key: cord-0629295-tyzmqdqa authors: Yang, Dingtong; Hyland, Michael F.; Jayakrishnan, R. title: Tackling the Crowdsourced Delivery Problem at Scale through a Set-Partitioning Formulation and Novel Decomposition Heuristic date: 2022-03-10 journal: nan DOI: nan sha: a7ac0d006d20e88c9fd3b41953c295f0378b4db2 doc_id: 629295 cord_uid: tyzmqdqa This paper presents a set-partitioning formulation and a novel decomposition heuristic (D-H) solution algorithm to solve large-scale instances of the urban crowdsourced shared-trip package delivery problem. The D-H begins by dividing the packages between shared personal vehicles (SPVs) and dedicated vehicles (DVs). For package-assignment to SPVs, this paper enumerates the set of routes each SPV can traverse and constructs a package-SPV route assignment problem. For package-assignment to DVs and routing, the paper first obtains DV routes by solving a conventional vehicle routing problem and then seeks potential solution improvements by switching packages from SPVs to DVs. The switching process is cost driven. The D-H significantly outperforms a commercial solver in terms of computational efficiency, while obtaining near-optimal solutions for small problem instances. This paper presents a city-scale case study to analyze the important service design factors that impact the efficiency of crowdsourced shared-trip delivery. The paper further analyzes the impact of three important service design factors on system performance, namely (i) the number of participating SPVs, (ii) the maximum detour willingness of SPVs, and (iii) the depot locations. The results and findings provide meaningful insights for industry practice, while the algorithms illustrate promise for large real-world systems. In recent years before the COVID-19 pandemic, convenient online shopping and other improvements in ecommerce resulted in high demand for package delivery services. E-commerce in the U.S grew at a rate of 16% from 2016 to 2017; compounded annually, this rate would double package deliveries every 5 years (Ivanov, 2018) . In 2020, the COVID-19 pandemic along with work-from-home requirements and preferences intensified the need for online shopping and front-door package delivery (Grashuis et al., 2020; Koch et al., 2020) . The package delivery trips stemming from e-commerce are likely to impact congestion and bring emissions of environmental consequence in transportation systems. However, designing new logistic systems in urban areas could potentially decrease the negative impacts of increased package delivery demand, while simultaneously saving logistics companies significant costs. As such, recent logistics research examines approaches that integrate package delivery systems within several transportation modes (Chen et al., 2017) , including public transit (Fatnassi et al., 2015) , taxis (B. Li et al., 2014) and private/personal vehicles (Archetti et al., 2016) . The private vehicle integration is often known as crowdsourced delivery, crowdshipping, or crowd logistics. Unlike traditional freight transportation that only involves dedicated trucks, crowdsourced delivery involves paying individuals from the general public to complete package delivery tasks, often with their own vehicles. Private firms and academic researchers are testing and evaluating various crowd-sourced delivery methods, respectively. A few programs have been launched by retailing, transportation, and logistics companies, such as Walmart Spark Driver, Amazon Flex, and Uber Eats. Alnaggar et al. (2021) provide a detailed summary of current crowdsource delivery programs. Researchers have also proposed various ways to formulate and solve problems related to crowd-shipping. Section 2 provides a detailed review of the current literature. Benefits of crowdsourced delivery include improved economic, social, and environmental sustainability (Rai et al., 2017) . From an economic perspective, crowdsourced delivery can reduce the fleet size and dedicated human resources involved in delivery logistics. As a result, crowdsourcing can reduce capital and labor cost for logistics companies (Qi et al., 2018) . From a social perspective, crowdsourced delivery could potentially lower traffic generated from freight delivery, thereby reducing congestion, and improving transportation efficiency (Archetti et al., 2016; Arslan et al., 2019) . Crowdsourced delivery also enhances the opportunity for same-day delivery, for both packages and food (Ulmer et al., 2021; Voccia et al., 2019) , thereby improving quality of life in urban areas. From an environmental perspective, reducing freight delivery traffic can decrease fuel consumption and harmful emissions (S. Lee et al., 2016) . This paper focuses on a specific type of crowdsourced delivery service where packages "share rides" with private vehicle drivers who have a trip of their own to complete. The underlying operational problem associated with this shared-ride crowdsourced delivery services, is called the "vehicle routing problem with occasional drivers" in Archetti et al. (2016) , 'crowdsourced delivery with ad hoc drivers' in Arslan et al. (2019) , and 'crowdsourced delivery with in-store customers' in Dayarian & Savelsbergh (2020) . The core idea behind this type of crowdsourced delivery is to utilize empty space inside personal vehicles, taking advantage of the enormous number of personal vehicle trips made each day with excess vehicle capacity. Therefore, we name this specific type of problem as the crowdsourced shared-trip delivery problem. Figure 1 displays a diagram of the crowdsourced shared-trip delivery system. A central manager is responsible for delivering packages from the distribution center to locations throughout the service area. The packages are small-to medium-sized parcels that can easily fit inside the trunk or the seats of normal sedans. The packages are required to be delivered within certain time windows, which are relatively loose. Crowdsourced space-sharing vehicles (named "shared personal vehicles" and abbreviated as SPVs), which are private/personal vehicles, travel to the distribution center, pick up one or more packages, and drop off the packages at locations in the service area. However, the distribution center location and the package drop-off location(s) should not require any SPV driver to detour too much from the shortest path between their trip origin and destination. The SPV drivers are compensated based on the number of packages they deliver and their total detour distance, which of course requires the drivers to communicate to the platform their personal trip's origin and destination. The central manager also has a fleet of trucks/vans (named "dedicated vehicles" and abbreviated as DVs) available to deliver parcels. The objective of the central manager is to minimize total delivery cost. The decision levers available to the central manager include the partitioning of packages between the SPVs and the DV fleet, the assignment of packages to specific vehicles in each set of vehicles, and the routing of each vehicle. The models and algorithms in this paper aim to address large-scale crowdsourced shared-trip delivery problems, which are rarely attempted in previous studies. Large-scale instances, particularly in terms of the number of packages and SPVs in the system, allow for the quantification of the benefits of bringing more SPVs into the system/platform. The large-scale problem instances are also valuable for long-term planning decisions related to DV fleet size and distribution center sizing and siting. In order to solve large-scale instances, the paper proposes a novel decomposition heuristic (D-H). The D-H handles the assignment of packages to SPVs and DVs separately but attempts to improve the solution by switching packages between SPVs and DVs. The paper includes a real-world city-scale case study to demonstrate the practical value of crowdsourced shared-trip delivery and to ascertain the effectiveness of the modeling approach and the solution algorithm. The contributions of this paper are as follows. First, the study proposes and models an urban package delivery system that combines shared personal vehicles and dedicated delivery vehicles. The proposed delivery system aims to leverage the considerably large volume of personal vehicle trips made each day to provide package delivery in urban and suburban areas. Even if a small percentage of these vehicles opt into a crowdsourced delivery system, they can deliver a large volume of packages with minimal detour time, distances, or cost. Second, this study models the proposed urban package delivery system and the underlying operational problem as both a mixed integer program based on the Vehicle Routing Problem (VRP) and a set partitioning problem. Third, based on the set partitioning formulation, the paper introduces a novel D-H that can solve large-scale problem instances. Fourth, the study provides valuable insights into the design of the proposed urban crowdsource delivery system with SPVs and DVs through new models, a solution approach, and case studies. The relevant design aspects include the DV fleet sizes, as well as the impact of parameters such as the cost of DVs, the cost of SPVs, the maximum detour distance for SPVs, and the distribution center location. The remainder of this paper is organized as follows. Section 2 reviews the literature relevant to crowdsourced freight delivery. Section 3 present the VRP-based formulation and set partitioning problem formulation. Section 4 describes the novel D-H solution algorithm. Section 5 introduces a real-world numerical case study and describes instances and parameters that are used. Section 6 summarizes major findings, concludes the paper, and discusses future research. This section reviews related research in the literature and delineates the unique contribution of the current study relative to the existing literature. Previous research related to crowdsourced logistics has been wideranging in terms of research methodology. This section reviews crowdsourced logistics research that employs (i) empirical methods to model crowdsourced delivery behavior and demand functions, (ii) optimization methods to model, design, and analyze crowdsourced logistics systems/services, and (iii) other methods including analytical models and simulation models. Rougès & Montreuil (2014) study 18 startups in the crowd shipping industry and claim that the businessto-consumer (B2C) crowdsourced delivery works best for intra-urban deliveries due to the need for partnerships with retailers and population density in urban areas. Punel et al. (2018) analyze the determinants of using crowd shipping after collecting 800 responses from a web-based survey. Their results indicate that crowd shipping package users believe the major advantages of crowd shipping to be environmental benefits and better vehicle utilization instead of affordability of crowd shipping items. The current paper applies optimization techniques to model, analyze, and design a crowdsourced delivery service; hence, the related literature is reviewed in detail here. The study of crowdsourced delivery problem as an optimization problem has been conducted using both static and dynamic problem settings, and literature review on both types of problems follow next. Static problems usually treat the crowdsource delivery problem as a multi-vehicle routing problem (m-VRP) or multi-vehicle pickup and delivery problem (m-PDP). Archetti et al. (2016) model the crowdsource delivery as an extension of classic VRP. Their model assumes a maximum of one task per SPV driver and they apply a multi-start heuristic by first assigning all packages to dedicated trucks and then solving a series of small-scale bi-partite matching problems to assign packages to SPVs. Macrina et al. (2017) extend the problem to a VRP with time-windows (VPRTW) and allow SPVs to carry multiple packages. Dahle et al. (2019) formulate the problem with consideration of pickups and drop-offs and formulate the problem as a pickup and delivery problem with time windows (PDPTW). The focus of Dahle et al. (2019) is to compare different compensation schemes and they argue that compensation needs to be large enough to exceed the threshold of the driver's willingness to deliver. The paper concludes that crowdsourced delivery would reduce total costs for logistic companies and that the savings would be around 10-15%. A dynamic version of the crowdsourced delivery problem has also been addressed in literature. Arslan et al. (2019) provide a dynamic way of modelling the problem. In addition to the case of a single store as the depot, the paper also considers in-store customers' willingness to travel to another depot for pickup and delivery. To solve the problem, the paper proposes a rolling horizon approach that employs a matching problem solution technique. Dayarian & Savelsbergh (2020) model the dynamic crowdsourcing problem with the assumption of a maximum of one task per SPV. The paper proposes and compares decision strategies, namely, myopic assignment and sample scenario planning. Gdowska et al. (2018) model the problem as a bi-level stochastic problem. They consider the possibility that in-store customers reject a matching found for delivering a package. The paper proposes a cost-driven heuristic technique to solve the problem. Formulating the crowdsource delivery problem based on the VRP or PDP is quite natural. However, due to the NP-hard nature of routing problems, and despite the rich literature on VRPs (Cordeau & Laporte, 2003; Golden et al., 2008; Laporte, 1992; Laporte et al., 2000) , obtaining optimal routes is difficult when the number of dedicated trucks exceeds just 10-15 and there are no SPVs. Problems with hundreds of potential SPVs in addition to dedicated trucks, such as the one addressed in this study, would be near-impossible to solve. Moreover, the decision to assign a package to a DV or a SPV is not simple, as it is ultimately driven by costs. However, the marginal cost of assigning a package to an SPV or DV is hard to estimate precisely and accurately a priori. Table 1 provides a comparison of the major optimization-related papers in crowdsourced urban logistics, along several dimensions. Notably, the current paper is the first to formulate the problem as a set partitioning problem in addition to a routing problem. Additionally, the D-H algorithm and set partitioning formulation handle more tasks and significantly more SPVs than studies in the existing literature. Studies applying approaches other than empirical analysis and optimization include Chen & Chankov (2018) who employ an agent-based simulation and Qi et al. (2018) who uses an analytical approach. The simulation results in Chen & Chankov (2018) indicate that the maximum willingness-to-detour affects the service level and the number of packages served by SPVs the most. Qi et al. (2018) develop a continuous approximation model for the open vehicle routing problem of SPV drivers. The study points out that the major economic benefit of crowdsourced delivery is reducing fleet size and offering operational flexibility. A set of package delivery orders (PDOs) that require delivery is defined as . Each PDO ( ) may have multiple packages. The study assumes that all packages are small-to medium-sized and easily fit in a normal sedan. Each PDO has a designated drop-off location, an earliest pickup time , and latest delivery time, . Two types of vehicles are used for delivery in the crowdsourced shared-trip delivery system, sharedpersonal vehicles (SPVs, usually family size sedans or wagons) and dedicated vehicles (DVs, usually vans or trucks) for delivery. Let be the set of all vehicles, be the set of SPVs and be the set of DVs; hence, = { ∪ }. An individual SPV is represented as ( ∈ ). The driver of an SPV may indicate the maximum number of PDOs they are capable or willing to carry/serve, and the parameter is denoted . Each SPV has its own origin and destination pair. If any PDOs are assigned to an SPV, the SPV must travel from its origin to the depot first, pick up the PDO(s), deliver all PDO(s), and lastly travel to its own destination. Let denote the earliest time an SPV can pick up PDOs at the depot and let denote the latest arrival time that an SPV should arrive at its own destination. A DV is represented as ( ∈ ). Without loss of generality, the study assumes that all DVs are identical and have a maximum number of stops they can make, denoted as . This parameter value would be set by the logistics service provider and would depend on factors such as the size and range of the vehicle, the maximum consecutive working hours for a driver, and the maximum driving distances of the driver. DVs are required to return to the depot/hub after completing delivery tasks. The service network is defined on a graph = ( , ). is the set of nodes, including the hub, all PDO drop-off locations, and all origins and destinations of SPVs. is the arcs/links connecting nodes, represented by tuple ( , ), where , are nodes. The departure and arrival hubs of DVs are represented as 0 ℎ (physically they are both the depot). The drop-off location of each PDO is represented as . The designated destination of each SPV is represented as . The monetized travel cost of a link ( , ) is represented as and for SPVs and DVs respectively. The travel time of a link ( , ) is represented as . An SPV driver is compensated by both the number of PDOs completed and the total detour distance from delivery. The compensation per PDO is represented by . The detour distance calculation is demonstrated in Figure 2 . The monetized cost for each SPV to travel from its origin to the depot is represented as 0 . The monetized cost for each SPV to travel directly from its origin to destination is represented as . Therefore, the total detour cost (compensation) for an SPV is calculated as 0 + − . The fixed cost of each DV, , includes the storage, insurance, and per vehicle overhead costs. All notations are summarized in Appendix 1 Notation Table. The natural way to formulate a delivery problem involves exploiting VRP formulations, since the delivery problem has the features of an "unrepeated route" and "returning to depot". The crowdsource delivery problem is a special variant of the original VRP. First, this problem needs to route two general types of vehicles, SPVs (sedans) and DVs (trucks). Therefore, the problem is considered as a heterogeneous fleet or mixed fleet VRP (Baldacci, Battarra, et al., 2008; Irnich et al., 2014) . In addition, in this crowdsourced shared-trip problem, SPVs do not return to the depot, which makes the problem similar to another variant of VRP, the so-called Open VRP (F. Li et al., 2007) . This paper refers to the combined variant as a Mixed Fleet Open Capacitated Vehicle Routing Problem with Time Windows (MFOCVRPTW) . The decision variables are introduced as follows: ,1}, ∀( , ) ∈ , ∀ ∈ . = 1, if arc ( , ) is visited by vehicle k. • ∈ ℝ + , ∀ ∈ , ∀ ∈ . Arrival time of vehicle at node . The formulation is presented as follows. In this formulation, the objective function aims to minimize the total cost of delivery using SPVs and DVs. The term, ( 0 + ∑ ( , )∈ − ) , represents the total detour cost of an SPV . The term (∑ ( , )∈ − 1) is the compensation for PDOs completed by SPV . The first term is multiplied by , the indicator variable for whether an SPV is used. If is 0, the SPV travels directly from its origin to its destination. The second term, ∑ ∑ ( , )∈ ∈ , is the total delivery routing cost of all DVs. The last term, ∑ ∈ , calculates the total fixed cost for using DVs. Constraints (2) to (6) are routing constraints. Constraints (2) and (3) indicate that every node must be visited once and only once by each vehicle (of either type). The constraints in Eqn. (4) are the flow balance constraints at each node. The constraints in Eqn. (5) indicate that an SPV must arrive at its designated destination. The constraints in Eqn. (6) are for DVs, and they guarantee that if a DV leaves the depot, it must return to the depot. The constraints in Eqn. (7) state that if an SPV delivers no PDOs on its way to destination, then = 0. The constraints in Eqn. (8) represent the DV usage constraint, meaning that only DVs that are activated can serve requests. The constraints in Eqn. (9), together with the constraints in Eqn. (7), ensure that when = 0, the only activity for the SPV k is to travel to its own destination ( ), i.e., this vehicle is not used for package delivery. Additionally, the constraints in Eqn. (9) regulate the number of delivery locations that an SPV can visit and is the so-called "capacity" constraint. Similarly, the constraints in Eqn. (10) regulate the number of delivery locations a DV can visit. Constraints (11) to (15) are time window constraints. The constraints in Eqn. (11) guarantee that the trip for an SPV starts after its earliest departure time. The constraints in Eqn. (12) guarantee that an SPV's arrival time at its destination must be no later than its latest arrival time. The constraints in Eqn. (13) ensure that a PDO is only picked up after its earliest pickup time. The constraints in Eqn. (14) ensure that a PDO must be delivered no later than its latest delivery time. The constraints in Eqn. (15) indicate that if = 1 (i.e., node is visited right after node by vehicle ), the arrival time of vehicle at node must be later than the arrival time of vehicle at node plus the necessary travel time of arc ( , ). The constraints in Eqn. (15) also serve as sub-tour elimination constraints. The constraints in Eqn. (16) to (19) are the binary and no-negativity constraints for decision variables. Formulating the crowdsourced shared-trip delivery problem from a vehicle routing perspective enables us to solve the problem by leveraging the rich literature on VRP. Exact methods include Branch-and-bound (Christofides & Eilon, 1969; Little et al., 1963) and branch-and-cut or generating cuts (Baldacci, Christofides, et al., 2008; Baldacci et al., 2012; Laporte et al., 1985) . Heuristics include the Clarke and Wright saving heuristic (Clarke & Wright, 1964) and multiple meta heuristics (Gendreau & Potvin, 2005; Hansen et al., 2001; Nikolaev & Jacobson, 2010; Prins, 2004) . The MFOCVRPTW is an NP-hard problem and using exact methods for large-scale problem is computationally infeasible. Therefore, for large-scale problems, heuristics are preferable. Inspired by the literature, this study constructs a Decompositionheuristic (D-H) algorithm to solve the problem. The basis for the decomposition is a set partitioning formulation, presented in the following subsection. The m-VRP can be reformulated from a set partitioning perspective (Baldacci, Christofides, et al., 2008; Desrosiers et al., 1992; Laporte, 1992 ; Y. H. Lee et al., 2008; Ropke & Cordeau, 2009 ). This paper treats the collection of all PDO locations ( ) as a set of nodes to be covered/contained by a collection of route sets (the collections of vehicle routes). Then the objective is to assign the origin and destination locations of each PDO to one feasible vehicle route while minimizing the total cost of the collection of vehicle routes. Like the approaches of Baldacci, Christofides, et al. (2008) and Ropke & Cordeau (2009) , the study formulates the crowdsourced shared-trip delivery problem as a set partitioning problem. Let , be a binary decision variable and represent whether the ℎ feasible route of is used. Correspondingly, let binary variable , represent whether the ℎ feasible route of has been used. Let , and , be the cost of travelling on of shared and dedicated ℎ respectively. Let , , and , , be two binary parameters. When , , or , , is 1, it is feasible for of shared or dedicated vehicle to service package delivery order . The set partitioning formulation of the problem is written as: subject to The objective function (20) is similar to the MFOCVRPTW formulation and minimizes the total cost. The first term, ∑ ∑ ( , − ) , , is the total detour cost of SPVs, and the second term ( ∑ ∑ ∑ , , , ) is the total "compensation per PDO completed" for SPVs. The third term is the total routing cost of DVs, and the last term is the total fixed cost of using DVs. The constraints in Eqn. (21) ensure that each PDO must appear once and only once on all vehicle routes. The constraints in Eqn. (22) guarantee that one and only one feasible route for each SPV is selected in the optimal solution. The constraints in Eqn. (23) state that no more than one feasible route for a DV should be used. The constraints in Eqn. (24) are binary constraints for the decision variables. A set partitioning problem can easily be converted to a bi-partite matching problem. Therefore, Formulation 2 provides a new approach for solving the crowdsourced shared-trip delivery problem as a matching/assignment problem between the PDOs and vehicle routes. Since the bi-partite matching problem has the feature of total unimodularity (Yannakakis, 1985) , it allows a linear relaxation of an integer problem. However, a major challenge still remains. To ensure optimality, it is necessary to enumerate all possible routes for each SPV and DV. Doing so for SPVs is challenging, whereas, doing so for DVs is computationally infeasible. Hence, most research relies on generating a 'sufficient' number of promising routes for each vehicle (Ryan et al., 1993) . The first obstacle is to enumerate a large number of routes for SPVs. The second is that even with a huge number of SPV routes, some PDO locations may still be unvisited, and therefore the DV routes must cover all unvisited locations to guarantee the feasibility of problem. To cope with these challenges, the paper introduces a Decomposition-Heuristic (D-H), which handles the routing and assignment of SPVs and DVs separately and considers potential PDO switching for solution improvement. As described in the previous section, the D-H handles PDO assignment to SPVs and DVs separately. Like Arslan et al. (2019) , this paper also makes the reasonable assumption that the average cost of using SPVs for package delivery is lower than that of using DVs. The major algorithmic steps include SPV route generation, PDO-SPV assignment, DV routing, and PDO switching between DVs and SPVs. The solution procedure is described as follows: o Generate a set of feasible routes for each SPV in 0 . ▪ Most of these routes can be generated 'offline'  described below. • The reason for implementing an incremental approach of adding a subset of SPVs at every iteration, is to avoid being trapped in a local minimum. The cause for such a local minimum is the non-convexity of truck routing, which can lead to inefficient PDO switching between SPVs and DVs in the case where all SPVs are included in 0 from the beginning. Without the incremental approach, it was found that there were instances when the total cost increased as the number of SPVs in the system increased, indicating that the solution algorithm did not necessarily improve as the solution space expands. For the incremental approach, the fewer SPVs in every batch, the more batches, the higher the solution quality but the longer the computational time. The advantages of applying the D-H are as follows. First, generating routes for an SPV is a relatively straight-forward task since the route length is bounded by the detour willingness of the SPV. Hence, SPV route generation can be conducted off-line. With day-to-day operations, the "promising" SPV routes identified or 'learned' from prior days can be stored and retrieved as needed. Additionally, under the assumption that, on average, package delivery via SPV is cheaper than delivering PDOs via DV, it makes sense to initially assign as many PDOs to SPVs as possible, which the algorithm does. Separating the SPVs and DVs in the routing process enables a straight-forward assignment problem between PDOs and the set of SPV routes. On the other hand, serving the remaining non-SPV PDOs with DVs are also straightforward given the wide range of exact, approximate, and heuristic solution algorithms for the VRPTW available in the literature. Moreover, using an insertion-based heuristic algorithm for DV routing also simplifies the procedure of obtaining the marginal cost of serving an additional PDO. Based on extensive empirical analysis, the SPVto-DV package-switching step significantly improves the overall solution quality. The main reason for the improvement is that even though the average delivery cost for SPVs is lower than the average delivery cost for DVs, once a DV is put into operation, the marginal cost of serving an additional PDO via DV is smaller than the marginal cost of serving most PDOs via an SPV, until the DV nears its capacity limits. This section describes the first step of the D-H for which we present an algorithm that generates k-shortest paths with budget constraints. We generate the possible routes from the depot to different SPV destination locations with various levels of detour willingness. The quality of k-shortest paths is the main determinant of the solution quality of the PDO-SPV route assignment problem, and we attempt to exhaustively generate all possible routes for an SPV under a travel budget constraint. Unlike the other steps in the D-H, it is conceivable that a large combination of SPV routes can be computed once (at the beginning of the month or year) and then stored by the logistics provider, as opposed to needing to recompute the routes every time the logistics provider solves the crowdsourced shared-trip delivery problem. Hence, we say that this subproblem can be addressed 'offline'. The k-shortest paths with budget constraints problem could be described as the follows. Given a graph = ( , ), find all possible paths between a start node and a target node that are within the travel time/cost/budget of . It is worth noting that the budget in this section is not equivalent to the maximum detour willingness of SPV drivers, although they are related. The maximum detour willingness of an SPV driver is defined as the maximum time the driver is willing to delay arriving at their destination given the departure time from their origin. In contrast, in this step, we generate routes for SPVs from the depot to their destinations (the blue paths in Figure 2 ), and therefore the budget for an SPV equals the maximum detour willingness of the SPV ( ) minus the shortest path travel time from the SPV origin to the depot ( 0 ) and possible PDO pickup and drop-off time ( ), represented as To solve the k-shortest path with budget constraints problem, we first present an intuitive recursion algorithm. The recursion algorithm starts at node , searches all neighboring nodes and terminates if it reaches the target node . Otherwise, the algorithm continues searching from neighboring nodes until target is reached or the budget is exhausted. The pseudocode for the recursion algorithm is in Algorithm 3 below. The recursion algorithm has the advantage of being intuitive and easy to code. The complexity of the algorithm depends on the number of nodes and the connectivity of the network. It has reasonable computational time for sparse networks. In the numerical example, presented in Section 6, we test the recursion algorithm on the City of Irvine network. For the 2000 SPV case, the computational time ranges from 12 mins (10 mins detour willingness) to 6 hours (30 mins detour willingness). For a network with a high level of connectivity, Algorithm 2 would be slow since it has a complexity of ( !). However, city street networks are not highly connected as each node typically has 4 or fewer outgoing edges. The City of Irvine test results presented later in this paper show that the algorithm is somewhat practical if the detour willingness is low. In order to cope with the computational complexity issue, we also provide Algorithm 3 below that adapts Yen's algorithm (Yen, 1971 ) by adding budget constraints to it. The pseudocode of the budgeted Yen's algorithm is displayed below. The complexity of original Yen's algorithm is ( ( + log ), with number of paths to be generated. Algorithm 3 keeps the heap structure of storing paths and applies Dijkstra's algorithm for shortest path finding, which is ( + ). In the worst-case scenario, when the network is fully connected and the budget is huge, along the spur path, all nodes will be visited, Dijkstra's algorithm would be called × | | times if we treat budget as a large constant. The overall complexity for Algorithm 3 is ( ( + log ). Based on computational experiments, Algorithm 3 outperforms Algorithm 2. For the same 2000 SPV case as described above, the computational time for Algorithm 3 is around 5 minutes compared to 12 minutes with Algorithm 2 for the 10-min detour willingness case. For the 30-min detour case, Algorithm 3 takes roughly 3.5 hours compared with 6 hours for Algorithm 2. For the 2000 SPV case, the computational time ranges from 12 mins (10 mins detour willingness) to 6 hours (30 mins detour willingness). The next step in the D-H is to match the PDOs with SPVs. To be more specific, instead of matching PDOs to individual SPVs, the decomposition algorithm attempts to match PDOs to SPV routes that were generated from the previous step. The following is a formulation of the PDO-SPV route assignment problem: subject to: The objective function (25) maximizes the total benefit of matching PDOs to SPV routes. To encourage successful matchings, (a large number) is introduced as a reward term. This operationalizes the strategy to match as many PDOs to SPVs as possible initially. In the objective function, is a binary parameter that indicates whether the ℎ route/path of SPV can feasibly serve PDO . The constraints in Eqn. (26) guarantee that a PDO is served by at most one route and at most one vehicle. The constraints in Eqn. (27) ensure each SPV only travels on at most one path through the depot. The constraints in Eqn. (28), ensure a PDO is only assigned to route if a SPV k is assigned to route . If equals zero, SPV does not use route , and therefore, no PDO should be served by vehicle on route . Moreover, if equals one, then vehicle does use route and the total PDOs carried by the vehicle should not exceed the maximum number of PDOs a SPV is willing to serve. Constraint (28) also acts as a linking constraint between decision variables and . It is worth noting that without Constraints (27) and the decision variable , Formulation 3 becomes a bipartite matching problem, which is solvable in polynomial time and has a complexity of ( 3 ). To take advantage of the complexity of the bi-partite matching problem, we implement a Benders decomposition to Formulation 3 for large-scale cases when the detour willingness of SPVs is high. The formulation and procedure for performing Benders decomposition are presented as follows. Master Problem (MP) = (30) subject to subject to The dual subproblem is formulated as follows: subject to Formulation 4 presents the master problem (MP) and subproblem (SP) of the PDO-SPV route assignment problem. The subproblem is a linear assignment problem with a time complexity of ( 3 ). To obtain optimal cuts (corner solutions), the study solves the dual sub-problem (DSP). When initialized with a feasible solution from the master problem, the subproblem is always feasible because the linear assignment problem always has a solution given its parameter settings are valid. Hence, the dual subproblem is never unbounded, and one does not need to generate feasibility cuts (extreme rays) for the master problem. The solution procedure is a standard Benders decomposition procedure, displayed in Algorithm 4 below. A feasible solution ̃, ; = −∞, = +∞. In Formulation 4, the subproblem is convex and is a restricted problem of the original problem (Formulation 3). Therefore, the subproblem is an underestimation of the optimal value for the original problem and solving subproblem gives a lower bound (LB) for the original problem. On the other hand, the master problem is non-convex with all integer constraints, and it is a relaxation of the original problem. Solving the master problem gives an upper bound (UB). Compared with the original problem, the problem size (number of constraints and variables) is significantly smaller. Adding cuts generated from the subproblem gradually restricts the master problem and alleviates the gap between LB and UB. In Algorithm 1, the D-H, we solve a VRP to obtain DV routes. To solve the problem, we employ the insertion algorithm described in Campbell & Savelsbergh (2004) . The reasons for choosing an insertion algorithm for solving the VRP are as follows. First, Algorithm 1 includes a step that compares the marginal price of serving a PDO if the algorithm switches the order from an SPV to a DV. An insertion algorithm provides an intuitive way of estimating the marginal cost. The following section, Section 4.5 elaborates the usage of marginal cost in detail. Moreover, as suggested by Campbell & Savelsbergh (2004) , the efficient insertion heuristic achieves a time complexity of ( 3 ), and therefore can handle large-scale problems. The insertion algorithm is used for single vehicle routing ( Step 3) and multi-vehicle routing (Step 5) in Algorithm 1. Step 3 performs a single vehicle routing in order to obtain an estimate of overall delivery cost for the PDOs that are to be served by DVs. More importantly, a single route that is formed by the all the DV-served PDOs provides a direct estimate of the insertion cost if any PDOs are to be moved from the SPV-served set. The marginal cost ( ) for a DV to serve a previously SPV-served PDO is calculated as follows. Assume the original link in a DV route is link ( , ), and the node to insert is node . = , + , − , (41) Step 5 applies the insertion algorithm for a multi-vehicle routing, the pseudo code for the insertion algorithm is presented in Algorithm 5. Step 4 of the Algorithm 1 involves a procedure for switching PDOs from the set to be served by SPVs to the set to be served by DVs. This subsection describes the rationales and the detailed procedure of performing a PDO switch. In Step 2 of D-H, we attempt to assign as many PDOs to SPVs as possible. However, this step was done 'myopically' without considering the routing of DVs. After Step 2 and 3, we obtain an estimate of DV delivery cost for the rest of the PDOs and the number of DVs needed for the tasks. If DVs are not at their full load, it becomes attractive to switch additional PDOs from the SPVs to the DV. The reason is that some PDO delivery locations assigned to SPVs in Step 2 may be close to the delivery locations along DVs routes created in Step 3. Therefore, gradually switching PDOs assigned to SPVs in Step 2 to DVs and updating the DV routes could reduce the total cost. The switching terminates when either the marginal benefit of switching a PDO from SPVs to DVs becomes negative, or all DVs are at full truck load. We conduct a neighboring search for all SPV served locations to find adjacent DV served locations. The detailed steps are as follows: Step 4.0 Obtain the DV route from single VRP as in Step 3. Step 4.1 For every delivery location to be served by SPVs (location ), find the nearest location that is to be served by DVs (location ). The DV round-trip cost (2 × ( , ) ) between provides an upper bound cost of using any DV to serve the location . If 2 × ( , ) < (the cost of serving location by SPV), it is more cost efficient to use DVs to serve the delivery location . Location is the nearest neighbor for location . The study uses the insertion cost of either inserting into link ( − , ) or link ( , + ) as the estimate of insertion cost of SPV location to the DV route. Step 4.2 Calculate the potential cost saving for location if PDOs in location are switched from SPVs to DVs. Compare over all locations to be served by SPVs. Find the minimum one. If the load of DV after switching exceeds a truck load, an additional DV is required, and therefore the additional DV's fixed cost needs to be added to the cost saving estimate. Step 4.3 Switch the PDO, update the DV route and SPV location list. Repeat Step 4.1 and 4.2. Terminate when all cost savings are negative. The pseudo code for PDO switching is displayed in Algorithm 6. The section describes the D-H and the details of the algorithms that is used in every step. The procedure of the D-H involves solving different subproblems, and different solution techniques are required for every subproblem. For the first subproblem, the budgeted k-shortest paths problem, the study proposes a recursion algorithm (Algorithm 2) that obtains SPV routes. For the PDO-vehicle route assignment problem, the paper presents the formulation and explains the procedure for a Benders decomposition when the problem size is large. For the vehicle routing problems, the study applies an insertion heuristic, which handles the problem efficiently and provides important cost reference for the comparison in the next solution step. For the PDO switch procedure, the paper utilizes a cost-driven approach that effectively decides the PDOs to be shifted from the SPV set to the DV set. In the next two sections, we conduct a real-world city-scale case study of crowdsourced shared-trip delivery and compare the computational results of using the D-H with the exact method of solving the MFOCVRPTW. We conduct a numerical case study using the road network of the City of Irvine, CA, USA. The network contains 442 nodes and 648 links. Two nodes are selected as potential depots for package delivery. Depot 1 is on the boundary and Depot 2 is close to the center of city. In the benchmark case, the study uses Depot 1. In the sensitivity analysis, the study compares the performance under Depot 2 with the benchmark case. PDOs and a maximum of 1200 SPVs. The PDO locations are randomly chosen, uniformly distributed across the network. Multiple packages may be in the same PDO, and multiple PDOs may share the same physical delivery location. Figure 3 displays the City of Irvine network marked with nodes, links, depots, and PDO destinations. Each PDO has a latest delivery time (time window). The latest delivery time for a PDO is randomly chosen from 12:00pm, 4:00 p.m. or 8 p.m. The SPV trips are generated from the California State Travel Demand Model (CSTDM). Each SPV has a maximum number of stops that it will make. Each driver has an earliest starting time (EST) from their origin and a latest arrival time (LAT) at their destination. The maximum detour willingness (by time) of a vehicle is calculated as − − ℎ ℎ . In the sensitivity analysis, the study tests detour willingness of 10, 15, 20 and 25 minutes. The SPVs are assumed to travel at a speed of 40 miles per hour and DVs at an average speed of 30 miles per hour. The study also assumes that once a PDO is matched to an SPV driver, the driver will not reject the assignment. The SPV drivers are compensated based on the number of PDOs completed (a fixed amount of $1.5 per PDO) and the detour distance incurred. For the latter, the study uses IRS' standard mileage rate for business travel, $0.56/mile (Internal Revenue Service, 2021). There are also several identical DVs available at the depot. The DVs are responsible for delivering the PDOs not served by the SPVs. We refer to Amazon logistics (2021) for the normal size of vans and trucks used for urban last mile delivery, which are usually 9' to 12' vans or trucks. We also refer to U-Haul website's (2021) for the cost of using such vehicles. The DV fixed cost in this study is set to $120 that incorporates labor costs for the professional driver, overhead costs associated with an addition DV, and other miscellaneous costs. The DV variable cost is set at $1.5/mile, of which $0.4/mile is from fuel and $1.1/mile is from depreciation, maintenance, and wear/tear. The next section compares solving the crowdsourced shared-trip delivery problem with the D-H and solving Formulation 1 using a commercial solver. The metrics that are used for comparison are the computational time and optimality gap. To evaluate the effectiveness of a crowdsourced delivery system, this study compares the total delivery cost and vehicle miles travelled (VMT) of using crowdsourced delivery and the conventional DV-only delivery system. The study also discusses the impact of SPV quantity by varying the number of candidate SPVs and conducting a sensitivity analysis on the impact of the SPV maximum detour willingness. In the sensitivity analysis of the SPV maximum willingness to detour, we define the maximum detour willingness as the latest time the driver is willing to arrive at their destination minus their earliest departure time. Therefore, the period of this maximum detour willingness covers the travel time from the SPV origin to the depot (orange segment in Figure 2 ), possible PDO pickup/drop-off time (preset to be 10 mins) and the travel time from the depot to the destination with necessary detour to deliver PDOs (blue segments in Figure 2 ). In the sensitivity study, we assign the same maximum detour willingness to all SPVs. Since each SPV has different travel times from their origins to the depot, the SPVs effectively have heterogenous travel time budgets from the depot to their destinations. In addition, the paper conducts a sensitivity analysis on the depot location to understand its impact on total delivery cost and VMT. The case study uses the numerical case where the depot is at Node 156288 (Depot 1 in Figure 3 ) and the SPV maximum detour willingness equals 30 minutes as the benchmark situation. The numerical study obtains results for cases where the number of SPVs range from 0 to 1,200. This subsection compares the computational results between the D-H and exact method of multi-vehicle routing problem to assess the effectiveness and efficiency of the D-H. The D-H is implemented in the Python 3.7 language, and the exact solution is obtained using the commercial solver Gurobi (version 9.1). All numerical cases are computed on a 2.20 GHz Intel Xeon Server with 128 GB RAM. The numerical experiment is conducted by using the aforementioned network of the City of Irvine. In total, the number of PDOs range from 10 to 100, and PDOs are distributed uniformly in the study area. In addition, a number of SPV samples are generated randomly. The numerical experiment regulates that every SPV has a fixed detour time of 30 minutes. The depot is Depot 1 in Figure 3 . The experiments compare the computational time and the optimality gaps. The computational time limit for Gurobi is set at 1,200 seconds. If Gurobi does not finish the optimization process after 1,200 seconds, the optimality gap between the primal and dual problems is reported along with the optimality gap between D-H and exact solutions. The experiment contains both small-scale problems (e.g., 10 PDOs, 10 SPVs) and larger-scale problems (e.g., 100 PDOs, 1000 SPVs). The results are summarized in Table 3 . The study first compares the computational results for small-scale problems. In the four cases of 10 PDOs, the optimality gap between the D-H and exact method is less than 3%. The optimality gap is ( − )/ . In the four cases of 20 PDOs, D-H solves the problem faster than the exact method. Moreover, the optimality gap is less than 5.2% in all cases. Hence, the D-H can obtain solutions to small problem instances that are comparable in quality to exact methods, while using considerably less computational effort. For large-scale problems, the exact method in the commercial solver begins to slow down considerably. For the case of 50 PDOs and 50 SPVs, the exact method provides a solution better than D-H, but the optimality gap is only 2%. However, the exact method uses up all 1200 seconds while the D-H method only requires 3.5 seconds. For the case of 50 PDOs and 100 SPVs, the D-H actually finds a better solution than the commercial solver when it stopped at 1200s. Moreover, for cases with more than 50 PDOs and 500 SPVs, the commercial solver cannot even determine the dual gap within 1200s, meaning that it does not even find a feasible solution. Conversely, the D-H algorithm can find solutions to these large problem instances in a relatively short period of time (i.e., less than 3 minutes in nearly all cases). In fact, the results indicate that computational time does not increase dramatically with problem size for the D-H algorithm. While the 100 PDOs and 100 SPVs case takes 11.7 secs, the 100 PDOs and 1000 SPVs case uses 173 secs). In summary, the study finds that the D-H algorithms is scalable. For small-scales problems, the algorithm achieves close solutions to an exact method in a fraction of time. For large-scale problems, the algorithm outperforms the exact method within the time limit. The first metric to compare in this study is the number of PDOs served by SPVs. According to Section 4, the PDOs are matched to the SPVs first (an initial matching), and then possible PDOs are switched from SPVs to DVs (final matching). This section presents the matching number for both initial matching and final matching. Figure 4 displays the initial and final matching numbers against the number of SPVs available. The blue line in Figure 4 represents the initial number of PDOs matched with SPVs. Given the structure of the algorithm, the initial matching number is the largest possible number of PDOs that can be served by SPVs. Unsurprisingly, the blue dash line shows that the number of PDOs that can be served by SPVs increases with the number of SPVs. Similarly, the orange line shows that the final number of PDOs served by SPVs increases with the number of SPVs in the fleet. Moreover, the dashed blue-line serves as an upperbound on the orange line. It is also worth noting that even when the number of available SPVs reaches a relatively high level, the PDOs are not fully covered by SPVs (matching rate at around 75%). If the company can keep increasing the number of SPVs, it is possible that all the PDOs can be served by SPVs, and no truck will be needed. However, the total number of SPVs required may be very high, which leads to a question whether the logistics company can obtain sufficient supply of SPVs to fully replace all DVs with SPVs. Hence, in nearly all cases, at least one DV is needed to serve PDOs. The orange line in Figure 4 represents the number of SPV-served PDOs at the optimal or near-optimal cost; the orange line forms a step function. The green line represents the number of PDOs served by DVs, which is also a step function. These findings indicate that the optimal solution involves utilizing DVs to full capability but also minimizing the number of DVs in operation. When the number of SPVs increases to a level where the feasible SPVs could serve an additional truckload of PDOs, the number of DVs required to PDOs drops by one, and the total cost drops to a lower level. This section analyzes the SPV (route) usage from the supply perspective to understand how the SPVs are used. The study defines an SPV with minimum detour as an SPV that travels directly from its origin to the depot on the shortest path and then travels from the depot to its destination on the shortest path while delivering at least one PDO. Since the SPV does not detour at all from the depot to its destination, we call this a minimum detour route. Figure 5 displays a histogram of the detour distances for SPVs assigned to serve PDOs. The histogram values come from computational experiments in Figure 4 , where the number of SPVs range from 100 to 1200. In Figure 5 , on the x-axis, "0" indicates the route is a minimum detour route, and the other integer numbers are the travel time differences between the used route and the minimum detour route. The average total detour distance for SPVs with delivery tasks is 1.06 miles. The left-most bar in the histogram shows that 14% of SPVs use the minimum detour route. Moreover, half of the SPVs traverse a route from the depot with less than 2 minutes of detour. From the PDO perspective, 17% of the PDOs are delivered by SPVs on a minimum detour route. Almost 90% of the SPVs use a route with less than 10 mins of detour. The reason why most of the detours are small is that the objective function penalizes the costs associated with detour distance. The implications of the results in Figure 5 are two-fold. First, from a service design perspective, increasing the maximum willingness-to-detour is unlikely to increase the number of PDOs served by SPVs. Second, algorithmically, the budget in the budgeted k-shortest path algorithm can be small and not have a significant impact on the solution quality. Reducing the k-shortest path budget can significantly improve computational complexity. Of course, it must be noted that the results in Figure 5 are dependent on the network structure and spatial distribution of package deliveries and SPV trips. Total delivery cost is the most critical metric as it is the objective function for the original problem formulation. Figure 6 displays SPV cost, truck cost, and total delivery cost as a function of the number of SPVs available. Figure 7 displays the per PDO costs for SPVs, DVs, and all vehicles combined. Figure 6 and Figure 7 illustrate that the total delivery cost (and, by definition, average cost) decreases as the number of available SPVs increases. This finding is expected given more SPVs represent a larger feasible region or more options for PDOs to be delivered by SPVs. Consistent with the results in Figure 4 , the total delivery cost in Figure 6 reduces in steps as the number of SPVs increases. This is, again, the result of reducing the number of DVs necessary to deliver the PDOs. The blue SPV cost line in Figure 6 shows an increasing trend while both DV (orange line) and total cost (green line) are decreasing. SPV costs are increasing with the number of SPVs due to the increase in PDOs served by SPV, while DV costs are decreasing due to the decrease in PDOs served by DVs or more accurately the decrease in number of DVs needed to serve PDOs. This result also indicates that the savings from crowdsourced share delivery are mainly the result of DV fixed cost reduction. Table 4 shows that the total cost slowly decreases between SPV size 400 to 900. The decrease stems from more SPVs allowing better matching (i.e., shorter detour distances) between SPVs and PDOs. However, compared to the decreases between 300 and 400 and 900 and 1000 when the DV fleet size decreases by one in both cases, the decreases in costs between 400 and 900 are relatively small. Overall, Table 4 shows that crowdsourced shared-trip delivery significantly reduces cost compared to an all DV fleet. The cost saving ranges from 15% to 30% as the number of SPVs available increases. Besides the total cost, the average cost of delivering a PDO decreases as the number of available SPVs increases. Both Figure 7 and Table 4 demonstrate the trend. It is worth noting that while the overall average delivery cost decreases with available SPVs, the average DV delivery cost increases slightly, and the average SPV delivery cost remains about the same. However, because the average SPV delivery cost is always lower than the average DV delivery cost, moving more packages with SPVs decreases the overall average delivery cost. The results in this section indicate that if properly managed, logistics companies that employ crowdsourced delivery service can reduce the number of DVs they purchase as well as the associated storage, maintenance, insurance, and other overhead costs associated with each DV. However, since it is difficult to serve all PDOs using only SPVs, it is hard to eliminate DV service completely, and the cost savings associated with SPVs are unlikely to increase by more than 30% to 40%, even for higher numbers of available SPVs. This section analyzes total vehicle miles traveled (VMT), a proxy for both congestion and environmental impact. Figure 8 and Table 5 present the total VMT resulting from delivery activities for SPVs, DVs, and all vehicles. The results indicate that total VMT tends to increase with the number of SPVs, although non-monotonically. This finding is unsurprising given that DV routes are more VMT efficient than most SPV routes, and as the number of available SPVs increase, the number of PDOs assigned to SPVs increases. Given that VMT increases with the number of SPVs, the question then becomes, does the crowdsourced delivery system increase congestion and worsen the environmental impacts of package delivery? The answer for congestion is most likely, 'yes', unless the DVs cause substantially more blocking of traffic lanes in dense urban areas compared to PSVs, which is unlikely. The answer for environmental impact is more nuanced. Since SPVs are small-to medium-sized sedans, the SPVs themselves are significantly more energy efficient and environmentally friendly on a per mile basis than most medium-duty vehicles used for dedicated delivery services. Therefore, crowdsourced shared-trip delivery may even reduce environmental emissions relative to exclusive DV delivery, unless the DVs are fully electric. Final answers on these questions can be found only from detailed simulation of vehicle movements, which is beyond the scope of this paper. Also worth mentioning is that the VMT estimate for each individual SPV in Figure 8 includes SPV travel from the SPV driver's origin to the depot. If the logistics company only considers SPVs from people already shopping at the store (at or near the depot), which is the case in almost all previous studies (Archetti et al., 2016; Arslan et al., 2019; Dayarian & Savelsbergh, 2020) , then total VMT is likely to decrease with the number of SPVs. In fact, Figure 9 displays the results of this setting-all SPVs are initially at the store/depot, therefore they do not need to detour to reach the store/depot. In Figure 9 , all the solid lines that are marked with "1" (indicating Case 1, where SPVs have non-depot origins) are the same as Figure 8 , while the dash lines represent the Case 2 where all SPV drivers are at the store/depot. While the VMT gap between DVs is relatively small between the two cases, the gap between the VMT from SPVs is quite dramatic. Hence, the VMT savings from assuming all SPVs originate at the store/depot drives the overall savings in VMT between the two green lines. The dashed green line is consistent with much of the literature and indicates that share-ride delivery can decrease VMT. This section presents a sensitivity analysis to assess the impact of the detour willingness of SPVs. As explained in both Section 4.2 and Section 5, the maximum willingness-to-detour of an SPV includes the travel time from its origin to the depot, the necessary PDO pickup/drop-off time, and the travel time from the depot to the destination. The benchmark case has a maximum detour willingness of 30 minutes for every SPV. The study uses 20, 25 and 35 minutes of maximum detour willingness to generate results and compare them with the benchmark. Figure 10 shows the impact of the maximum detour distance on the total delivery cost. The results are consistent with expectations, namely, the total delivery cost decreases as the maximum willingness to detour increases. The impact is quite significant under certain conditions related to the number of available SPVs. At 400 available SPVs, the total delivery cost is over $650 for the 20-min maximum detour case, whereas the total delivery costs are between $525 and $550 for the other three cases. Looking at the cost axis, at $550, the 35-minute and 30-min cases require only 100 available SPVs to achieve a total cost around $550, while the 15-min case needs at least 300 available SPVs and the 10-min case requires around 500 available SPVs. The longer detour times allow SPVs to travel to more 'unpopular' nodes that may have a PDO destination. These 'unpopular' nodes are quite inefficient for DVs and most SPV routes to serve. The extra flexibility clearly has a positive impact on the crowdsourced delivery service operations. Additionally, Figure 10 indicates that for the maximum detour willingness of 20 and 25 minutes, increasing the number of available SPVs beyond 500, has effectively no impact on the total delivery cost. Conversely, as the number of available SPVs increases in the 30-min and 35-min maximum detour cases, the total delivery cost continues to decrease. This further illustrates the value of detour flexibility. Figure 11 displays the relationship between the number of PDOs that can be served by SPVs (y-axis) and the total number of available SPVs (x-axis) with different cases depending on the maximum willingness to detour (different color lines). Figure 11 shows that with the same number of SPVs, the 35-min detour can serve 4 to 6 times more PDOs than the 20 or 25-min cases. Moreover, compared to the 25-min case, the 30min case can significantly increase the number of potential PDOs served. On the contrary, the marginal increase in possible PDOs served when the maximum detour increases from 30-min to 35-min is much smaller than the marginal increase when maximum detour increases from 25-min to 30-min. The results indicate that, at least for the Irvine network and the spatial demand in the scenarios in this study, there is a strong business case for incentivizing drivers to accept a 30-min detour rather than a 25-min detour. A related parameter of interest is the percentage of SPVs that can deliver at least one PDO without violating any time-window constraints, i.e., ( . / )% , as a function of the maximum willingness to detour. According to computational results for the Irvine case study with a 20-min detour willingness, only 2% ~3% of SPVs are feasible SPVs. While with a 25-min detour, the percentage increases to 18%. For the 30-min and 35-min cases, the feasible SPV percentages are 40% and 52%. This result substantiates the finding that a decent detour willingness significantly increases the potential of the crowdsourced shared-trip delivery service. Without at least a 25-or 30-min detour willingness for SPVs, it becomes difficult to serve a substantial number of PDOs. We also compare SPV usage across different scenarios in Figure 12 , which displays histograms for detour time under four different maximum detour scenarios. Figure 12 indicates that under all the maximum detour willingness scenarios, the majority of PDOs delivered by SPVs are carried by SPVs with a short detour route, meaning that most SPVs do not need to detour a lot from the depot to their own destinations to deliver PDOs. Combining the findings of Figure 11 and Figure 12 , we may conclude that a higher maximum detour willingness allows more SPVs (especially for those SPVs whose shortest origin to destination paths are not close to the depot) to participate in the crowdsourced delivery system. However, for individual SPVs who are matched with PDOs, the actual detour distance is quite small in most cases. Therefore, in Step 2 of Algorithm 1, a large budget may be unnecessary when enumerating SPV routes from the depot to the respective SPV destinations, which can potentially save computational time. This subsection presents the results of another sensitivity analysis, namely related to the impact of depot location on several performance measures. The section compares two cases: Case 1 where the depot is at the service region boundary; and Case 2 where the depot is at the center of the city. The study first compares the total cost and total VMT based on the depot location. Figure 13 shows that, in general, the center depot delivery costs are lower than the boundary depot costs. The reason is that a center located depot attracts more SPVs than a boundary-located one. However, when the number of SPVs is high, the difference in the total cost between a center depot and a boundary depot is relatively small (5.4%). This finding indicates that a higher number of SPVs can mitigate the total cost deficit caused by depot location selection. Hence, from a managerial perspective, an interesting and important conclusion emerges: the location of depots may be less important in terms of system cost, within a crowdsourced shared-trip delivery system than a conventional DVs-only system. For the total VMT metric (Figure 14) , in most cases, a center located depot results in less VMT than a boundary located vehicle, since more SPVs do not need to detour as much to a central depot. The VMT difference is significant (25%) for the SPV = 1200 case. Additionally, the depot location affects the percentage of SPVs that can feasibly serve at least one PDO. According to Figure 15 , the center located depot tends to produce more feasible SPV-PDO matches than a boundary depot. The percentage of feasible drivers under a 30-min detour willingness assumption for the center located depot case is 60%, while the same metric for a boundary located depot is only 40%. The higher percentage of feasible drivers also leads to a higher matching rate of PDOs by the SPV. Section 6 includes a real-world city-scale numerical case study of crowdsourced shared-trip delivery. The section presents results related to computational time between the D-H and a commercial solver as well as results for total cost, total VMT, number of PDOs served by vehicle type for various crowdsourced sharedtrip delivery system configurations. Although the results are generated based on the geometrical features and typology of the City of Irvine, they do lead to interesting findings and significant implications. The computational experiments show that the D-H algorithm outperforms the exact solver on computational time in all cases. In the cases where SPVs and PDOs are small, the D-H solution quality is comparable to the exact solver. In large cases, where the solver cannot find solutions in 1200 seconds using an exact method, the D-H can solve the problem in a few minutes. The ability to solve large-scale problem instances in crowdsourced delivery problem enables the inclusion of more SPVs into the system and provides flexibility of modeling crowdsourced delivery problem. The case study also shows a near linear (but stepwise) relationship between number of available SPVs and PDOs served by SPV (Figure 4) . However, the total cost of delivering PDOs has a less than linear relation with the number of participating SPVs. Using SPVs in a crowdsourced shared-trip delivery system saves cost compared to conventional dedicated delivery (from 15% to 40%), where the major cost savings stems from reducing the DV fleet size. However, while completely replacing DVs is possible, it would require a very large number of participating SPVs to achieve such a goal. Additionally, the total VMT of crowdsourced delivery highly depends on the origins of SPVs. If SPV origins are located near the depot, the VMT savings from crowdsourced delivery compared to dedicated DVs is about 10% to 20%. When the SPV origins are distinct from the depot, and SPVs need to drive to the depot for PDO pickup, then the crowdsourced shared-trip delivery produces more VMT than the 'all DVs' case, and the amount of VMT increases as more SPVs are used. To further understand the impact of the maximum willingness to detour on crowdsourced delivery, we conduct a sensitivity analysis. The results indicate that, on one hand, when a higher maximum detour willingness is imposed, the system has more participating SPVs, and higher detour willingness leads to lower total cost. On the other hand, the SPVs which are matched with PDOs are likely not required to detour a lot the depot to their destinations. This finding could further inspire the improvement on the D-H by only generating short detour routes from the depot to the destinations. Finally, we analyze the impact of depot location on various performance metrics. The results indicate, unsurprisingly, that a center located depot has lower total cost and VMT compared to a boundary located depot. This study presents new mathematical models and algorithms to address large-scale crowdsourced sharedtrip delivery problem instances. Following the literature on crowdsourced delivery, the paper first models the crowdsourced shared-trip delivery as an MFOCVRPTW (mixed fleet open capacitated vehicle routing problem with time window). This formulation captures the generalized features of crowdsourced sharedtrip delivery. However, finding an optimal solution based on this formulation is computationally expensive. To solve the problem in an efficient manner, the paper reformulates the problem as a set partitioning problem. The alternative set partitioning formulation also inspires a new solution approach, referred to as the decomposition heuristic (D-H). The novel D-H solution algorithm decomposes the problem by vehicle type, shared vehicles (SPVs) and dedicated vehicles (DVs). The D-H algorithm matches packages to each vehicle type separately and routes each vehicle type separately. The D-H also solves 4 subproblems, namely, the budgeted k-shortest paths problem, the large-scale matching problem, the PDO switching problem, and the multiple vehicle routing problem. Solution algorithms for each subproblem are also discussed. The D-H is compared with an exact method for solving the crowdsourced shared-trip delivery problem. The D-H can obtain solutions with a 1.5% optimality gap for small-scale cases and the D-H is much faster than the exact method allowing it to handle large problem instances. The models and algorithms are applied to the City of Irvine. The paper analyzes major factors that impact the efficiency of a crowdsourced shared-trip delivery. The analysis includes the metrics such as total cost, total VMT, and the number of PDOs delivered by SPVs. Moreover, the study presents a sensitivity analysis on various performance metrics with respect to changes in the maximum detour willingness and the study analyzes the impact of depot location on performance. Subsection 6.8 presents a summary of the major findings and their implications. Overall, the paper significantly advances the state-of-the-art through novel, computationally efficient and operationally effective mathematical models, and associated heuristic solution algorithms. Moreover, the paper contributes to logistics and transportation practice by addressing real-world problem instances (through numerical case studies) and provides valuable managerial insights and useful algorithms. The research presented in this study also opens several additional lines of research. First, while this study assumes the central logistics operator has full information about the origin and destination of each SPV (information that significantly impacts the fee paid to SPVs), obtaining this information in the real-world would not be straightforward. Hence, developing truth-telling mechanisms to obtain this information from SPV drivers may prove to be important. Moreover, a future crowdsourced delivery system may include connected automated vehicles (CAVs) that are owned by individuals but rented out to logistics providers during certain portions of the day. Therefore, further research could combine the usage of shared CAVs for people trips and PDO delivery to achieve better objectives, such as reducing the carbon emission and the total cost. Another research direction involves integrating crowdsourced delivery with other modes of freight and passenger transport, such as transit vehicles or drones. The "internet of things" may enable such a multimodal package delivery system that has the potential to decrease the carbon footprint of package delivery. Moreover, from an algorithm perspective, the D-H could be further improved by applying learning algorithms to generate promising candidate routes for SPVs. Via learning promising candidate routes, it may be possible to reduce computational time further. Table Notation Description 0 Depot Binary, whether a PDO could be served by ℎ route of SPV k , , Binary, whether a node j could be visited by ℎ route of SPV k , , Binary, whether a node j could be visited by ℎ route of DV k Travel budget (in minutes) The maximum willingness to detour (in minutes) , Crowdsourced delivery: A review of platforms and academic literature Amazon Logistics The Vehicle Routing Problem with Occasional Drivers Crowdsourced delivery-a dynamic pickup and delivery problem with ad hoc drivers Routing a Heterogeneous Fleet of Vehicles An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts Recent exact algorithms for solving the vehicle routing problem under capacity and time window constraints Efficient Insertion Heuristics for Vehicle Routing and Scheduling Problems Crowdsourced delivery for last-mile distribution: An agent-based modelling and simulation approach Multi-hop driver-parcel matching problem with time windows An Algorithm for the Vehicle-dispatching Problem Scheduling of Vehicles from a Central Depot to a Number of Delivery Points A tabu search heuristic for the static multi-vehicle dial-a-ride problem The pickup and delivery problem with time windows and occasional drivers Crowdshipping and Same-day Delivery: Employing In-store Customers to Deliver Online Orders A New Optimization Algorithm for the Vehicle Routing Problem with Time Windows Planning and operating a shared goods and passengers ondemand rapid transit system for sustainable city-logistics Stochastic last-mile delivery with crowdshipping Tabu Search The vehicle routing problem: Latest advances and new challenges Grocery shopping preferences during the COVID-19 pandemic Variable neighborhood decomposition search IRS issues standard mileage rates for 2021 Chapter 1: The Family of Vehicle Routing Problems Growth of E-Commerce and Ride-Hailing Services is Reshaping Cities Innovative Goods Delivery Solutions for Cities of the Future Online shopping motives during the COVID-19 pandemic-lessons from the crisis The traveling salesman problem: An overview of exact and approximate algorithms Classical and modern heuristics for the vehicle routing problem Optimal Routing under Capacity and Distance Restrictions Smart logistics: distributed control of green crowdsourced parcel services A heuristic for vehicle fleet mix problem using tabu search and set partitioning The Share-A-Ride Problem: People and parcels sharing taxis The open vehicle routing problem: Algorithms, large-scale test problems, and computational results An Algorithm for the Traveling Salesman Problem The Vehicle Routing Problem with Occasional Drivers and Time Windows Simulated Annealing A simple and effective evolutionary algorithm for the vehicle routing problem Studying determinants of crowd-shipping use Shared mobility for last-mile delivery: Design, operational prescriptions, and environmental impact. Manufacturing and Service Operations Management Crowd logistics: an opportunity for more sustainable urban freight transport? Branch and cut and price for the pickup and delivery problem with time windows Crowdsourcing Delivery: New Interconnected Business Models to Reinvent Delivery Extensions of the petal method for vehicle routing U-Haul Truck Rentals The restaurant meal delivery problem: Dynamic pickup and delivery with deadlines and random ready times The same-day delivery problem for online purchases On a Class of Finding the K Shortest Loopless Paths The authors declare that we have no conflicts of interest. Yang et al (2022)