Multi-Objective Analysis of Ridesharing in Automated Mobility-on-Demand

Transcription

1 Robotics: Science and Systems 018 Pittsbugh, PA, USA, June 6-30, 018 Multi-Objective Analysis of Rideshaing in Automated Mobility-on-Demand Michal Čáp and Javie Alonso-Moa Dept. of Cognitive Robotics, 3ME, TU Delft, the Nethelands Abstact Self-diving technology is expected to enable the ealization of lage-scale mobility-on-demand systems that employ massive ideshaing. The technology is being celebated as a potential cue fo uban congestion and othes negative extenalities of individual automobile tanspotation. In this pape, we quantify the potential of ideshaing with a fleet of autonomous vehicles by consideing all possible tade-offs between the quality of sevice and opeation cost of the system that can be achieved by shaing ides. We fomulate a multi-objective fleet outing poblem and pesent a solution technique that can compute Paeto-optimal fleet opeation plans that achieve diffeent tadeoffs between the two objectives. Given a set of equests and a set of vehicles, ou method can ecove a tade-off cuve that quantifies the potential of ideshaing with given fleet. We povide a fomal optimality poof and demonstate that the poposed method is scalable and able to compute such tade-off cuves fo instances with hundeds of vehicles and equests optimally. Such an analytical tool helps with systematic design of shaed mobility system, in paticula, it can be used to make pincipled decisions about the equied fleet size. I. INTRODUCTION Ubiquitous connectivity and apid advances in automation ae expected to evolutionize tanspotation of both goods and people. In paticula, uban mobility is being apidly tansfomed due to the emegence of new foms of on-demand tanspotation options, exemplified by sevices such as Ube o Lyft. In the nea futue, self-diving technology is expected to enable the ealization of lage-scale automated mobilityon-demand (AMoD) systems that povide pesonal point-topoint tanspotation that is as comfotable and affodable as taveling by pivate ca, but uses a smalle fleet of shaed vehicles, which tanslates to eduction of paking capacity equiements [6, 1, 17]. The benefits of automated on-demand mobility can be futhe inceased by employing ideshaing, whee multiple passenges taveling in a simila diection can be matched and tanspoted in one vehicle. By employing fewe vehicles to seve fixed tanspotation demand, ideshaing has potential to educe enegy consumption, congestion and taffic-elated pollution [16, ]. Yet, ou ability to quantify the extent of those benefits and to identify unde what cicumstances can those benefits be achieved is vey limited. We will efe to mobility-on-demand systems that use both autonomous vehicles and ideshaing as Shaed Automated Mobility-on-Demand (SAMoD) systems. When designing a SAMoD system, we ae typically inteested in two main pefomance metics. On the one hand, the uses of the system ae inteested in the quality of sevice - uses of the system desie to minimize sevice discomfot, i.e, they pefe to be deliveed to thei destination as fast as possible. On the othe hand, the entity opeating the system is inteested in the minimization of the opeation cost - this usually tanslates in the aim to minimize the fleet size and the total enegy consumed by the system. The sevice discomfot and opeation cost objectives ae usually in conflict, i.e, both cannot be minimized simultaneously. Instead, impovement in one citeia must be taded fo degadation in the othe citeia. Fo example, on the one hand, use discomfot is minimized by matching each equest with a dedicated vehicle. And, on the othe hand, the opeation cost can be educed by matching multiple equest to a single vehicle, which in tun inceases the use discomfot. Even though the two objectives ae fundamentally intetwined on multiple levels, the majoity of the existing wok in SAMoD only consides one of the objectives individually and assumes a fixed fleet with a known numbe of vehicles. In paticula, thee is no pincipled study on how the two objectives inteact on the opeational level. In this pape we model ideshaing as a multi-objective vehicle fleet outing poblem with two citeia: to imize sevice quality and to minimize opeation cost. In contast to single objective optimization, multi-objective optimization poblems typically do not have one optimal solution. Instead, we ae inteested in a set of Paeto-optimal solutions [10]. A solution is called Paeto-optimal if thee is no othe solution that would achieve bette pefomance in both consideed objectives simultaneously. When the Paeto-optimal solutions ae epesented on the objective plane, with one axis epesenting the value of discomfot and the othe axis the opeation cost, we obtain a Paeto cuve that gaphically descibes the best attainable tade-offs between the two objectives. Such a tadeoff cuve epesents the fundamental limits of ideshaing fo a paticulaity poblem configuation at hand. In othe wods, thee is no ideshaing stategy that pojects below this cuve. A. Contibution This pape has two main contibutions. Fistly, we povide a fomalization of ideshaing as a vehicle fleet outing poblem with two competing objectives. Secondly, we design a scalable solution method fo the poblem. The method can geneate epesentative Paeto-optimal solutions fo poblem instances consisting of a set of equests and a set of vehicles and

2 consequently ecove the shape of the Paeto cuve. Moeove, we fomally pove the optimality of the method. Finally, we apply the method to compute the shape of Paeto cuves both fo synthetic poblems and fo a collection of 47 histoical taxi tips in Manhattan. The poposed analytical tool helps with systematic design of shaed mobility system, in paticula, it can be used to make pincipled decisions about the equied fleet size. Yet, the poposed method is geneal and not limited to the analysis of SAMoD systems. It could also be employed to aid systematic design of othe multi-obot multi-task assignment poblems that include outing of a lage vehicle fleet and tight pefomance constaints. B. Related Wok The ealy woks in AMoD focused on the development of models and algoithmic tools fo on-demand fleets with single-occupancy vehicles [6, 14, 17, 0, 15]. Howeve, one of the main pomises of on-demand systems is the ability to implement massive ideshaing. This is, to match multiple customes, which ae taveling in a simila diection, to a single vehicle. This tanslates to a significant eduction in the numbe of vehicles on the oads. Rideshaing was taditionally fomalized in the famewok of Vehicle Routing Poblems (VRP) [19], typically as a specific vaiant of VRP with Pickup and Delivey [4, 5, 3, 8] o a Dial a Ride Poblem (DARP) [7, 5]. Yet, the existing exact VRP methods focus on instances with tens of vehicles and equests [13, 9] and as such they ae not applicable to management o analysis of lage-scale fleets that often consist of thousands of vehicles and equests. The potential fo lage-scale ideshaing was studied using the shaeability netwok model [16] evealing that up to 80% of the taxi tips in Manhattan could be paiwise shaed such that the tavel time is inceased by no moe than a couple of minutes. The analysis was late extended to othe cities [18]. The model assumption of imum two passenges in a vehicle was late lifted in [], whee Alonso-Moa et al. poposed a scalable technique fo finding optimal assignment of equests to a given fixed fleet of vehicles such that the aveage tavel delay is minimized. The poblem of pedictive outing in MoD systems has been ecently also addessed [1, 11]. In this pape, we boow seveal algoithmic ideas fom [] to design a multi-objective fleet outing algoithm that can be used to study the tade-off between the opeation cost and the tavel discomfot expeienced by the uses of the system. Specifically, the vehicle-goup assignment (VGA) component pesented in Section IV can be seen as a moe concise efomulation of the method by Alonso-Moa et al. [] that, e.g., avoids RV and RTV gaph constuction, which makes the VGA algoithm simple to implement and to analyze. II. PROEM STATEMENT Conside a fleet of vehicles that has to sevice a given set of tanspotation equests. We study the poblem of finding a collection of Paeto-optimal system opeation plans that epesent vaying tade-offs between the sevice quality and the opeation cost. A. Vehicle Fleet and Tanspotation Demand Thee is a fleet of m 1 vehicles that can be used to seve the tanspotation demand. Each vehicle v stats at a given initial position o veh v at time t veh v. Fo convenience, we define the set of vehicle indices as V = {1,..., m}. The tanspotation demand is modeled as a set of n tanspotation equests. The i-th equest is a tuple i = (o i, d i, t i ), whee o i is the oigin of equest i, d i is the destination of equest i, and t i is the time when the equest i was announced. The set of all equest indices is denoted R = {1,..., n}. Let tt(x 1, x ) denote the tavel time fom point x 1 to x, whee each point epesents an oigin of a equest, a destination of a equest, o an initial position of a vehicle. B. Vehicle Plan A plan fo a vehicle v denoted π v = ((o veh v, t veh v ), o 1, o,...) encodes the initial spatio-tempoal position of the vehicle and a sequence of odes that the vehicle should follow, whee each ode o i is eithe to pickup a equest, o to dop-off a equest. Fo a plan to be valid, fo evey ode to pickup equest, the plan must contain an ode to dop-off the equest late in the sequence, and vice vesa, fo evey dop-off ode, the plan must contain a pickup ode ealie in the sequence. The set of all equests seved by a valid vehicle plan π v is denoted as eq(π v ). The set of all valid plans fo vehicle v is denoted by Π v. C. System Plan A system plan assigns a paticula plan to each individual vehicle. Fo a system plan to be valid, each vehicle plan must be valid and futhemoe, all equests must be seved by exactly one vehicle. Thus, the set of all valid system plans is denoted by Π and defined as { Π := (π 1 Π 1,..., π m Π m ) : eq(π v ) = R and } i, j V i j : eq(π i ) eq(π j ) =. Obseve that if thee ae no futhe constaints, then a valid system plan exists fo any tanspotation demand. Consequently, a system plan that minimizes any desied cost citeion is guaanteed to always exist. D. Optimization Citeia Thee ae two types of agents that have inteest in the choice of the system plan: a) the uses of the system and b) the opeato of the system. While each of the uses is inteested in imization of the quality of sevice, the opeato is inteested in the minimization of the opeating cost. Futhemoe, we will use the tem sevice discomfot to epesent the negative of sevice quality.

3 We assume that the discomfot peceived by use who issued equest depends only on the plan of the vehicle that seves the equest. Let q (π v ) be a chosen sevice discomfot metic that measues the discomfot expeienced by the custome who issued equest when the equest is seved by vehicle v that follows plan π v. To declutte the notation, we define the same metic ove system plans and use q (π) as a shothand fo q (π v ), whee π v π is the plan of vehicle that seves equest. We will define the discomfot metic as the time that elapses between the equest announcement and the dop-off at the desied destination, i.e., q (π v ) := t dopoff (π v ) t, whee t dopoff (π v ) is the time when the equest is dopped off unde plan π v. Futhemoe, let s v (π v ) denote the cost that the system opeato has the bea when vehicle v executes plan π v. Fo simplicity, we will define s v (π v ) to be equal to the time that the vehicle v spends in opeation when following the plan π v. E. Baseline Plan The system plan that minimizes the total use discomfot, o equivalently, the aveage dop-off time, is efeed to as a baseline imum comfot plan π and it is defined as π := agmin π Π q (π). R Futhemoe, let q := q (π ) be the discomfot that equest expeiences unde the baseline plan. Note that in some of ou taget scenaios, a baseline plan can be constucted tivially. Conside, fo example, a system with a depot stoing m n vehicles that need to seve a set of n equests. Then, the baseline plan is obtained by letting each equest to be seved by a dedicated vehicle. In a peeto-pee ideshaing scenaio, each passenge can eithe ealize the tip using his own pivate vehicle o ideshae and tavel in pivate ca of anothe passenge. Thus, fo n individual tips to be made, thee ae m = n available vehicles. The baseline plan then coesponds to a situation in which no two ides ae shaed. F. Tading Discomfot fo Opeation Cost We can now poceed to the fomalization of the poblem of finding the set of system plans that tade-off discomfot fo opeation cost. The discomfot induced by ideshaing fo a paticula equest can be measued by consideing the diffeence in discomfot elative to the baseline assignment. Let δ (π v ) := q (π v ) q epesent the induced discomfot that the equest expeiences unde vehicle plan π v. Now, we can define the sevice quality metic c s : Π R 0 as c s (π = (π 1,..., π m )) := δ (π v ) eq(π v) and opeation cost metic c o : Π R 0 as c o (π = (π 1,..., π m )) := s v (π v ). The above sevice quality metic aims to epesents socalled social optimum, i.e, it aims at minimization of total use discomfot, which is equivalent to minimization of aveage discomfot acoss all uses. This objective necessaily leads to solutions that distibute the discomfot unequally among individual equests. In esult, some customes suffe fom induced discomfot moe than othes. Howeve, in many pactical systems, the vaiance in induced discomfot as assigned to individual equests must be contolled. Fo example, human passenges ae paticulaly sensitive to discomfot and ae likely to switch to an altenative mode of tanspot if the induced discomfot is deemed significantly highe than the discomfot that othe uses need to bea. Theefoe, we also intoduce a bound on imum induced discomfot of each tanspotation equest and use δ to denote the bound on imum allowed induced discomfot assigned to a single tanspotation equest. Ou goal is to study the dynamics of the inteaction between the induced discomfot and opeation cost objectives subject to imum induced discomfot constaints on individual equests. In paticula, we would like to know what ae the best possible tade-offs between the two citeia that can be potentially achieved. This can be expessed in a famewok of multi-objective optimization as follows. Poblem 1 (Multi-objective Fleet Routing). Given a fleet of vehicles, a set of tanspotation equests, and a tavel time function, solve agmin π Π (c s (π), c o (π)) δ (π) δ, R. A solution to the above poblem is a set of all Paeto-optimal system plans, each epesenting a paticula tade-off between the sevice discomfot and opeation cost. III. SOLUTION APPROACH Finding all Paeto-optimal solutions fo a lage-scale discete multi-objective optimization poblem, such as the one fomulated in Poblem 1, is not feasible in pactice [10]. In ode to obtain an appoximation of the shape of the Paeto font fo ou poblem, we apply a popula solution technique known as scalaization [10]. In this appoach we solve a family of single-objective optimization poblems paametized by a weight paamete w, each asking fo a system plan that minimizes a convex combination of the two consideed objectives: Poblem (Single-objective Fleet Routing). Given a weight paamete w [0, 1] solve agmin π Π w c s (π) + (1 w) c o (π) δ (π) δ, R. An optimal solution of the above single-objective optimization poblem is a Paeto-optimal solution fo Poblem 1. By finding a solution to the scalaized vesion of the poblem fo a

4 sequence of weights w 0 = 0 < w 1 <... < w k 1 < w k = 1, we can ecove a collection of epesentative Paeto-optimal solutions that can be used to ecove the shape of the Paeto font. It should be noted, howeve, that although all solutions of scalaized poblem ae Paeto-optimal solutions, the opposite does not hold. In paticula, the scalaization technique is able to geneate all Paeto-optimal solutions that lie on the convex hull of the feasible set in the objective plane, but it is unable to geneate Paeto optimal solutions that lie stictly inside the convex hull. Howeve, such an appoximation is often sufficient, because it is capable of descibing the dynamics of the inteaction between the two citeia and moeove, thee is typically a good choice of epesentative solutions on the convex hull to choose fom. Most impotantly, using scalaization, one can efficiently ecove the shape of Paetofont even fo lage poblem instances, as we will discuss in the following section. IV. VEHICLE-GROUP ASSIGNMENT METHOD The single-objective optimization poblem stated in Poblem is a specific vaiant of a vehicle outing poblem with multiple vehicles and time windows, a class of poblems that ae known to be NP-had. In esult, it cannot be solved efficiently in geneal. In this section, we intoduce a method, that we will efe to as Vehicle-Goup Assignment Method (VGA), that can solve many lage-scale eal-wold instances of the poblem optimally in pactical time. Vehicle-goup assignment method finds optimal solution to Poblem by geneating all possible goups of equests that each vehicle can seve and then by finding an optimal assignment of such goups to individual vehicles. We will efe to a set of equest as a goup. We say that a goup G R is feasible fo vehicle v if the vehicle can seve all equests fom the goup without violating the imum induced discomfot constaints. If a goup G is feasible, we use c(v, G) to denote the cost of minimum-cost plan fo vehicle v that seves all equests in goup G. The actual optimal plan fo vehicle v to seve equests in goup G is denoted as π(v, G). In ode to detemine the feasibility and cost of vehicle v seving all equests in goup G, one needs to solve a vehicle outing poblem fo a single vehicle stating at a given initial position and visiting pickup and destination positions of each equest G, such that a) the pickup point of each equest is visited befoe the dop-off point and b) the imum dop-off delay of each equest is not exceeded. Then, we can define F v P(R) to be a set of all feasible goups fo vehicle v, whee P(R) denotes the set of all subsets, i.e., the powe set, of the set R. A useful popety of goup feasibility, initially obseved in [], is the following: Fo a goup to be feasible, all its subgoups must be feasible as well. Moe fomally, fo any vehicle v, we have G F v only if G G : G F v. This obsevation can be exploited to design a pocedue that iteatively geneates the sets F 1 v, F v,... containing feasible Algoithm 1: Iteative geneation of goups fo vehicle v. The boolean-valued function feasible(v, G) evaluates to tue, if vehicle v can seve all equests in goup G without violating the imum induced discomfot constaints. 1 F 0 v { }; F 1 v ; 3 fo i R do 4 if feasible(v, {i}) then 5 F 1 v F 1 v {i}; 6 k = ; 7 while Fv k 1 do 8 Fv k ; 9 foall G Fv k 1, Fv 1 do 10 if G G {}, G = k 1 : G Fv k 1 and feasible(i, G {}) then 11 Fv k Fv k {G {}}; 1 k k + 1; 13 F v F 0 v F 1 v F v F k v ; goups of size 1,,... fo vehicle v. The pseudocode of the goup geneation pocedue is given in Algoithm 1. Afte feasible goups have been geneated fo all vehicles, we need to choose a single goup fo each vehicle such that evey equest is seved by exactly one vehicle. A vehicle-goup assignment, typically denoted by a, is a mapping V P(R). Fo example, an assignment a = {(1, {, 3}), (, {1}), (3, )} epesents the fact that vehicle 1 will seve equests and 3, vehicle will seve equest 1 and vehicle 3 will be idle. The minimum-cost vehicle-goup assignment a can be obtained by solving the following optimization poblem: Poblem 3 (Vehicle-Goup Assignment). Given a set of feasible goups and a goup cost function fo each vehicle, solve agmin a(1) F 1,...,a(m) F m c(v, a(v)) a(i) a(j) = i, j V, i j. Then, the optimal system plan π = (π 1,..., π m ) can be ecoveed by taking π 1 = π(1, a (1)),..., π m = π(m, a (m)). To solve Poblem 3 using off-the-shelf ILP solves, we can make the following staightfowad convesion to a binay intege linea pogam: agmin {x v,g } x v,g c(v, G) g F v G F v x v,g = 1 v V G F v x v,g 1 Fv () = 1 R x v,g {0, 1} v V, G F v,

5 whee 1 S (x) is the indicato function, i.e., 1 S (x) = 1 if x S and 1 S (x) = 0 othewise. The decision vaiables x v,g epesent all possible vehiclegoup assignments. The fist constaints enfoces that thee is exactly one goup assigned to each vehicle, while the second constaint enfoces that evey equest is assigned to exactly one vehicle. If x v,g = 1 in the optimal solution, then we have a (v) = G. The vehicle-goup assignment fomulation tuns out to be beneficial when the constaints on imum dop-off delay become tight. Such constaints effectively eliminate goups containing equests that ae fa away fom the vehicle, because the vehicle cannot aive to the equest in time. Futhemoe, in some settings, such constaints will also effectively eliminate fomation of lage goups of equests. V. THEORETICAL ANALYSIS The Vehicle-Goup Assignment method is an optimal solution algoithm fo Poblem, as stated by the following theoem. Theoem 4. If and only if a is a solution to Poblem 3, then (π(1, a (1)),..., π(m, a (m))) is a solution to Poblem. Poof: Recall the definition of Poblem : agmin π Π w c s (π) + (1 w) c o (π) δ (π) δ, R and define c w v (π v ) := w eq(v) δ (π v ) + (1 w) s v (π v ). Then, the objective citeion can be expessed as a sum of cost functions ove single-vehicle plans as follows: w c s (π) + (1 w) c o (π) = w eq(v) δ (π v ) + (1 w) s v(π v = (w ) eq(v) δ (π v ) + (1 w) s v (π v ) = cw v (π v ). Similaly, the constaint δ (π) δ, i R, can be equivalently expessed as δ (π v ) δ, eq(v) v V. Recall the definition of the set of all valid system plans Π and make the constaint focing that evey equest is seved by at most one vehicle explicit. We obtain the following efomulation of the above poblem: agmin π 1 Π 1,...,π m Π m c w v (π v ) δ (π v ) δ, eq(π v ) v V. eq(π i ) eq(π j ) = i, j V, i j. Define Π v (G) := {π Π v : eq(v) = G}, f v (G) := π Π v (G) : eq(π v ) : δ (π) δ, π v (G) := min π Π cw v (π) s.t. δ (π v ) δ, eq(π v ), v(g) c v (G) := c w v (π v (G)). Assume abitay patitioning of equests to m disjoint goups. That is, let G 1 R,..., G m R such that i, j V, i j : G i G j = and G v = R. Given such patitioning, the optimal system plan π(g 1,..., G m ) = (π 1,..., π m) is a solution to agmin π 1 Π 1(G 1),...,π m(g m) Π m c w v (π v ) δ (π v ) δ, eq(v) v V. Obseve that the optimization poblem is decoupled and thus it is feasible if and only if f v (G v ). If it is feasible, we can obtain the optimal value fo each optimization vaiable π v, v V independently as π v = agmin c w π v Πv(Gv) v (π v) δ (π v) δ, eq(v) = π v (G). The minimum of the objective value can be obtained as cw v (π v ) = c v(g v ). Now we pove the two diections of the equality. 1) By contadiction. Let π 1,..., π m be a solution to Poblem. Now assume that a is a solution to Poblem 3, but v V, such that eq(π v) a v. This implies that thee is anothe patitioning to goups that admits lowe value of objective function. This is impossible because the solution to Poblem 3 is a patitioning that minimizes the objective function. ) By contadiction. Let a 1,..., a m be a solution to Poblem 3. Now assume that π is a solution to Poblem, but v V, such that eq(π v) a v. Thee ae two possibilities: a) It holds that v V : eq(π v) = a v. This implies, that π is not an optimal plan given patitioning a. This is impossible because given a paticula feasible patitioning the two fomulations have been shown to be equivalent. b) It holds that v V : eq(π v) a v, i.e., the optimal solution to Poblem lies in diffeent patitioning than the patitioning coesponding to the solution to Poblem 3. This is impossible, because the poblem is fomulated such that all feasible patitionings ae exploed and the one containing minimum cost solution is selected. VI. EXPERIMENTAL ANALYSIS In this section, we demonstate the applicability of the poposed multi-objective optimization technique. We use the algoithm to obtain insights into the dynamics of inteaction between the quality of sevice and the opeation cost in a given tanspotation system. We fist apply the algoithm in context of an idealized tanspotation system opeating on an Euclidean plane and then demonstate the applicability of the method to solve a eal-wold ideshaing poblem. A. Rideshaing in Euclidean Plane We stat by analyzing the behavio of the algoithm using synthetic instances that epesent a fleet of holonomic vehicles moving at constant unit speed in the Euclidean plane and that have to sevice a collection of andomly geneated tavel equests in a ectangula egion of the plane. Moe specifically, we geneate n equests such that the oigin point and destination point ae sampled unifomly fom egion [0, 100] [0, 100] and the announcement time is sampled fom inteval [0, 50]. We conside the pee-to-pee ideshaing scenaio, i.e., each of the geneated equests is assumed to have its own vehicle available at the oigin of the equest at the time of announcement of the equests. In esult, each geneated instance has n vehicles and n equests.

6 Opeation cost, el. to baseline [%] Paeto cuve fo n=50 andomly geneated equests 100 A = 0.30q 95 = 0.60q B = 1.00q C A: n = 50 = 1.00q w = 1.00 B: n = 50 = 1.00q w = 0.70 C: n = 50 = 1.00q w = No of vehicles [-] Aveage induced discomfot [% of q ] o Veh. op. time: 56.0 (100%) Avg. delay: 0.0 (0%) Veh. used: 50 (100%) o Veh. op. time: 310. (91%) Avg. delay: 48. (%) Veh. used: 46 (9%) o Veh. op. time: (69%) Avg. delay: (3%) Veh. used: 30 (60%) Figue 1. Rideshaing in Euclidean Plane. Top-left: Paeto cuves in objective plane. Makes epesent individual Paeto-optimal solutions. The line epesent convex lowe-bound on the Paeto-font. Bottom-left: The dependency between the aveage induced discomfot and fleet size used by Paeto-optimal solutions. Right: Thee epesentative Paeto-optimal solutions fo δ = 1 q. The oigin of each equest is denoted by a ed cicle and its destination is denoted by a ed diamond. Blue boxes epesent vehicles. The outes of vehicles that cay only one vehicle ae shown by thin blue lines. The outes of vehicles that cay multiple equests ae highlighted in geen. Illustation of Solution Next, we illustate how can be the VGA method used the ecove the shape of Paeto font fo a andom synthetic poblem instance. Figue 1-A shows the andom instance in consideation. The instance consists of 50 andom equests with oigin points shown as ed cicles and destination points shown as ed diamonds and 50 vehicles depicted as blue squaes, initially located at the oigin of each equest. Figue 1-A also shows the baseline system plan: In this case, each vehicle picks-up its neaest equest and dives diectly to the dop-off point of the equest. To obtain a set of Paeto-optimal solutions, we solve Poblem fo a sequence of weight paamete values anging fom w = 0 to w = 1. We epeat the pocess fo thee diffeent bounds on induced discomfot δ. The thee esulting Paeto cuves, one fo each value of δ, ae shown in the top-left plot in Figue 1. Anothe paamete of inteest that can be measued fo each Paeto-optimal solution is the numbe of vehicles used in the system plan. The bottom-left plot in Figue 1 shows the dependency between the induced discomfot and the numbe of active vehicles. In Figue 1, plots A, B, and C, we show Paeto-optimal system plans coesponding to thee selected points on Paeto font fo δ = 1 q. We can see that the system plan A, that pojects to the point at the top-left end of the Paeto font, employs no ideshaing, since this plan optimizes solely the sevice quality. When we move in down along the Paeto cuve, the espective Paeto-optimal plans contain an inceasing numbe of tips that wee meged togethe and ae seved by a single vehicle as exemplified by system plan B. The system plan C coesponds to the bottom exteme point of the Paeto cuve and achieves the lowest opeating cost that can be achieved constaints on imum induced discomfot, in this case δ = 1 q. Even though the shown system plans ae Paeto-optimal with espect to opeation cost, that we define as total opeation time ove all vehicles, we can see that minimization of opeation cost also indiectly leads to eduction of the fleet. This is because the only way to minimize opeation cost is to shae moe ides, which in tun educes the numbe of active vehicles in the solution. Anothe phenomena that can be obseved in ou illustative example is that the Paeto cuve fo an instance with a induced discomfot bound δ = 0.6 q appoximately coincides with the Paeto font fo instance with elatively loose bound δ = 1 q at the top pat of the cuve, but diveges at its bottom end. This divegence esults fom the conflict between the sevice quality objective, that asks fo the minimization of the aveage induced discomfot, and the constaint that bounds the imum induced discomfot of individual equests. Some Paeto-optimal solutions fom the loosely constained instance ae not feasible in the case of moe tightly constained instances, because they distibute induced discomfot unequally among the individual equests and consequently violate the imum induced discomfot bounds. Application to Fleet Sizing In the pevious section, we have illustated how we can geneate a set of Paeto-optimal solutions fo a poblem instance consisting of a fixed set of equests and a fixed fleet of vehicles. Each such Paeto-optimal system plan implicitly pescibes how many and which specific vehicles fom the

7 Ope. cost, el. to baseline [%] No. of veh., el. to baseline [%] Samples of Paeto-optimal fonts. = 0.30q 50 equests 100 equests 00 equests Aveage induced discomfot [% of q ] Figue. Top: Paeto cuves fo instances with n {50, 100, 00} andomly geneated tavel equests in the same spatio-tempoal egion. The thick lines epesent the expected Paeto cuve fo given density. Bottom: The numbe of vehicles (100% epesents n vehicles) used in Paeto optimal solution coesponding to the individual Paeto cuves. fleet should seve the equests in ode to achieve a paticula Paeto-optimal tade-off between the opeation cost and sevice discomfot. The multi-objective fleet optimization appoach can theefoe be used fo the pupose of opeational fleet optimization, i.e., it answes the questions of how many and which of the cuently available vehicles should be used to achieve optimal pefomance when seving a given, deteministically known tanspotation demand. In the following, we would like to demonstate that the method can be also used to get insights in the issue of stategic fleet optimization, i.e, to detemine what is the optimal size and composition of a fleet of a tanspotation system befoe the tanspotation demand is evealed. Fo most tanspotation system, we have access eithe to histoical data o to statistical infomation about the tanspotation demand to be seved. Then, we can eithe use the histoical data o take a sample fom a demand model and use ou method to obtain a set of Paeto-optimal system plans. By computing such Paetooptimal system plans fo diffeent ealizations of demand to be seved, we can study which vehicles ae active in each Paetooptimal system plan. Then, we can estimate the distibution of diffeent fleet paametes, e.g., we can ecove the distibution of fleet sizes. Such infomation can be used to detemine the appopiate fleet size fo a system in hand. One can, fo example, choose the fleet size such that it is lage than the size of 95% of optimal fleets achieving discomfot of % and at the same time lage than 99% of optimal fleets achieving discomfot of 5%. To illustate the value of the above appoach, we will use it analyze how does the spatio-tempoal density of demand in the system influence the cost savings that can be ealized by employing ideshaing. We fist geneate a set Euclidean ideshaing instances with n = 50, n = 100, and n = 00 equests andomly geneated in a spatio-tempoal egion of fixed size as descibed in Section VI-A. Then, we compute a set of Paeto-optimal solutions fo each such instance. The Paeto cuves fo sampled instances ae shown by thin lines in the top plot in Figue, whee each demand density is plotted in a diffeent colo. The thick line epesents expected Paeto cuve fo an instance with the indicated numbe of equests, obtained by aveaging the values of the opeation cost and aveage induced discomfot ove the individual samples fo individual values of w = 0,..., 1. We can see that the Paeto cuves fo instances with diffeent numbe of equest (i.e., instances having diffeent demand density) occupy diffeent pats of the objective plane. We can futhe obseve that when demand-density is inceased, solutions having both lowe opeation cost and lowe use discomfot can be found. Moe specifically, fo 50 equests in the given space-time egion, one can educe opeation cost by 6% in exchange fo aveage % discomfot degadation. In contast, fo 00 equests in the same egion, one can educe opeation cost by 1% in exchange fo 1.8% discomfot. The bottom plot in Figue shows the numbe of vehicles used in sampled Paeto-optimal solutions fo diffeent demand densities. Again, when the demand density is highe, we can find moe favoable tadeoffs between the fleet size and induced discomfot. As we can see, such an expeiment povides a quantitative justification fo the intuition suggesting that the benefits of ideshaing ae best ealized in aeas with high density of tavel equests. B. Case Study: Rideshaing in Manhattan In this section, we demonstate the applicability of the poposed technique fo analysis of a eal-wold tanspotation system. Moe specifically, we analyze the potential of ideshaing among taxi passenges in Manhattan. We base ou analysis on the dataset eleased by NYC Taxi and Limousine Commission that contains a pickup time and oigin and destination geo-coodinates fo each passenge tip seved by any of the yellow taxis in New Yok City [1]. Fom this dataset, we select a 60-second slice of data fom Tuesday, May 7th 013 between 9:00:00 am and 9:01:00am, which consists of 47 equests acoss Manhattan. Futhemoe, we conside the complete oadgaph of Manhattan consisting of 4 09 nodes and edges. The tavel time along each edge is estimated using the method descibed in [16]. The tavel time between any two points on the map is then computed by finding the minimum-time path between the two given points on the oadgaph. We apply the poposed multiobjective optimization method to compute the Paeto-cuve that epesents best attainable tade-offs between opeation

8 A: n = 47 = 0.5q w = 1.00 B: n = 47 = 0.5q w = 0.85 C: n = 47 = 0.5q w = Veh. op. time: s (100%) Avg. delay: 0 s (0%) Veh. used: 4 (99%) Veh. op. time: 0984 s (93%) Avg. delay: 4 s (1%) Veh. used: 387 (91%) Veh. op. time: s (76%) Avg. delay: 48 s (9%) Veh. used: 9 (68%) Figue 3. Manhattan Case Study. Thee epesentative Paeto-optimal system plans fo δ = 0.5 q. The outes of vehicles that cay only single passenge ae plotted in semi-tanspaent gey. The outes of vehicles that cay multiple passenges ae plotted by thicke line in colo. Ope. cost, el. to baseline [%] No. of vehicles [-] Paeto cuve fo a slice of NYC taxi tips. (n=47) 100 A = 0.10q B = 0.15q = 0.0q = 0.5q Aveage induced discomfot [% of q ] Figue 4. Manhattan Case Study. Top: Paeto cuves in objective plane fo diffeent values of δ. The makes epesent pojections of epesentative Paeto-optimal solutions. Bottom: The numbe of vehicles used in each Paeto-optimal system plan. cost and induced discomfot fo such set of tavel equests. The esults fo diffeent bounds on imum individual discomfot ae shown in Figue 4. The system plans coesponding to thee diffeent points (denoted A, B, and C) on = 0.5 q ae shown in plots A, B, and C in Figue 4. The paths of taxis that cay one passenge ae shown in light gay, the paths of taxis that wee assigned multiple passenges ae highlighted in colo. the Paeto font fo δ We can see that fo δ = 0.5 q C, it is possible to educe the opeation cost of the fleet to 76% of the baseline opeation cost (i.e., each equest tavels alone) and educe the numbe of active vehicles fom 45 to 9, while the aveage induced discomfot pe equest would incease to 48 s, which coesponds to δ = 0.09 q. VII. CONCLUSION Uban mobility is being tansfomed by newly emeging foms of on-demand tanspotation. Self-diving technology, in paticula, is expected to enable the opeation of lage centally-contolled vehicle fleets. In this pape, we agued that the potential fo ideshaing aises when the system opeation cost can be taded-off fo use discomfot and we studied the dynamics of the inteaction between the two competing objectives. In paticula, we fomulated the poblem as a multiobjective fleet outing poblem and designed a computational method based on the idea of vehicle-goup assignment. The method can compute a set of epesentative Paeto-optimal system plans to achieve diffeent tade-offs between cost of opeation and use discomfot. We gave a fomal poof of optimality of the poposed method. Futhemoe, we showed that the method is emakably scalable and is capable of computing such tade-off cuves fo instances consisting of hundeds of equests and vehicles. In paticula, we applied the method to a set of 47 taxi equests that wee issued in Manhattan in a 60-second long time window. In futue wok, we will investigate how the poposed method can be adapted, possibly by intoducing appoximations o heuistics, to compute the tade-off between opeation cost and quality of sevice in even lage instances of the poblem. We will also study what is the best way to use the infomation povided by ou method to design shaed automated mobilityon-demand systems and to appopiately select the equied numbe of vehicles in the fleet. Acknowledgements: The wok pesented in this pape was suppoted by Amstedam Institute fo Advanced Metopolitan Solutions (AMS).

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