, we only change the value of one parameter, while the remaining parameters’ values remain in their baseline values.


Number of transshipment points To analyze the effects of TPs on the performance of the DRMTP, we consider the number of TPs \(\in \{1,2,3,4,5\}\). A higher number of TPs reduces the distances between the TPs and customer locations, and therefore, shorter trips for the trucks to pick up packages from the TPs. We expect that the truck travel distance and the total delivery time reduce as the number of TPs increases. Note that we only specify the number of TPs, while the locations of TPs and their allocated sets of customer locations are determined using the optimization model presented in Sect. 4.2. The distributions of transshipment points and their allocated customer locations are shown in Appendix A for all the scenarios with 1 to 5 TPs.


Number of trucks We consider five scenarios with a varying number of trucks. The number of trucks depends on the case study and IAT. We expect that as the number of trucks increases, the package delivery time is reduced.


Number of drones We consider scenarios with the number of drones in ranges 3-7 and 1-5 for the Boston and Pittsfield case studies, respectively. We expect that increasing the number of drones can enable faster replenishment of TPs, and therefore, can lead to a reduced total delivery time.


Drone speed We consider scenarios with drone speed \(\in \{40,45,50,55,60\}\) km/h. We expect that faster drones can improve the total delivery time because TPs can be resupplied faster. We assume that the truck speed is fixed due to city traffic regulations.


Drone capacity We simulate seven scenarios with drone capacity 4 to 9 packages. We expect that an increase in drone capacity leads to a reduction in the travel distance and the total delivery time.

4.4 Experimentation

All simulation experiments are run with a 70-hour warm-up period, and an 80-hour simulation length representing 10 days of service. The trucks and drones do not return to the depot after each 8-hour shift, that is, the 80-hour simulation run is nonstop. Each scenario is run for 100 replications to calculate the average values of the performance metrics. The Simulation starts with no package in the system.

4.5 Implementation

All modeling and experiments are performed using Simio 10.181 on a 64-bit Windows 10 operating system with a dual processor, 2.8 GHz CPU, and 16 GB RAM.

5 Experimental results

For each case study, we present the results of the simulation for the truck-only, DRMTP baseline scenario, and sensitivity analyses.

5.1 Case study 1: urban area in Boston

5.1.1 Truck-only model results for the case study in Boston

This model uses only trucks to deliver packages to customer locations from the DC. For the scenario with an IAT of 1 min and 0.5 min, we use 7 trucks and 14 trucks, respectively. Table 5 summarizes the results. The average total travel distance for the case with IAT = 0.5 minutes is almost twice its value in the case with IAT = 1 min. However, the average delivery time increases by 0.36 h. This indicates that even though the number of trucks is doubled, the trucks deliver more packages on each route. When IAT is 1 min, trucks deliver the total number of packages in approximately 45.1 routes per truck, while the number of routes is 57.2 per truck when IAT is 0.5 min.

Table 5 Results of the truck-only scenario for case study 1, Boston
Full size table

5.1.2 DRMTP results for the case study in Boston

Table 6 summarizes the results of the DRMTP for the baseline scenario. As the number of TPs increase, the package delivery time and the total truck travel distance decrease. It is expected because as the number of TPs increases, the average distance between the TP and customer locations reduces, resulting in a reduction in the total travel distance. We believe the slight increase in the total truck distance when the number of TPs increases from 1 to 2 is due to the TP location-allocation optimization model and also using a real-world non-symmetric road network for distance calculations between the customers and TP locations.

The results show that the DRMTP can effectively reduce the total delivery time and average total travel distance. A model with one TP and five drones reduces the delivery time by nearly 28% compared to the truck-only scenario. Similarly, the models with four TPs and five drones result in delivery time decreases of 66% and 73% for the scenarios with IATs of 1 min and 0.5 min, respectively. The models also reduce the truck travel distance. The models with four TPs and five drones can reduce the total truck travel distance by up to 11% depending on the inter-arrival time. Given the benefits of a higher number of TPs, the DRMTP with four TPs is used for the baseline scenario. We conduct several analyses, varying the number of trucks, number of drones, drone capacity, and drone speed.

Table 6 Results of the DRMTP for the case study in Boston
Full size table

5.1.3 Sensitivity analysis of the case study in Boston

We conduct a detailed analysis based on the underlying parameters of the model defined in Sect. 4.3.1. The results are shown in Figs. 7, 8, 9, 10 and 11. The model’s performance highly varies depending on the number of trucks and TPs. Increasing the number of TPs reduces the distance between the customer locations and TPs, resulting in a considerable reduction in delivery time (see Fig. 7). Among all the drone parameters, the average delivery time is impacted by the drone delivery speed (see Fig. 10). However, the impacts of the number of drones and the drone capacity are not significant, unless the system cannot reach a steady state due to a very low number of drones. Reducing the number of drones has a negligible impact on the results unless the model has three drones in which drones cannot transfer all the packages from the DC to the TPs. In this case, the simulation model does not reach the steady state, the packages are accumulated at the DC, and the delivery time increases tremendously when IAT is 0.5 min. The analysis of the drone speed shows that it has a significant impact on the average delivery time. For example, for the IAT of 0.5 min, the average delivery time reduces by 10% when drone speed increases from 40 km/hr to 60 km/hr. This reduction primarily resulted from faster deliveries from the DC to the TPs by drones. Due to the higher volume of packages that arrive at the TPs, the total packages that can be delivered to customers during the simulation is higher as well.

5.2 Case study 2: suburban area in Pittsfield, MA

5.2.1 Truck-only model results for the case study in Pittsfield

The results of the truck-only model are given in Table 7. With the reduction of IAT from 18 to 9 min, the average delivery time considerably increases. Nevertheless, a single truck is sufficient to reach the steady state, and deliver all the packages with a delivery time of maximum 3 h.

Table 7 Results of the truck-only scenario for case study 2, Pittsfield
Full size table

5.2.2 Truck-drone transshipment model results for the case study in Pittsfield

Table 8 Results of the DRMTP for the case study in pittsfield
Full size table

We analyze the baseline scenario using one truck and one drone (see Table 4). The model is simulated for two IATs = 18 and 9 min, and different numbers of TPs to compute the values of the performance metrics. The results are given in Table 8. According to the results for the scenarios with IAT = 18 min, the DRMTP with one drone and one TP reduces the average delivery time by 35% and the total truck travel distance by 23% compared to the Truck-only model. Similarly, for the scenario with IAT = 9 min, by using one TP in the DRMTP, the average delivery time and the truck travel distance reduce by 2% and 11%, respectively. In the scenario with one TP and one truck, the truck picks up the packages from the TP and delivers them to customer locations. However, as the number of TPs increases beyond one TP, the average delivery time increases. This happens because by increasing the number of transshipment points, the truck spends more time traveling between the TPs. It is expected because each TP is dedicated to serving a defined set of customers determined based on the optimization model in Sect. 4.2. A similar pattern can be observed for the performance metric travel distance. The results indicate that there is an optimal number of TPs to choose for this case study. That is, increasing the number of TPs does not necessarily improve the values of the performance metrics.

5.2.3 Sensitivity analysis of the case study in Pittsfield

The detailed results of the sensitivity analyses are shown in Figs. 12, 13, 14, 15 and 16. The average delivery time of the DRMTP in this case study is very sensitive to the IAT, resulting in approximately double the delivery time for the IAT = 9 min compared to the case with IAT = 18 min. The vehicle utilization is higher for IAT = 9 min, requiring more packages to be delivered on each truck route, and therefore, resulting in a longer average delivery time. This can be observed in almost all the sensitivity analysis results, except for the scenarios in which the number of trucks is increased. In those scenarios with a high truck utilization, especially IAT = 9 min, an additional truck substantially reduces the delivery time from 2.1 h to 0.7 h (see Fig. 13). However, increasing the number of trucks beyond 2 trucks does not reduce the delivery time further as the trucks are not highly utilized. For instance, the delivery time only increases from 0.5 h to 0.6 h for an IAT = 9 min compared to an IAT of 18 min. Increasing the number of trucks, however, increases the truck average travel distance. This is mainly due to the road network in the suburban area of Pittsfield in which most of the customer locations are located close to one of the three main roads. Even though the truck delivers more packages on each route, the road network infrastructure and customer locations result in approximately the same truck travel distance. Among the drone parameters, the drone speed affects the average delivery time (see Fig. 15). This is primarily because faster drones can arrive at the TPs earlier, supplying TPs with packages in a shorter period. On the other hand, the number of drones and the drone capacity do not affect the delivery time and travel distance. Because of the low number of customer orders in this region, drones are not fully loaded when they are dispatched to the TPs. Therefore, increasing the capacity and number of drones do not affect the performance metrics (see Figs. 14 and  16).

6 Conclusions and future research

In this paper, we proposed a hybrid truck-and-drone delivery model, named the Drone Resupply Model with Transshipment Points (DRMTP). The model uses drones to resupply transshipment points with packages, which are picked up by the trucks to be delivered to customers. An optimization model was formulated to determine the locations of the TPs along with the allocations of customer locations to TPs. A simulation model was developed to simulate the delivery operations and analyze the sensitivity of the model and performance metrics to the underlying parameters of the model. The model was evaluated in two case studies in Massachusetts, USA and compared with the truck-only model which does not use drones. The results show the effectiveness of the model in both case studies. The DRMTP with one TP, for example, could effectively reduce the total delivery time by 28% and 35% in the case studies in Boston and Pittsfield compared to the truck-only model, respectively. The results, however, are highly dependent on the parameter values of the problem, such as the volume of orders per day in the regions, and the parameters of the model, such as the number of TPs, number of trucks and drones, and drone speed and capacity. For instance, the optimal number of TPs for the DRMTP varies depending on the volume of customer orders in the region. The number of trucks can considerably reduce the average delivery time although the marginal effect reduces as the number of trucks increases. If the values of parameters the number of drones, drone capacity, and drone speed are sufficiently large, the effects of these parameters on the performance metrics are not significant.

Several opportunities for further extension of the DRMTP exist. Although we used the First-In-First-Out (FIFO) policy, the model could be further refined by optimizing the order of packages that are selected for loading on drones to deliver to the TPs. One could also consider the current truck routes to determine the order of packages that are loaded on the drones. Another direction for future research could be the comparison of this model with the drone resupply model which allows rendezvous of the truck and drones at customer locations instead of the TPs. The model would require the trips of the drones and trucks to be coordinated to prevent potential waiting of vehicles for each other. Another research can locate the TPs considering a limited flight range for drones. This assumption is expected to reduce the efficiency of the model compared to the results in this study. Finally, one can aim at other performance metrics, such as minimizing the total costs of transportation and the effect of payload on drone battery consumption.