Triangular Trade Map Assignment Drivers
In the year 1755, a black slave named Mark Codman plotted to kill his abusive master. A God-fearing man, Codman had resolved to use poison, reasoning that if he could kill without shedding blood, it would be no sin. Arsenic in hand, he and two female slaves poisoned the tea and porridge of John Codman repeatedly. The plan worked — but like so many stories of slave rebellion, this one ended in brutal death for the slaves as well. After a trial by jury, Mark Codman was hanged, tarred, and then suspended in a metal gibbet on the main road to town, where his body remained for more than 20 years.
It sounds like a classic account of Southern slavery. But Codman’s body didn’t hang in Savannah, Ga.; it hung in present-day Somerville, Mass. And the reason we know just how long Mark the slave was left on view is that Paul Revere passed it on his midnight ride. In a fleeting mention from Revere’s account, the horseman described galloping past “Charlestown Neck, and got nearly opposite where Mark was hung in chains.”
When it comes to slavery, the story that New England has long told itself goes like this: Slavery happened in the South, and it ended thanks to the North. Maybe
we had a little slavery, early on. But it wasn’t real slavery. We never had many slaves, and the ones we did have were practically family. We let them marry, we taught them to read, and soon enough, we freed them. New England is the home of abolitionists and underground railroads. In the story of slavery — and by extension, the story of race and racism in modern-day America — we’re the heroes. Aren’t we?
As the nation prepares to mark the 150th anniversary of the American Civil War in 2011, with commemorations that reinforce the North/South divide, researchers are offering uncomfortable answers to that question, unearthing more and more of the hidden stories of New England slavery — its brutality, its staying power, and its silent presence in the very places that have become synonymous with freedom. With the markers of slavery forgotten even as they lurk beneath our feet — from graveyards to historic homes, from Lexington and Concord to the halls of Harvard University — historians say it is time to radically rewrite America’s slavery story to include its buried history in New England.
“The story of slavery in New England is like a landscape that you learn to see,” said Anne Farrow, who co-wrote “Complicity: How the North Promoted, Prolonged, and Profited From Slavery” and who is researching a new book about slavery and memory. “Once you begin to see these great seaports and these great historic houses, everywhere you look, you can follow it back to the agricultural trade of the West Indies, to the trade of bodies in Africa, to the unpaid labor of black people.”
It was the 1991 discovery of an African burial ground in New York City that first revived the study of Northern slavery. Since then, fueled by educators, preservationists, and others, momentum has been building to recognize histories hidden in plain sight. Last year, Connecticut became the first New England state to formally apologize for slavery. In classrooms across the country, popularity has soared for educational programs on New England slavery designed at Brown University. In February, Emory University will hold a major conference on the role slavery’s profits played in establishing American colleges and universities, including in New England. And in Brookline, Mass., a program called Hidden Brookline is designing a virtual walking tour to illuminate its little-known slavery history: At one time, nearly half the town’s land was held by slave owners.
“What people need to understand is that, here in the North, while there were not the large plantations of the South or the Caribbean islands, there were families who owned slaves,” said Stephen Bressler, director of Brookline’s Human Relations-Youth Resources Commission. “There were businesses actively involved in the slave trade, either directly in the importation or selling of slaves on our shores, or in the shipbuilding, insurance, manufacturing of shackles, processing of sugar into rum, and so on. Slavery was a major stimulus to the Northern economy.”
Turning over the stones to find those histories isn’t just a matter of correcting the record, he and others say. It’s crucial to our understanding of the New England we live in now.
“The absolute amnesia about slavery here on the one hand, and the gradualness of slavery ending on the other, work together to make race a very distinctive thing in New England,” said Joanne Pope Melish, who teaches history at the University of Kentucky and wrote the book “Disowning Slavery: Gradual Emancipation and ‘Race’ in New England, 1780-1860.” “If you have obliterated the historical memory of actual slavery — because we’re the free states, right? — that makes it possible to turn around and look at a population that is disproportionately poor and say, it must be their own inferiority. That is where New England’s particular brand of racism comes from.”
Dismantling the myths of slavery doesn’t mean ignoring New England’s role in ending it. In the 1830s and ’40s, an entire network of white Connecticut abolitionists emerged to house, feed, clothe, and aid in the legal defense of Africans from the slave ship Amistad, a legendary case that went all the way to the US Supreme Court and helped mobilize the fight against slavery. Perhaps nowhere were abolition leaders more diehard than in Massachusetts: Pacifist William Lloyd Garrison and writer Henry David Thoreau were engines of the antislavery movement. Thoreau famously refused to pay his taxes in protest of slavery, part of a philosophy of civil disobedience that would later influence Martin Luther King Jr. But Thoreau was tame compared to Garrison, a flame-thrower known for shocking audiences. Founder of the New England Anti-Slavery Society and the newspaper The Liberator, Garrison once burned a copy of the US Constitution at a July Fourth rally, calling it “a covenant with death.” His cry for total, immediate emancipation made him a target of death threats and kept the slavery question at a perpetual boil, fueling the moral argument that, in time, would come to frame the Civil War.
But to focus on crusaders like Garrison is to ignore ugly truths about how unwillingly New England as a whole turned the page on slavery. Across the region, scholars have found, slavery here died a painfully gradual death, with emancipation laws and judicial rulings that either were unclear, poorly enforced, or written with provisions that kept slaves and the children born to them in bondage for years.
Meanwhile, whites who had trained slaves to do skilled work refused to hire the same blacks who were now free, driving an emerging class of skilled workers back to the lowest rungs of unskilled labor. Many whites, driven by reward money and racial hatred, continued to capture and return runaway Southern slaves; some even sent free New England blacks south, knowing no questions about identity would be asked at the other end. And as surely as there was abolition, there was “bobalition” — the mocking name given to graphic, racist broadsides printed through the 1830s, ridiculing free blacks with characters like Cezar Blubberlip and Mungo Mufflechops. Plastered around Boston, the posters had a subtext that seemed to boil down to this: Who do these people think they are? Citizens?
“Is Garrison important? Yes. Is it dangerous to be an abolitionist at that time? Absolutely,” said Melish. “What is conveniently forgotten is the number of people making a living snagging free black people in a dark alley and shipping them south.”
Growing up in Lincoln, Mass., historian Elise Lemire vividly remembers learning of the horrors of a slaveocracy far, far away. “You knew, for example, that families were split up, that people were broken psychologically and kept compliant by the fear of your husband or wife being sold away, or your children being sold away,” said Lemire, author of the 2009 book “Black Walden,” who became fascinated with former slaves banished to squatter communities in Walden Woods.
As she peeled back the layers, Lemire discovered a history rarely seen by the generations of tourists and schoolchildren who have learned to see Concord as a hotbed of antislavery activism. “Slaves [here] were split up in the same way,” she said. “You didn’t have any rights over your children. Slave children were given away all the time, sometimes when they were very young.”
In Lemire’s Concord, slave owners once filled half of town government seats, and in one episode town residents rose up to chase down a runaway slave. Some women remained enslaved into the 1820s, more than 30 years after census figures recorded no existing slaves in Massachusetts. According to one account, a former slave named Brister Freeman, for whom Brister’s Hill in Walden Woods is named, was locked inside a slaughterhouse shed with an enraged bull as his white tormentors laughed outside the door. And in Concord, Lemire argues, black families were not so much liberated as they were abandoned to their freedom, released by masters increasingly fearful their slaves would side with the British enemy. With freedom, she said, came immediate poverty: Blacks were forced to squat on small plots of the town’s least arable land, and eventually pushed out of Concord altogether — a precursor to the geographic segregation that continues to divide black and white in New England.
“This may be the birthplace of a certain kind of liberty,” Lemire said, “but Concord was a slave town. That’s what it was.”
If Concord was a slave town, historians say, Connecticut was a slave state. It didn’t abolish slavery until 1848, a little more than a decade before the Civil War. (A judge’s ruling ended legal slavery in Massachusetts in 1783, though the date is still hotly debated by historians.) It’s a history Connecticut author and former Hartford Courant journalist Anne Farrow knew nothing about — until she got drawn into an assignment to find the untold story of one local slave.
Once she started pulling the thread, Farrow said, countless histories unfurled: accounts of thousand-acre slave plantations and a livestock industry that bred the horses that turned the giant turnstiles of West Indian sugar mills. Each discovery punctured another slavery myth. “A mentor of mine has said New England really democratized slavery,” said Farrow. “Where in the South a few people owned so many slaves, here in the North, many people owned a few. There was a widespread ownership of black people.”
Perhaps no New England colony or state profited more from the unpaid labor of blacks than Rhode Island: Following the Revolution, scholars estimate, slave traders in the tiny Ocean State controlled between two-thirds and 90 percent of America’s trade in enslaved Africans. On the rolling farms of Narragansett, nearly one-third of the population was black — a proportion not much different from Southern plantations. In 2003, the push to reckon with that legacy hit a turning point when Brown University, led by its first African-American president, launched a highly controversial effort to account for its ties to Rhode Island’s slave trade. Today, that ongoing effort includes the CHOICES program, an education initiative whose curriculum on New England slavery is now taught in over 2,000 classrooms.
As Brown’s decision made national headlines, Katrina Browne, a Boston filmmaker, was on a more private journey through New England slavery, tracing her bloodlines back to her Rhode Island forebears, the DeWolf family. As it turned out, the DeWolfs were the biggest slave-trading family in the nation’s biggest slave-trading state. Browne’s journey, which she chronicled in the acclaimed documentary “Traces of the Trade: A Story from the Deep North,” led her to a trove of records of the family’s business at every point in slavery’s triangle trade. Interspersed among the canceled checks and ship logs, Browne said, she caught glimpses into everyday life under slavery, like the diary entry by an overseer in Cuba that began, “I hit my first Negro today for laughing at prayers.” Today, Browne runs the Tracing Center, a nonprofit to foster education about the North’s complicity in slavery.
“I recently picked up a middle school textbook at an independent school in Philadelphia, and it had sub-chapter headings for the Colonial period that said ‘New England,’ and then ‘The South and Slavery,’ ” said Browne, who has trained park rangers to talk about Northern complicity in tours of sites like Philadelphia’s Liberty Bell. “Since learning about my family and the whole North’s role in slavery, I now consider these things to be my problem in a way that I didn’t before.”
If New England’s amnesia has been pervasive, it has also been willful, argues C.S. Manegold, author of the new book “Ten Hills Farm: The Forgotten History of Slavery in the North.” That’s because many of slavery’s markers aren’t hidden or buried. In New England, one need look no further than a symbol that graces welcome mats, door knockers, bedposts, and all manner of household decor: the pineapple. That exotic fruit, said Manegold, is as intertwined with slavery as the Confederate flag: When New England ships came to port, captains would impale pineapples on a fence post, a sign to everyone that they were home and open for business, bearing the bounty of slave labor and sometimes slaves themselves.
“It’s a symbol everyone knows the benign version of — the happy story that pineapples signify hospitality and welcome,” said Manegold, whose book centers on five generations of slaveholders tied to one Colonial era estate, the Royall House and Slave Quarters in Medford, Mass., now a museum. The house features two carved pineapples at its gateposts.
By Manegold’s account, pineapples were just the beginning at this particular Massachusetts farm: Generation after generation, history at the Royall House collides with myths of freedom in New England — starting with one of the most mythical figures of all, John Winthrop. Author of the celebrated “City Upon a Hill” sermon and first governor of the Massachusetts Bay Colony, Winthrop not only owned slaves at Ten Hills Farm, but in 1641, he helped pass one of the first laws making chattel slavery legal in North America.
When the house passed to the Royalls, Manegold said, it entered a family line whose massive fortune came from slave plantations in Antigua. Members of the Royall family would eventually give land and money that helped establish Harvard Law School. To this day, the law school bears a seal borrowed from the Royall family crest, and for years the Royall Professorship of Law remained the school’s most prestigious faculty post, almost always occupied by the law school dean. It wasn’t until 2003 that an incoming dean — now Supreme Court Justice Elena Kagan — quietly turned the title down.
Kagan didn’t publicly explain her decision. But her actions speak to something Manegold and others say could happen more broadly: not just inserting footnotes to New England heritage tours and history books, but truly recasting that heritage in all its painful complexity.
“In Concord,” Lemire said, “the Minutemen clashed with the British at the Old North Bridge within sight of a man enslaved in the local minister’s house. The fact that there was slavery in the town that helped birth American liberty doesn’t mean we shouldn’t celebrate the sacrifices made by the Minutemen. But it does mean New England has to catch up with the rest of the country, in much of which residents have already wrestled with their dual legacies of freedom and slavery.”
Francie Latour is an associate editor at Wellesley magazine and a former Globe reporter.
© Copyright 2010 Globe Newspaper Company.
Ride-sharing services can provide not only a very personalized mobility experience but also ensure efficiency and sustainability via large-scale ride pooling. Large-scale ride-sharing requires mathematical models and algorithms that can match large groups of riders to a fleet of shared vehicles in real time, a task not fully addressed by current solutions. We present a highly scalable anytime optimal algorithm and experimentally validate its performance using New York City taxi data and a shared vehicle fleet with passenger capacities of up to ten. Our results show that 2,000 vehicles (15% of the taxi fleet) of capacity 10 or 3,000 of capacity 4 can serve 98% of the demand within a mean waiting time of 2.8 min and mean trip delay of 3.5 min.
Ride-sharing services are transforming urban mobility by providing timely and convenient transportation to anybody, anywhere, and anytime. These services present enormous potential for positive societal impacts with respect to pollution, energy consumption, congestion, etc. Current mathematical models, however, do not fully address the potential of ride-sharing. Recently, a large-scale study highlighted some of the benefits of car pooling but was limited to static routes with two riders per vehicle (optimally) or three (with heuristics). We present a more general mathematical model for real-time high-capacity ride-sharing that (i) scales to large numbers of passengers and trips and (ii) dynamically generates optimal routes with respect to online demand and vehicle locations. The algorithm starts from a greedy assignment and improves it through a constrained optimization, quickly returning solutions of good quality and converging to the optimal assignment over time. We quantify experimentally the tradeoff between fleet size, capacity, waiting time, travel delay, and operational costs for low- to medium-capacity vehicles, such as taxis and van shuttles. The algorithm is validated with ∼3 million rides extracted from the New York City taxicab public dataset. Our experimental study considers ride-sharing with rider capacity of up to 10 simultaneous passengers per vehicle. The algorithm applies to fleets of autonomous vehicles and also incorporates rebalancing of idling vehicles to areas of high demand. This framework is general and can be used for many real-time multivehicle, multitask assignment problems.
New user-centric services are transforming urban mobility by providing timely and convenient transportation to anybody, anywhere, and anytime. These services have the potential for a tremendous positive impact on personal mobility, pollution, congestion, energy consumption, and thereby quality of life. The cost of congestion in the United States alone is roughly $121 billion per year or 1% of GDP (1), which includes 5.5 billion hours of time lost to sitting in traffic and an extra 2.9 billion gallons of fuel burned. These estimates do not even consider the cost of other potential negative externalities such as the vehicular emissions (greenhouse gas emissions and particulate matter) (2), travel-time uncertainty (3), and a higher propensity for accidents (4). Recently, the large-scale adoption of smart phones and the decrease in cellular communication costs has led to the emergence of a new mode of urban mobility, namely mobility-on-demand (MoD) systems, led by companies such as Uber, Lyft, and Via. These systems are able to provide users with a reliable mode of transportation that is catered to the individual and improves access to mobility to those who are unable to operate a personal vehicle, reducing the waiting times and stress associated with travel.
One of the major inefficiencies of current MoD systems is their capacity limitation, typically restricted to two passengers. Our method applies not only to shared taxis but also to shared vans and minibuses. A recent study in New York City showed that up to 80% of the taxi trips in Manhattan could be shared by two riders, with an increase in the travel time of a few minutes (5). However, the method and analysis of ref. 5 was (i) limited to two riders for an optimal allocation (three with heuristics), (ii) intractable for larger number of passengers, and (iii) did not allow for allocation of additional riders after the start of a trip. There are no studies of this scale that quantify the benefits of larger-scale ride pooling, mainly due to the lack of efficient and scalable algorithms for this problem, both of which we address in this work.
Much of the fleet management literature for MoD systems considers the case of ride-sharing without pooling requests, focusing on fluid approximations (6), queuing based formulations (7), case studies in specific regions [e.g., Singapore (8)], and operational considerations for fleet managers (9). With the growing interest and rapid developments in autonomous vehicles, there is also an increasing focus on autonomous MoD systems (6, 9, 10). However, none of these works considered the ride-pooling problem of servicing multiple rides with a single trip. The ride-pooling problem is more related to the vehicle-routing problem and the dynamic pickup and delivery problem (11⇓⇓⇓–15), where spatiotemporally distributed demand must be picked up and delivered within prespecified time windows. A major challenge when addressing this problem is the need to explore a very large decision space, while computing solutions fast enough to provide users with the experience of real-time booking and service.
Here, we consider the problem of using a fleet of vehicles with varying passenger capacities, and, in contrast to ref. 5, we address both the problems of assigning vehicles to matched passengers and rebalancing—or repositioning—the fleet to service demand. We show how the unified problem of passenger and vehicle assignment can be solved in a computationally efficient manner at a large scale, thereby demonstrating the capability to operate a real-time MoD system with multiple service tiers (shared-taxi, shared-vans, and shared-buses) of varying capacity.
Whereas previous approaches to this problem have focused on heuristic-based solutions (16⇓–18), we present a reactive anytime optimal algorithm. That is, an algorithm that efficiently returns a valid assignment of travel requests to vehicles and then refines it over time, converging to an optimal solution. If enough computational resources are available, the optimal assignment for the current requests and time would be found; otherwise, the best solution found so far is returned.
Traditional approaches that rely on an integer linear program (ILP) formulation, such as ref. 19, also provide anytime guarantees for the multivehicle-routing problem. However, in contrast to our approach, their applicability is limited to small problem instances, which in ref. 19 was 32 requests and 4 vehicles, with a computation cost of several minutes. We also rely on an ILP formulation, but because we do not explicitly model the edges of the road network in the ILP, our approach scales to much larger problem instances. We observe that instances such as New York City, with thousands of vehicles, requests, and road segments, can be solved in real time.
Our approach decouples the problem by first computing feasible trips from a pairwise shareability graph (5) and then assigning trips to vehicles. We show that this assignment can be posed as an ILP of reduced dimensionality. The framework allows for flexibility in terms of prescribing constraints such as (but not limited to) maximum user waiting times and maximum additional delays due to sharing a ride. We also extend the method to proactively rebalance the vehicle fleet by moving idle vehicles to areas of high demand. In summary, we present a framework for solving the real-time ride-pooling problem with (i) arbitrary numbers of passengers and trips, (ii) anytime optimal rider allocation and routing dependent on the fleet location, and (iii) online rerouting and assignment of riders to existing trips.
We quantify experimentally the performance tradeoffs between fleet size, capacity, waiting time, travel delay, and operational costs for low- and medium-capacity vehicles (such as taxis, vans, or minibuses) in a large urban setting. Detailed experimental results are presented for a subset of ∼3 million rides extracted from the New York City taxicab public dataset. We show that 3,000 vehicles with a capacity of 2 and 4 could serve 94 and 98% of the demand with a mean waiting time of 3.2 and 2.7 min, and a mean delay of 1.5 and 2.3 min, respectively. To achieve 98% service rate, with comparable waiting time (2.8 min) and delay (3.5 min), a fleet of just 2,000 vehicles with a capacity of 10 was required. This fleet size is 15% of the active taxis in New York City (Movie S1). We also show that our approach is robust with respect to the density of requests and could therefore be applied to other cities.
Our system runs in real time and is particularly suited to autonomous vehicle fleets that can continuously reroute based on real-time requests. It can also rebalance idle vehicles to areas with high demand and is general enough to be applied to other multivehicle, multitask assignment problems.
Passenger Assignment and Vehicle Routing
We consider a fleet of vehicles of capacity , the maximum number of passengers each vehicle can have at any given time. We address the problems of both optimally assigning online travel requests to vehicles and finding optimal routes for the vehicle fleet. Each travel request consists of the time of request, a pickup location and a drop-off location.
We propose an anytime optimal algorithm for batch assignment of a set of requests to a set of vehicles , which minimizes a cost function , satisfies a set of constraints , and allows for multiple passengers per vehicle. A passenger is a past request that has been picked up by a vehicle and that is now en route to its destination. We denote by the set of passengers for vehicle . In a second step, the method also allows to rebalance the fleet of vehicles by driving idle vehicles to areas of high demand, where those vehicles are likely to be required in the future. A schema of the method is shown in Fig. 1.
Schematic overview of the proposed method for batch assignment of multiple requests to multiple vehicles of capacity . The method consists of several steps leading to an integer linear optimization that provides an anytime optimal assignment. (A) Example of a street network with four requests (orange human, origin; red triangle, destination) and two vehicles (yellow car, origin; red triangle, destination of passenger). Vehicle 1 has one passenger, and vehicle 2 is empty. (B) Pairwise shareability RV-graph of requests and vehicles. Cliques of this graph are potential trips. (C) RTV-graph of candidate trips and vehicles which can execute them. A node (yellow triangle) is added for requests that cannot be satisfied. (D) Optimal assignment given by the solution of the ILP, where vehicle 1 serves requests 2 and 3 and vehicle 2 serves requests 1 and 4. (E) Planned route for the two vehicles and their assigned requests. In this case, no rebalancing step is required because all requests and vehicles are assigned.
Our formulation is flexible with respect to physical and performance-related constraints that might need to be added. In our implementation, we consider the following. (i) For each request , the waiting time , given by the difference between the pickup time and the request time , must be below a maximum waiting time , for example, 2 min. (ii) For each passenger or request the total travel delay must be lower than a maximum travel delay , for example, 4 min, where is the drop-off time and is the earliest possible time at which the destination could be reached if the shortest path between the origin and the destination was followed without any waiting time. The total travel delay includes both the in-vehicle delay and the waiting time. Finally, (iii) for each vehicle , we consider a maximum number of passengers, , for example, capacity 10.
We define the cost of an assignment as the sum of delays (which includes the waiting time) over all assigned requests and passengers, plus a large constant for each unassigned request. Given an assignment of requests to vehicles, we denote by the set of requests that have been assigned to some vehicle and the set of unassigned requests, due to the constraints or the fleet size. Formally,
This constrained optimization problem is solved via four steps (Fig. 1), which are: computing a pairwise request-vehicle shareability graph (RV-graph) (Fig. 1B); computing a graph of feasible trips and the vehicles that can serve them (RTV-graph) (Fig. 1C); solving an ILP to compute the best assignment of vehicles to trips (Fig. 1D); and rebalancing the remaining idle vehicles (Fig. 1E).
Given a network graph with travel times, we consider a function for single-vehicle routing. For a vehicle , with passengers , this function returns the optimal travel route to satisfy requests . This route minimizes the sum of delays subject to the constraints (waiting time, delay, and capacity). For low-capacity vehicles, such as taxis, the optimal path can be computed via an exhaustive search. For vehicles with larger capacity, heuristic methods such as Lin–Kernighan (20), Tabu search (21), or simulated annealing (22) may be used. Fig. 2, Right shows the optimal route for a vehicle with four passengers and an additional request.
(A) Snapshot: 2,000 vehicles, capacity of 4 ( min, Wednesday, 2000 hours). Vehicle in the fleet are represented at their current positions. Colors indicate number of passengers (0: light blue; 1: light green; 2: yellow; 3: dark orange; 4: dark red); 39 rebalancing vehicles are displayed in dark blue—mostly in the upper Manhattan returning to the middle. (B) Close view of the scheduled path for a vehicle (dark red circle) with four passengers, which drops one off, picks up a new one (blue star), and drops all four. Drop-off locations are displayed with inverted triangles. See Movie S1 for a complete simulation.
The RV-graph (Fig. 1B) represents which requests and vehicles might be pairwise-shared and builds on the idea of shareability graphs proposed by ref. 5 but also includes the vehicles at their current state. Two requests and are connected if an empty virtual vehicle starting at the origin of one of them could pick up and drop off both requests while satisfying the constraints . A cost is associated to each edge . Likewise, a request and a vehicle are connected if the request can be served by the vehicle while satisfying the constraints , as given by . The edge is denoted by .
Next, the cliques of the RV-graph—or regions for which its induced subgraph is complete—are explored to find feasible trips and compute the RTV-graph (Fig. 1C). A trip is a set of requests to be combined in one vehicle. A trip is feasible if all of the requests can be picked up and dropped off by some vehicle, while satisfying the constraints .
This step computes feasible trips. There might be several trips of varying size that can service a particular request. In addition, more than one vehicle might be able to service a trip. The assignment step will later ensure that each request and vehicle are assigned to a maximum of one trip. The RTV-graph contains two types of edges: (i) edges , between a request and a trip that contains request (i.e., ), and (ii) edges , between a trip and a vehicle that can execute the trip (i.e., is feasible). The cost , sum of delays, is associated to each edge e(T,v).
The algorithm to compute the feasible trips and edges proceeds incrementally in trip size for each vehicle, starting from the request-vehicle edges in the RV-graph (SI Appendix, Algorithm 1). For computational efficiency, we rely on the fact that a trip only needs to be checked for feasibility if there exists a vehicle for which all of its subtrips (obtained by removing one request) are feasible and have been added as edges to the RTV-graph.
Next, we compute the optimal assignment of vehicles to trips. This optimization is formalized as an ILP, initialized with a greedy assignment obtained directly from the RTV-graph. To compute the greedy assignment , trips are assigned to vehicles iteratively in decreasing size of the trip and increasing cost (sum of travel delays). The idea is the maximize the amount of requests served while minimizing the cost (SI Appendix, Algorithm 2).
The optimization problem is formulated in Algorithm 1. A binary variable is introduced for each edge between a trip and a vehicle in the RTV-graph. If , then vehicle is assigned to trip . We denote by the set of indices for which an edge exists in the RTV-graph, i.e., the set of possible pickup trips. An additional binary variable is introduced for each request . These variables are active, i.e., , if the associated request can not be served by any vehicle and is ignored. The set of variables is then
The cost terms are the sum of delays for trip and vehicle pickup (stored in the edge of the RTV-graph) and is a large constant to penalize ignored requests.
Two types of constraints are included. Line 3 in Algorithm 1 imposes that each vehicle is assigned to one trip at most. Line 4 in Algorithm 1 imposes that each request is assigned to a single vehicle or ignored. In these constraints, three sets appear. The set of trips that can be serviced by a vehicle , or edges , is . The set of trips that contain request , or edges , is . The set of vehicles that can service trip , or edges