Assignment and Pricing of Shared Rides in Ride-Sourcing using Combinatorial Double Auctions
Renos Karamanis, Eleftherios Anastasiadis, Panagiotis Angeloudis, Marc Stettler
11 Assignment and Pricing of Shared Rides inRide-Sourcing using Combinatorial DoubleAuctions
Renos Karamanis, Eleftherios Anastasiadis, Panagiotis Angeloudis and Marc Stettler
Abstract —Transportation Network Companies employ dy-namic pricing methods at periods of peak travel to incentivisedriver participation and balance supply and demand for rides.Surge pricing multipliers are commonly used and are appliedfollowing demand and estimates of customer and driver tripvaluations. Combinatorial double auctions have been identified asa suitable alternative, as they can achieve maximum social welfarein the allocation by relying on customers and drivers statingtheir valuations. A shortcoming of current models, however, isthat they fail to account for the effects of trip detours thattake place in shared trips and their impact on the accuracy ofpricing estimates. To resolve this, we formulate a new shared-rideassignment and pricing algorithm using combinatorial doubleauctions. We demonstrate that this model is reduced to amaximum weighted independent set model, which is known tobe APX-hard. A fast local search heuristic is also presented,which is capable of producing results that lie within 10% ofthe exact approach for practical implementations. Our proposedalgorithm could be used as a fast and reliable assignment andpricing mechanism of ride-sharing requests to vehicles duringpeak travel times.
Index Terms —Transportation Network Companies, Ride-Sourcing, Ride-Sharing, Combinatorial Double Auctions ©2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, includingreprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, orreuse of any copyrighted component of this work in other works.
I. I
NTRODUCTION T HE recent proliferation of Transportation Network Com-panies (TNCs) has been facilitated by an increasingdemand for efficient, economic and personalised modes ofurban mobility. TNCs have quickly captured significant shareof the urban mobility market, by providing a service thatis usually cheaper than taxis, more convenient than publictransport, and an effective alternative to private car ownership.Their success has been underpinned by the use of powerfulalgorithms and analytics, which helped reduce waiting timesand increase fleet utilisation [1]–[3].To maintain a balance between the supply and demand forrides, TNCs frequently apply dynamic pricing measures [4]usually taking the form of variable surge tariff multipliers.Such measures can motivate drivers to attend under-servedareas, dampen demand by eliminating requests from riders thatare delaying their departure, and also incentivize shared ridesbetween customers or the use of public transport [5].Through these methods, TNCs effectively operate a two-sided market, to the benefit of both the drivers (supply) and
R. Karamanis, E. Anastasiadis, P. Angeloudis and M. Stettler are withthe Department of Civil and Environmental Engineering, Imperial CollegeLondon, UK, e-mail: [email protected]. the riders (demand). As many major TNCs consider to deployAutonomous Vehicles (AVs) in the near future, their platformsare likely to be transformed into one-sided markets, where theywill enjoy complete control of the supply [6]. Previous workby Karamanis et al. [7] demonstrated that such platforms canstill incentivise the use of shared rides or public transport whileremaining profitable.Existing dynamic pricing methods suggest new equilibriumprices to customers without having prior knowledge of theirtrip valuations. If these are considered, market equilibriumprices could be identified without approximation, thereforetransforming this process into an auction. Previous work onride pricing using auction theory [8]–[11] focused on the inter-actions between riders (bidders) and drivers (sellers) who areexpected to declare their valuations and costs for prospectiverides.A TNC platform, taking the role of the auctioneer, would beresponsible for determining the winner of each auction [11].Possible auction settings might involve one or multiple driversthat are assigned to customers sequentially or simultaneously.Manipulations of the auctions by either side can be avoidedthrough the use of mechanism design theory, and the analysisof participation incentives.Combinatorial Double Auctions (CDAs) [12] can be usedto allocate multiple drivers to riders simultaneously and effi-ciently using linear programs that are commonly referred to aswinner determination problems (WDP) [9], which are knownto be NP-hard. WDPs can be formulated as set packing prob-lems that maximise the auctioneer’s revenue (or social welfare)while taking into account the utilities of the participants [13].Research in dynamic ride-sharing (DRS) (carpooling), isparticularly relevant, with several studies exploring the ap-plicability of auction models between commuting driversand riders [8], [14], [15]. In [8], the authors propose aCDA discounted trade reduction mechanism for DRS as-signment and pricing. The proposed mechanism is found tobe incentive-compatible , individually rational and weaklybudget-balanced . A system of parallel DRS auctions wasproposed in [14] aiming to identify rider-driver matches thatminimise detours. A DRS model using mechanism design was Economically efficient auction allocations maximise social welfare. Incentive-compatible mechanisms ensure that every participant is incen-tivised to be truthful. Individual rationality ensures that no participant incurs a loss. [16] Weakly budget-balanced mechanisms ensure that auctioneer will not incura loss [15] a r X i v : . [ c s . CC ] A p r presented in [15], demonstrating that maximum social welfarecannot be feasibly reached while incentivising the participationof commuters and truthful reporting of trip reservation prices.Lam [9] models the allocation of AV seats to customersas a combinatorial auction, using the Vickrey-Clarkes-Grove(VCG) mechanism to sequentially assign customers to vehiclesand determine prices. Three types of service are considered:private rides, shared rides and requests split over multiplevehicles. A separate study developed a CDA model for dial-a-ride AV fleets [11] where multiple customers and AV operatorssubmit bids, while a platform determines allocations that max-imise social welfare. The model is applied for three types ofservice as in [9], with prices computed using a relaxed versionof the problem with Lagrangian multipliers. The algorithmis shown to be NP-hard, but optimal solutions can still beobtained for realistic problem instances in reasonable times.Another proposed technique [10], [17] involves a truthfulDRS mechanism based on a second-price auction with reserveprices.The majority of studies on ride-sharing auctions (TableI) use two-dimensional models that perform one-to-one as-signments between buyers and sellers. Nonetheless, DRSoutputs inherently consist of one-to-many assignments fortrips that contain at least three participants (one driver, tworiders), whose trip-time utilities are interdependent. This lim-itation was partially addressed by previous studies [9], [11]which, however, did not consider detour effects. An alternativeapproach [10] utilised sequential rider-vehicle matches, butwithout accounting for the effect of detours on valuations,assignments and pricing estimates.To address this literature gap, we develop a mathematicalmodel that considers the effects of shared-ride detours througha winner determination process. This implements a sealed-bidCDA, with simultaneous driver-rider assignments that seek tomaximise the total trade surplus. To reduce the problem searchspace, we build upon the concept of shareability networks[18], and transform the formulation into a Maximum WeightedIndependent Set Problem (MWIS), which is known to be APX-hard.Our contribution is summarised as follows:1) We propose a WDP model for DRS assignment, imple-menting a CDA while considering the effect of detourson the valuations of auction participants.2) We provide a local search algorithm which producesapproximate results in polynomial time using greedyheuristic solutions as initializers.3) We identify the effects of shill bidding on our proposedCDA and suggest a robust trip price determinationmethodology.The paper is structured as follows: Section II outlinesour proposed assignment and pricing methodology for sharedrides. An exact implementation of the model and an approx-imate heuristic are described in section III alongside a casestudy for a hypothetical TNC in New York. Findings andrecommendations for future work are provided in section IV. TABLE I: Auction studies on ride-sharing assignment Study Problem Auction Assignment Detours[8] DRS CDA one-to-one No[14] DRS Vickrey one-to-one Yes[15] DRS CDA one-to-one Yes[9] DARP VCG one-to-many No[11] DARP CDA one-to-many No[10], [17] DARP VCG one-to-one Yes
II. M
ETHODOLOGY
Our model assumes that travellers request shared ridesthrough a central TNC platform that operates its own vehiclefleet. Alongside origin/destination coordinates, travellers alsosubmit their trip valuations. Vehicles have a fixed per-minutecost rate that is known in advance by the platform. Theobjective of the model is to maximise the trade surplus, definedas the sum of differences between traveller valuations andvehicle costs.Assignments are performed in intervals with duration ∆ -given the larger pool of possible matches; this quasi-onlineapproach is expected to outperform a possible first-in-first-out (FIFO) alternative ( [11], [18]). Two assignment typesare considered: the first is between riders willing to share atrip (i.e. rider-rider), and the latter pertains to vehicles thatwould like to offer trips (vehicle-riders). In both cases, thealgorithm seeks to identify potentially combinable requests,therefore establishing shareability networks [18] that serve asinputs to the CDA model alongside rider trip valuations. Anyvehicles or travellers that are not matched by the CDA aredeferred to later model executions alongside any requests thatmight have emerged in the meantime. A. Pre-matching
The pre-matching stage is used to filter incompatible vehicle-rider and rider-rider combinations before the executionof the CDA, therefore reducing instance sizes without penal-ising solution quality. Quality indices δ w and δ d are used toreflect the maximum allowable rider wait time, and detour respectively. Let R represent a set of ride requests and K aset of vehicles operated by the platform.For each vehicle k ∈ K we seek to obtain a subset N k ⊆ R that the vehicle can access within a period with approximateduration δ w . Conversely, for each ride request r ∈ R , we seekto identify a subset A r ⊆ K that can be picked up within δ w .A ride request r is placed in N k and a vehicle k is placed in A r according to Algorithm 1 if condition C (eq. (1)) is met,where T ( (cid:104) k, r (cid:105) ) is the travel time from the current location ofvehicle k to the origin of request r , and T ( c ) is the executiontime of a stop sequence c . C : T ( (cid:104) k, r (cid:105) ) ≤ δ w (1) Incompatible combinations produce large wait and/or detour times forriders in the combination. Detour is defined as the additional in-vehicle time of a shared trip from aprivate trip that a rider might experience.
Algorithm 1
Prematching check: Vehicle-Rider for k ∈ K dofor r ∈ R doif C then N k ← N k ∪ rA r ← A r ∪ k end ifend forend for In the case of rider-rider matching, we obtain the subset ofsecond requests I r ⊆ R \ r that can be matched with a request r ∈ R and executed with a detour lasting δ d or less. We alsoobtain a subset of requests J r ⊆ R \ r that can be matchedwith r as the second rider in the vehicle, also with a detourof δ d or less. As such, for every request pair i, j ∈ R, i (cid:54) = j where i and j are the first and second rider, respectively, thereexists a set of origin-destination combinations (cid:104) o i , o j , d i , d j (cid:105) and (cid:104) o i , o j , d j , d i (cid:105) . The following conditions apply: C : T ( (cid:104) o i , o j , d i (cid:105) ) ≤ P i + δ d (2) C : T ( (cid:104) o i , o j , d i , d j (cid:105) ) ≤ P j + δ d (3) C : T ( (cid:104) o i , o j , d j , d i (cid:105) ) ≤ P i + δ d (4) C : T ( (cid:104) o i , o j , d j (cid:105) ) ≤ P j + δ d (5)In eq. (2)-(5), P r represents the travel time for a privatetrip r ∈ R . Algorithm 2 is used to prematch rider pairs - sincethese are obtained alongside vehicle-rider pairs the complexityof these operations relates to the cardinality of set R and is O ( | R | ) [18]. The maximum possible total detour and waitingtime for any rider r ∈ R once the assignment is confirmed is δ w + δ d due to pre-matching. Algorithm 2
Prematching check: Rider-Rider for i ∈ R dofor j ∈ R \ i doif ( C ∧ C ) ∨ ( C ∧ C ) then I i ← I i ∪ jJ j ← J j ∪ i end ifend forend for The resulting adjacency subsets N k , A r , I r and J r can bevisualised using a network where nodes represent vehicles orride requests. A link from a vehicle k to rider r exists if r ∈ N k (and consequently k ∈ A r ), whereas a link between riders i and j exists if j ∈ I i (and consequently i ∈ J j ) or vice-versa. Figures 1a and 1b illustrate the auction participants’initial locations and the result of pre-matching respectively, ina randomly generated problem instance of 20 vehicles and 40riders in Manhattan, New York City. | S | denotes the cardinality of any set S in this paper. (a) (b) Fig. 1: Problem instance before (a) and after (b) pre-matching.
B. Combinatorial Double Auction
Our auction model builds upon [11] by introducing a tradinggood and applying a shareability network to reduce searchspace. Furthermore, it takes into account the quality of sharedtrips and the proximity of vehicles to achieve higher timesavings. As a result, riders would obtain different overall tripvaluations when matched to different passengers or vehicles,while the pool of potential assignments would be further honeddue to the use of a trip compatibility network. Without loss ofgenerality, we assume that individual trip requests only consistof single riders. This assumption can be relaxed to extend themodel to cater for larger passenger groups.We consider a set of riders R and a set of vehicles K . Each rider r ∈ R is identified as a 6-element tuple (cid:104) F r , C r , P r , I r , J r , A r (cid:105) , where F r is the maximum reservationprice, C r is the time valuation, P r is an 1D array of vehicletravel times required for a private trip (pick-up to drop-off),while I r , J r and A r are the adjacency subsets obtained throughpre-matching (Section II-A).The 3D array S i,j,n represents the remaining vehicle traveltime for matched riders i and j , once the final passenger ispicked up, with the pick-up sequence in the order (cid:104) o i , o j (cid:105) . Weuse 3 dimensions for S i,j,n to account for i and j havingdifferent remaining travel times once j is picked up. Forexample, if i is dropped off first, the remaining time for i might be T ( (cid:104) o j , d i (cid:105) ) , whereas the remaining time for j couldbe T ( (cid:104) o j , d i , d j (cid:105) ) . At the same time, the remaining travel timefor the vehicle would be T ( (cid:104) o j , d i , d j (cid:105) ) . Using the proceduredescribed in Algorithm 2, we obtain the assignment withthe shortest total vehicle time. Finally, the index n can takevalues between [1 , , , denoting whether S i,j,n refers to thefirst or final passenger to be picked up, or the vehicle itself,respectively.The array W i,j is used to represent the vehicle travel timefrom initial vehicle locations or rider origins i to rider origins j , for i ∈ K ∪ R , j ∈ R . The binary decision variable x i,j ∈{ , } is used to indicate if a vehicle or request i is assigned by the action to request j , such that i ∈ K ∪ R , j ∈ R . Thenlet: T r, = (cid:88) i ∈ I r (cid:2) x r + | K | ,i (cid:0) W r + | K | ,i + S r,i, (cid:1)(cid:3) (6) T r, = (cid:88) i ∈ J r (cid:2) x i + | K | ,r (cid:0) W i + | K | ,r + S i,r, (cid:1)(cid:3) (7)denote the driving times from the pick-up location of the firstpassenger to drop-off location of the first and second passengerrespectively. Similarly, let: T r, = (cid:88) i ∈ I r (cid:2) x r + | K | ,i (cid:0) W r + | K | ,i + S r,i, (cid:1)(cid:3) (8)be the driving time from the pick-up location of the firstpassenger to the drop-off location of the last passenger. Thetotal service time t r of each request r is therefore defined asfollows : t r = (cid:88) k ∈ A r (cid:104) x k,r W k,r + x k,r T r, (cid:105) + T r, (9)Using the waiting and travel time from (9) we can definethe reservation price f ( r ) for rider r as follows: f ( r ) = F r − C r t r (10)The utility u r of a rider with respect to request r is: u r = (cid:40) f ( r ) − (cid:80) k ∈ K p k,r ( t r ) if r can be served otherwise (11)where p k,r ( t r ) in (11) is the corresponding service charge forrider r when is assigned to vehicle k , as a function of the traveltime t r . Its value is determined by the platform and is equal tozero if vehicle k is not assigned to request r . Each availablevehicle k ∈ K , is described as a 3-tuple (cid:104) B k , Q k , N k (cid:105) ; where B k is its marginal operational cost, Q k is its capacity beforeassignment and N k is a subset defining riders in its vicinity(calculated as per Section II-A). We define the travel time d k to serve a particular set of riders for vehicle k , from startingto travel to the first rider until the delivery of the last rider asfollows: d k = (cid:88) r ∈ N k (cid:104) x k,r W k,r + x k,r T r, (cid:105) (12)Using eq. (12), we define the cost of serving the ridersassigned to each vehicle k as: b ( k ) = B k d k (13)As such, the total utility for vehicle k when included in theauction process is defined by: µ k = (cid:40)(cid:80) r ∈ R p k,r ( t r ) − b ( k ) , if k serves any ride , otherwise. (14) The wait time from the initial vehicle location to first passenger pickupwhich the second passenger experiences is omitted for complexity reasons.
To identify the winners of the auction and the assignment ofvehicles to riders, we adopt a WDP methodology that simulta-neously considers all rider bids and vehicle costs. To achievethis, we modify the structure of the existing formulation toensure that utilities equal to zero if rider r cannot be servedor vehicle k is not assigned, for rider and vehicle utilitiesrespectively.Since t r and d k both equal to zero if rider r or vehicle k are not included in any assignments, the versions of the riderutility u r and vehicle utility µ k are transformed as follows: u r = X r F r − C r t r − (cid:88) k ∈ K p k,r ( t r ) (15) µ k = (cid:88) r ∈ R p k,r ( t r ) − b ( k ) (16)where the term X r = (cid:16) (cid:80) k ∈ A r x k,r + (cid:80) i ∈ I r x i + | K | ,r (cid:17) indicates whether rider r is in the auction either as a firstor as a second client. The model aims to maximise the totalutility of all the participants (vehicles and riders), with theobjective function defined as follows: SW = (cid:88) r ∈ R u r + (cid:88) k ∈ K µ k = (cid:88) r ∈ R (cid:16) X r F r − C r t r (cid:17) − (cid:88) k ∈ K b ( k ) (17)where SW indicates the value of social welfare. Observethat the service charges cancel out in the summation ofthe participants’ utilities. The optimisation problem is thenformulated with the following set of constraints: Model 1 (Winner Determination Problem for Ride Sharing): maximize SW (18a) subject to x k,r + (cid:88) i ∈ I r x r + | K | ,i ≤ Q k , ∀ k ∈ K, ∀ r ∈ N k , (18b) (cid:88) k ∈ A r x k,r + (cid:88) i ∈ J r x i + | K | ,r ≤ , ∀ r ∈ R, (18c) (cid:88) k ∈ A r x k,r − ≤ M (cid:16) − (cid:88) i ∈ I r x r + | K | ,i (cid:17) , ∀ r ∈ R, (18d) − (cid:88) k ∈ A r x k,r ≤ M (cid:16) − (cid:88) i ∈ I r x r + | K | ,i (cid:17) , ∀ r ∈ R, (18e) (cid:88) r ∈ R x i,r ≤ , ∀ i ∈ K ∪ R, (18f) (cid:88) i ∈ K ∪ R x i,r ≤ , ∀ r ∈ R, (18g) (cid:88) k ∈ A r x k,r − (cid:88) i ∈ I r x r + | K | ,i = 0 , ∀ r ∈ R, (18h) x i,j ∈ { , } , ∀ i, j ∈ K ∪ R (18i)Eq. (18b) ensures that the number of assigned riders toeach vehicle k is at most equal to the vehicle capacity Q k if assigned with a rider r . (18c) guarantees that if rider r isassigned, it is either the first rider or the second passenger to board. Eqs. (18d) and (18e) utilize the Big M method [19] toensure that if any two riders are matched, the first rider r inthe matching has to be picked up by a vehicle k . M is definedas a sufficiently large positive number.Eq. (18f) ensures that each vehicle or rider is assignedas a starting point towards a rider at most once. Eq. (18g)ensures that each rider is assigned as a destination from avehicle location or a rider no more than once. Finally, eq.(18h) ensures that if a vehicle is connected to a rider, therewould be an additional rider in the trip.Note that eqs. (9) and (12) feeding into the objectivefunction, include non-linear terms. We therefore introducevariables y k,r ∈ R + and z k,r ∈ R + , to replace the non-linear terms in equations (9) and (12) respectively as shownin equations (19) and (20). t r = (cid:88) k ∈ A r (cid:16) x k,r W k,r + y k,r (cid:17) + T r, (19) d k = (cid:88) r ∈ N k (cid:0) x k,r W k,r + z k,r (cid:1) (20)Consequently the objective function in equation (17) trans-forms into the following: SW L = (cid:88) r ∈ R u r + (cid:88) k ∈ K µ k = (cid:88) r ∈ R (cid:16) X r F r − C r t r (cid:17) − (cid:88) k ∈ K B k d k = (cid:88) r ∈ R (cid:34) X r F r − C r (cid:20) (cid:88) k ∈ A r (cid:16) x k,r W k,r + y k,r (cid:17) + T r, (cid:21)(cid:35) − (cid:88) k ∈ K B k (cid:20) (cid:88) r ∈ N k (cid:16) x k,r W k,r + z k,r (cid:17)(cid:21) (21)where SW L denotes the value of the objective after lineariza-tion. To ensure that the variable y k,r equals its desired value,we introduce the following linearization constraints: y k,r ≤ M x k,r (22) y k,r ≤ T r, (23) y k,r ≥ T r, − M (1 − x k,r ) (24) y k,r ∈ R + (25)for every r ∈ R and every k ∈ A r . In a similar fashion, weintroduce the following linearization constraints for variable z k,r : z k,r ≤ M x k,r (26) z k,r ≤ T r, (27) z k,r ≥ T r, − M (1 − x k,r ) (28) z k,r ∈ R + (29)for every k ∈ K and every r ∈ N k .By incorporating the additional variables and constraints inequations (19)-(29), our optimisation methodology for Model 1 transforms to the following Mixed Integer Linear Program(MILP):
Model 2 (Transformed WDP for Ride Sharing) maximize SW L subject to (18 b ) - (18 i ) , (22) - (29) C. Reduction to Maximum Weighted Independent Set
To assess the complexity of
Model 2 , we present a reductionto the MWIS problem. We assume that in the largest instance,all vehicles can be matched to all requests, and all requests aresharing-compatible. In that scenario, with K and R being thesets of vehicles and requests, respectively, we let C denote theset of all possible combinations, where |C| = |K||R| −|K||R| .Assuming that all vehicles will be assigned, the set of allpath-vehicle allocations is (cid:0) |K||R| −|K||R||K| (cid:1) . To prove the APX-hardness of Model 2 , we use an approximation-preservingreduction from MWIS.
Theorem 1.
Model 2 is NP-HardProof.
We reduce an instance of MWIS, a known APX-hard problem [20], to an instance of Model 2 . Given a weightedgraph G = ( V, E, w ) , the MWIS objective is to find a set ofpairwise disjoint nodes S ⊆ V with maximum total weight.Let the tuple ( k, i, j ) denote the ride-sharing trip of Model 2 with vehicle k in which the first passenger is i and the secondis j , ∀ k ∈ K , i, j ∈ R and i (cid:54) = j . Also let u i ( k, i, j ) and u j ( k, i, j ) denote the utilities of riders i and j respectively,for the trip ( k, i, j ) and u k ( k, i, j ) denote the utility of thevehicle.Consider now the following representation; let G =( V, E, w ) be a graph where each vertex represents a combina-tion c = ( k, i, j ) . An edge exists between vertices c n and c m if and only if the trip combinations c n and c m have a commonelement, i.e. a common vehicle or rider. Let: w c = u k ( k, i, j ) + u i ( k, i, j ) + u j ( k, i, j ) (30)denote the weight of vertex c = ( k, i, j ) . We note thatchanging the order of two riders in a combination can result ina different weight for the corresponding vertex. That is becausethe detour or the wait time after the reordering can exceedeither of the thresholds δ d , δ w set during pre-matching, thusresulting in a different value of rider utilities.We now prove the correctness of the above transformation.Let OP T ( I (cid:48) ) denote an optimal solution to a Model 2 instance I (cid:48) . For any two trip combinations c, c (cid:48) that either have a com-mon rider or vehicle, at most one of them will be in OP T ( I (cid:48) ) and the vertices representing these trips will be connected byan edge in graph G . As a result OP T ( I (cid:48) ) is represented by aset of independent nodes in G and since the solution is optimalwith cost (cid:80) r ∈ R u r + (cid:80) k ∈ K µ k = (cid:80) c ∈ V w c by equation (30) APX is the complexity class of optimization problems that cannot beapproximated within some constant factor unless P (cid:54) = NP this corresponds to an independent set of maximum weight in G .Conversely, suppose we have an optimal solution OP T ( I ) on an instance I of MWIS in G . Since OP T ( I ) is inde-pendent, no pair of nodes will be connected, so no pairof trips from WDP will have a common element. Againaccording to eq. (30), the total weight of the selected tripsis maximised.We notice that the above reduction preserves the approx-imation [20]. Let f be the (polynomial time) transformationfrom an instance I (cid:48) of Model 2 to an instance I of MWIS asdescribed above i.e. I = f ( I (cid:48) ) and let g be the (polynomialtime) algorithm that produces a solution to I given a solutionto I (cid:48) . Let also α = 1 and β = 1 . Using transformation f , the optima of I and I (cid:48) satisfy the following inequality OP T ( I (cid:48) ) ≤ αOP T ( I ) . Furthermore, having a solution withweight w (cid:48) for any instance I (cid:48) , we can construct a solution for I with weight w such that | w − OP T ( I ) | ≤ β | w (cid:48) − OP T ( I (cid:48) ) | using algorithm g . Corollary 1.
Model 2 is APX-Hard.
Many greedy approximation algorithms have been previ-ously proposed, with their approximation ratio expressed asa polynomial in terms of the average or maximum nodedegree in the graph [21]. We note that in the fully connectedscenario, the average/maximum degree of node c is ∆ c = | R | ( | R | − − | K | − | R | − . To demonstrate this,if we consider a combination ( k, i, j ) , there exist additional | R | ( | R | − − trip combinations with vehicle k . For everyother vehicle from the remaining | K |− , there exist | R |− trip combinations including rider i and an additional | R |− including rider j , which are not already accounted. Thus,simplifying ( | K | − | R | −
1) + 2( | R | − results to ( | K | − | R | − . D. Local Search Algorithm using Greedy Search Initialisers
We established earlier that solving the MWIS problem fora fully connected CDA scenario, would involve finding aMWIS in graphs with |C| = |K||R| − |K||R| nodes withan average/maximum node degree of ∆ c = | R | ( | R | − − | K | − | R | − . Considering a small localised examplewith 10 vehicles and 20 potential riders, that would generatea network with 3800 nodes with an average/maximum nodedegree of 1045.An exact solution would, therefore, be impractical, as ex-isting solution algorithms are slow even for a few hundreds ofvertices [22]. We propose a local search algorithm based onsimulated annealing (SA), a technique that has been shown toperform very well for the maximum clique problem (a similarpremise, as it is the opposite of an independent set) [23].Simulated Annealing (SA) was initially proposed as aprobabilistic method to solve difficult optimisation problems[24]. It aims to bring a system from an arbitrary initial stateto an eventual state of minimum energy. Most SAs use anenergy measure that is inversely proportional to the qualityof the solution and is minimised using an iterative process.Starting from a seed solution, SA iterations generate several neighbouring solutions, which are accepted in accordancewith a stochastic process. The process continues until the”temperature” of the problem reaches a user-defined minimum.A high-level structure of our SA algorithm for the MWISproblem is presented in Algorithm 3. Algorithm 3
SA for the Independent Set ProblemGenerate initial solution S for graph G Set initial and minimum temperatures T , T min S old = S E old = energy ( S old , G ) S best = S old E best = E old T ← T while T > T min do S new = neighbour ( S old , G ) E new = energy ( S new , G ) if E new < E best then S best = S new E best = E new end if S old , E old = select ( S old , S new , E old , E new , T ) T = αT , (where α is a constant and α < ) end while Output: S best , E best Algorithm 3 utilizes a graph G , constructed to identify allpossible vehicle-rider-rider combinations by representing themas a set of nodes. Each node in the set is a 3-tuple, (cid:104) c, w c , N c (cid:105) . c refers to the combination of vehicle-rider-rider in the formof (cid:104) k, i, j (cid:105) , w c refers to the weight of the node as defined inSection II-C and N c is a list of neighbouring nodes. It can beeasily shown that the degree of each vertex is equal to | N c | .To construct the graph we set N c = ∅ and iterate throughthe network nodes to populate N c for each vertex. As withAlgorithm 4, this process requires | K || R | iterations (fullyconnected scenario) to create the set of vertices V . Populating N c for each vertex (and creating the edge set E ), requires | V | iterations (Algorithm 5). Since | V | = ( | K || R | ) , thecomplexity of the worst case scenario for network generationis O ( | K | | R | ) . This process, however, can be easily paral-lelised.A set of greedy heuristics with known lower bound perfor-mance [21] is used to obtain an initial solution S , consistingof an ordered set of vertices in V . These operate by sortingvertices in a descending order with respect to w c , / | N c | , w c / | N c | and w c / (cid:80) i ∈ N c w i , respectively. The best solutionamong these four is identified through inspection.To calculate the energy of a solution (Algorithm 6), weiterate through the ordered vertex sequence S . At each step,we add the next vertex in S to the independent set I andremoving its neighbours from S . Iterations continue until S isempty. The energy of the solution is, therefore, equal to thenegative sum of all values w c , for each vertex within I .When it comes to the generation of neighbouring solutions,we randomly select two vertices in the independent set I ofthe old solution S old and switch their positions in S old toproduce sequence S new . This approach increases the chance Algorithm 4
Vertex Generation Process V ← ∅ for k ∈ K dofor i ∈ N k dofor j ∈ I i do w c = u i ( k, i, j ) + u i ( k, i, j ) + u k ( k, i, j ) if w c ≥ then c = (cid:104) k, i, j (cid:105) N c = ∅ V ← V ∪ (cid:104) c, w c , N c (cid:105) end ifend forend forend for Output: V Algorithm 5
Edge Generation ProcessNon-empty set VE ← ∅ for i ∈ V dofor j ∈ V \ i doif c i ∩ c j (cid:54) = ∅ then N c i ← N c i ∪ jN c j ← N c j ∪ iE ← (cid:104) i, j (cid:105) end ifend forend for Output: G = ( V, E ) that sequence S new will produce a different independentset and energy than S old . Finally, we form our stochasticselection method on defining an acceptance probability forevery new solution, which is calculated using E old , E new andtemperature T as shown in Algorithm 7. Better solutions arealways accepted, whereas worse solutions have less chance ofbeing accepted as the iterations progress (i.e. as temperature T decreases). E. Trip Price Determination
Optimal solutions of the WDP in CDAs produce efficientoutcomes which are individually rational. That is, assuming
Algorithm 6
Energy CalculationNon-empty ordered sequence S Graph G = ( V, E ) I ← ∅ while S (cid:54) = ∅ do i = S (1) I ← I ∪ iS ← S \ ( S ∩ ( N c i ∪ i )) , (obtain N c i from G ) end while E = − (cid:80) i ∈ I w c i , (obtain w c i from G )Output: E Algorithm 7
Selection ProcessInputs: S old , S new , E old , E new , Tp = X , (where X ∼ U (0 , ) if E new < E old then p a = 1 else p a = e ( E old − E new ) /T end ifif p a > p then S old = S new E old = E new end if Outputs: S old , E old participants in the auction are truthful about their valuations.There is, however, no guarantee that auction participants(bidders) will state their true valuations. [16] explains thisproblem with an example of three bidders. We will extendthis example to our CDA, to illustrate how untruthful bids canarise.Let us consider a CDA scenario involving three riders(bidders) and one vehicle. Let us also assume that from thesix possible allocation combinations, the following three yielda positive value for total trade surplus: f ( (cid:104) , (cid:105) ) = 10 , f ( (cid:104) , (cid:105) ) = 8 , b ( (cid:104) , (cid:105) ) = 10 (31) f ( (cid:104) , (cid:105) ) = 7 , f ( (cid:104) , (cid:105) ) = 9 , b ( (cid:104) , (cid:105) ) = 11 (32) f ( (cid:104) , (cid:105) ) = 5 , f ( (cid:104) , (cid:105) ) = 10 , b ( (cid:104) , (cid:105) ) = 12 (33)In eqs. (31)-(33), f r ( (cid:104) S (cid:105) ) and b k ( (cid:104) S (cid:105) ) represent total valu-ation and cost for a rider r and a vehicle k , respectively, fora trip with a pickup sequence S . Using Model 2 , the platformallocates the trip with the only vehicle servicing riders and in the sequence (cid:104) , (cid:105) as it is the combination producingthe highest trade surplus. Note that riders and , assumingeveryone bids truthfully, can report a lower value per time andstill win the auction with the same combination.The inclusion of additional riders will give rise to morecomplex bidding strategies. In the case that riders and reduce their bids excessively, they might lose in the auction.This characteristic CDA property is known as the thresh-old problem [25] and refers to the implication of valuationmisreporting thresholds for individual participants, which canmotivate bidders to employ perverse bidding strategies [26].Pricing in VCG auctions, where bidders pay the differ-ence of welfare in their absence with the welfare of otherswhen they are included in the auction, is incentive-compatible[16]. Furthermore, incentive-compatible payments have beenderived through the solution of dual relaxed linear problems(LPs) of the WDP [27]. Previous studies [11], [28], [29],used relaxed dual WDP problems to identify allocation andpricing in double auctions, with Lagrangean multipliers to be considered as prices. It has been shown that optimal dualvariables in LP coincide with VCG payments [30].However, the use of near-optimal CDA solutions does notpreserve incentive compatibility [31]. Negligible variationsfrom the optimal objective can have significant consequenceson the payments to be made by bidders [32]. As such, an ap-proximate WDP solution would inhibit the use of VCG or dualLP relaxations that would guarantee incentive-compatibility.The NP-hardness of our proposed CDA prohibits the identi-fication of exact WDP solutions in practical implementations,thereby we omit the use of VCG or dual LP relaxations forprice determination.Instead, we propose a model which resembles a GeneralisedFirst Price (GFP) auction for trip pricing. A GFP mechanismis an untruthful auction mechanism, where participants bidfor the allocation of a limited amount of slots. Participantspay their bid values in case they are assigned to a slot.Previous research outlined deficiencies in the GFP mechanismby strategically employed shill bidding which destabilizes theauction [33]. Subsequent work in [34] attributes these GFPdeficiencies to the auction interface and argues that GFPauctions can be robust by allowing expressiveness of theparticipants using multidimensional bids.In the conventional GFP, an individual i submits a singlebid f i , which is multiplied by s ≥ s ≥ ... ≥ s k , k beingthe last available slot. The expressive version of GFP dictatesthat an individual i submits a different bid f ik for each slot k which is multiplied by s ≥ s ≥ ... ≥ s k accordingly. Ourproposed CDA resembles an expressive GFP, as travellers bidfor a limited number of vehicle seats (slots) and by submittinga valuation per time C r , they might obtain a different valuation f ( r ) for each potential vehicle-rider-rider assignment.To limit the effect of untruthful bids on the auction outcome,we propose that each rider only submits the valuation per time C r . The platform in turn identifies and privately informs therider of its maximum reservation price F r , so that if matched,the payment will comprise of a discounted static price for thetime of the trip attributing to P r and an additional variable rateattributing to C r ¯ δ r , where ¯ δ r is the wait and detour time savedby choosing the platform, instead of the rest of the market.Consequently, the maximum reservation price F r is derivedby the platform using the following generalised cost equation: F r = p b + P r p t + C r ( P r + δ w + δ d ) (34)where p b is the flat fee and p t is the discounted price perminute for a shared trip, lasting P r minutes if private, asspecified by the platform. δ w and δ d refer to the guaranteedmaximum wait and detour times respectively, which are usedin pre-matching by the platform.By introducing this format, it is straight-forward to deduceby observing equations (10), (13) and (17) that in the eventwhere bidders submit per time valuations C r which are veryclose to zero, our proposed CDA converts to an optimal 3Dassignment problem where the sum of detours is minimised,if the following inequality holds for any vehicle k and riders i , j prior to the auction: b ( k ) ≤ f ( i ) + f ( j ) (35)By introducing this condition with equation (35), we ensurethat the auction always returns an assignment if a pre-matchinginstance exists as any rider payments in the GFP instance willalways cover the vehicle costs. We thereby need to choose theappropriate value for the flat fee p b , such that equation (35)holds. In doing so, we assume that the total rider paymentper vehicle equals its cost. We also assume that vehicle costs B k are uniform across the fleet (i.e. B k = B ∀ k ∈ K ) andextend the functions as per equations (10) and (13): Bd k = F i − C i t i + F j − C j t j (36)Using equation (34), and by substituting δ w + δ d with δ , wereach to the following: Bd k = 2 p b + p t ( P i + P j ) + C i ( P i + δ − t i ) + C j ( P j + δ − t j ) (37)In the minimal total bid scenario, both C i and C j inequation (37) would be zero. We also know that d k is equalto max ( t i , t j ) . By setting the total wait and detour timeexperienced by each rider r as δ r , we can replace t r by P r + δ r . Therefore, with C i and C j set to zero we arriveto the following equation: Bmax ( P i + δ i , P j + δ j ) = 2 p b + p t ( P i + P j ) (38)The maximum vehicle cost in equation (38) for any valuesof P i and P j , occurs if max ( P i + δ i , P j + δ j ) = max ( P i , P j )+ δ , for δ as introduced above, being the maximum total waitand detour time guarantee by the platform for an individualrider. Assuming the value of p b is zero and that p t is setby the platform such that p t ≥ B , if min ( P i , P j ) ≥ δ thecondition in (35) always holds. If however min ( P i , P j ) <δ , a flat fee p b is required to ensure the condition in (35).As such, assuming both P i , P j → , and max ( δ i , δ j ) = δ ,using equation (38), the flat fee for our proposed GFP interfaceshould be as follows: p b = Bδ B ( δ w + δ d )2 (39)III. D ISCUSSION
Our methodology was implemented using Python and testedon a workstation with an Intel i7-4790 CPU (3.6GHz) and8GB RAM. Exact solutions were obtained using the Branchand Cut algorithm provided by IBM ILOG Cplex OptimizationStudio 12.7.1.To test the algorithm, we create a case study network setin Manhattan, NY. The underline road network and traveltimes were obtained using the OSMnx library [35]. To accountfor congestion, we applied a 20% penalty to the free-flowspeeds in residential and motorway link segments, and 40%elsewhere. Rider origin-destination pairs, as well as vehiclelocations, were sampled uniformly in space to create CDAinstances. Only trips with travel time that is greater than 5
TABLE II: Performance comparison of SA and BC.
Totalvehicles Totalriders BCsolution BCruntime[sec] SAsolution SAruntime[sec] Error[%]4 8 48.199 0 48.199 0.002 0.008 8 77.714 0.13 77.714 0.008 0.005 10 120.523 0.02 120.523 0.004 0.0010 10 127.936 0.13 127.936 0.027 0.005 12 108.327 0.03 108.327 0.004 0.006 12 136.996 0.05 136.996 0.007 0.0012 12 145.248 0.11 144.866 0.026 0.266 14 122.487 0.05 122.487 0.021 0.007 14 131.537 0.28 128.81 0.029 2.077 15 208.895 0.5 208.895 0.034 0.008 16 171.665 0.27 170.613 0.053 0.617 17 160.367 0.06 160.253 0.024 0.079 18 236.672 1.66 235.918 0.076 0.328 20 204.96 0.28 204.93 0.061 0.0110 20 236.125 1.45 234.785 0.149 0.5711 22 295.669 3.45 291.632 0.209 1.3712 24 331.237 29.95 321.797 0.598 2.8510 25 326.057 39.86 321.967 0.64 1.2514 28 377.018 103.5 373.223 2.702 1.0115 30 397.518 130.92 387.628 2.374 2.4915 20 285.793 18.14 283.101 0.407 0.9420 20 270.432 58.24 267.002 0.746 1.2712 25 324.687 26.64 321.524 1.239 0.9715 25 323.191 42.3 316.062 1.399 2.2116 25 341.845 82.19 333.836 1.496 2.3417 25 330.884 124.78 321.94 1.573 2.7014 30 419.626 49.59 415.906 2.193 0.8913 26 329.992 63.22 324.955 2.587 1.5316 32 431.772 1003.55 408.41 3.423 5.4117 34 469.94 4126.59 445.309 4.456 5.24
Fig. 2: Run-time for BC and SA methods.minutes were considered, while δ w and δ d were both set to 10and 15 minutes respectively.For this study, we used UK-based estimates of workingtime valuations [36] for the derivation of rider valuations.Vehicles were assumed to have a capacity of two customers,with their operating costs B k uniformly set to 12.96 GBP/hour.Conversely, customer time valuations C r were sampled froma log-normal distribution with a mean of 17.69 GBP/hour and σ = 0 . . The discounted price per minute p t was set to 0.75GBP/min.Table II provides a performance comparison of the Simu-lated Annealing (SA) and the Branch and Cut (BC) algorithmsfor a range of instances. As can be seen in the table and infigure 2, the runtime for the BC approach grows exponentiallyas more vehicles and riders are considered in the instance,thereby increasing the node count of the MWIS instance,whereas the runtime for the SA remains relatively short.The APX-complete nature of our problem is also signified Fig. 3: Percentage error of approximation against node count.Fig. 4: Solution values for BC and SA methods against nodecount.in the solution comparison between BC and SA as observedin figure 3, as the percentage error gradually increases witha larger instance size. However, as shown in figure 4, theapproximation error is relatively low for instances of such size.A visual comparison of the results obtained by the BC and SAalgorithms is provided in figures 5a and 5b, respectively, foran instance involving 10 vehicles, 20 customers and two edgesper match outlines the similarities between solutions obtainedusing the two approaches. (a) (b) Fig. 5: Visualisation of (a) BC and (b) SA solutionsTo strengthen the argument for the inclusion detour calcu-lations on CDAs for ride-sharing, we conducted a comparisonanalysis between exact solutions of our
Model 2 and the Fig. 6: Average total wait and detour time per request for
Model 2 and CDA in [11].algorithm in the state-of-the-art which mostly resembles ourproblem statement, namely the CDA model in [11]. We createdinstances from 10 to 22 requests, with the assumption ofone seat per request. For each instance, we assumed thereare just enough vehicles to cover the demand (i.e. half thenumber of requests). To run the CDA in [11], we convertedthe distance-based methodology to time-based to match
Model2 and omitted private rides. We used a vehicle capacity of tworides for all vehicles in both models.As observed in Figure 6, since the CDA in [11] omitsdetours and wait times in their calculation, the resultingassignment creates much higher detour and wait times onaverage for each instance. Consequently, ignoring the effectof detours and wait times in ride-sharing CDAs can produceassignments which might not be acceptable by the users of theservice. Taking the time dimension into account we can indeedmassively improve the convenience of the service as observed.However, we achieve this with an increase in computationalcomplexity, as discussed in II. Nonetheless, a reduction of thesolution space can be achieved via the pre-matching stage, asshown in II-A.
A. Trade surplus implications
A large number of problem instances were considered, withfleet sizes ranging between 3 and 60 vehicles, and a customerbase of 10 to 60 riders. From the range of greedy heuristics thatwere considered for SA initialisation described in section II-D,weight-based approaches were found to yield the best results(Figure 7). Figures 8 and 9 illustrate the relationship betweenproblem sizes and algorithm run times, which is found to bein polynomial time.We define the trade surplus index (TSI) as the ratio of theobjective value and the number of assigned vehicles in eachinstance. Figures 10 and 11 illustrate its relationship with thefleet coverage index (FCI), defined as the ratio of vehiclesavailable against the number of vehicles required to serve allrequests. An interesting feature of our approach (as shown inFigures 10 and 11) is that the TSI is inversely proportional tothe FCI for values of the latter between 0 and 1, and remainsconstant beyond that point.This pattern can be explained by considering a scenariowith 1 vehicle and 10 riders. In this case, the node with thehighest weight will be the solution in the MWIS problem. Theaddition of a new vehicle (with the same cost), assuming that Fig. 7: Performance comparison of greedy initialisersFig. 8: Network creation time against node count.it is included in the MWIS solution, will lead to a reduction inthe average node weight. This trend will persist with furtherincreases in the size of the fleet, as riders with lower valuationsare accommodated and gradually reduce the overall TSI. Assuch, once
F CI > the TSI will on average remain constant,consistent with the notion of market equilibrium while supplyincreases beyond current demand levels. B. Practical Implementation
To investigate the practical implementation of our proposedmethodology, we analysed ride-sharing data provided by theTaxi and Limousine Commission (TLC) of New York City(NYC). Specifically, we exported the high volume for-hire ve-hicle trip records provided in [37] and identified typical dailyweekday trip count profiles which originate and terminate inthe island of Manhattan NYC for the entirety of the ride-sharing market.By recording the MWIS node count for a varying requestinput, we were able to grasp the effect of ride requests onthe problem size for an FCI equal to one (supply=demand),as shown in figure 12. Using the identified runtime trendsoutlined in figures 8, 9 and 12, we compiled table III, whichreports the time performance of our proposed methodology forvarying request inputs. By assessing the request performancelevels in table III we chose fifty requests as the practical limitin Manhattan, since our proposed methodology produces ride-sharing solutions approximately within one minute, which weregarded as acceptable.By examining the typical per-minute shared ride countin Manhattan in figure 13, we observe that the demandsurpasses the cutoff of fifty requests only during three distinct Fig. 9: SA runtime against node count.Fig. 10: Trade surplus per serving vehicle and fleet coverage.demand peaks, specifically during the morning, afternoon andevening. As such, since the highest peak narrowly exceeds ahundred shared trip rides, our proposed methodology can bepractically implemented during peak hours when demand forrides exceeds supply (FCI ≤
1) with an assignment durationinterval ∆ of thirty seconds for the entire Manhattan sharedride market.Nonetheless, the choice of a practical request cutoff valuealso depends on the error of the SA solution when comparedto the exact solution. As observed in figure 12, the numberof nodes increases almost in a cubic rate with an increasingnumber of requests. Also, the percentage error increasesapproximately in a linear fashion with an increasing numberof nodes, as observed in figure 3. As such, an instance of100 requests might have a comparable SA total utility valuewhen split in three instead of two instances of 50 requests. Forreference, by running BC instances of 50 requests for FCI = 1 ,using the upper bound and the best integer solution provided,we pinpoint the SA percentage error within − of theexact solution. An exact solution for such an instance size was prohibitive due tocombinatorial explosion. As such, we used a long-run upper bound and bestinteger solution of the BC algorithm before termination.
TABLE III: MWIS instance node count and runtimes againstvarying request inputs.
Requests Nodes Networkruntime[sec] SAruntime[sec] Totalruntime[sec]40 2500 5 10 1545 3500 10 20 3050 5000 15 50 6555 6000 25 80 10560 8000 45 120 165
Fig. 11: Trade surplus per serving vehicle fleet coverage andnode numbersFig. 12: MWIS node count for a varying request input inManhattan, NYC.Our practical implementation recommendations above as-sume pervasiveness of autonomous vehicles similar to thelevels of current conventional ride-sharing platforms. Nonethe-less, the adoption rate of autonomous vehicles in commercialride-sharing is a conjecture. As such, plausible scenarios couldinvolve mixed fleets and custom rider requirements. Even so,our algorithm is still applicable for such customisation asone could screen any preferences in the pre-matching stage(Section II-A). IV. C
ONCLUSION
In this paper, we considered the problem of ride-sharingassignment and pricing in TNC platforms with autonomousvehicles. Our proposed assignment and pricing approachutilises a local search algorithm that solves a WDP MILPvariant approximately in polynomial time by computing three-dimensional assignments to maximise trade surplus. By inves-tigating the robustness of our proposed model, we derived aGFP auction interface which conveniently reduces to a stablethree-dimensional assignment with minimal detours if ridersreport untruthful bids. We demonstrated the practicability ofour proposed assignment and pricing method in a large urbansetting such as Manhattan, NYC.Our suggestions for future research in this area are twofold.First, we believe that both computational complexity andaccuracy improvements are possible in exploring the breadthof meta-heuristics and machine learning algorithms in solv-ing the proposed problem of ride-sharing auctions. Spatialclustering of requests, for example, could split much largerinstances than the ones tested into parallel problems, whichcould be solved in a reasonable time, without compromising Fig. 13: Typical Weekday Per-Minute Shared-Trip Count inManhattan, NYC.much of the efficiency of the algorithm. Secondly, agent-basedmodelling studies which focus on analysing heterogeneous bidbehaviours in our proposed methodology (or a variant of it)could be useful. Such studies could produce large data-sets ofsolutions and aid in better assessing the effects of shill biddingin such a combinatorial auction setting.R
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Computers, Environmentand Urban Systems Eleftherios Anastasiadis received a BSc in infor-matics and telecommunications from the Universityof Athens in 2011. He received an MSC and a PhDin theoretical computer science from the Universityof Liverpool in 2012 and 2017 respectively.In 2017 he worked as quality assurance engineerin Rosslyn Data Technologies Ltd. Since 2018 he isa postdoctoral researcher at the Transport Systemsand Logistics Laboratory in the department of Civiland Environmental Engineering at ImperialCollegeLondon. His research interests include approxima-tion algorithms and mechanism design for network optimisation problems,and agent-based simulation for autonomous vehicle fleets.
Panagiotis Angeloudis is a Senior Lecturer andDirector of the Transport Systems and LogisticsLaboratory, part of the Centre for Transport Studiesand the Department of Civil & Environmental En-gineering at Imperial College London. He receivedan MEng in Civil & Environmental Engineering in2005 and a PhD in Transport Operations in 2009,both from Imperial College London.His research interests lie on the field of transportsystems and networks operations, with a focus onthe the efficient and reliable movement of peopleand goods across land, sea and water. He was recently appointed by the UKDepartment for Transport to the Expert Panel for Maritime 2050 and was amember of the UK Government Office of Science Future of Mobility reviewteam. He is affiliated with the Centre for Systems Engineering and Innovation,the Institute for Security Science and Technology, the Grantham Institute andthe Imperial Robotics Forum.
Marc Stettler is a Senior Lecturer in Transportand the Environment in the Centre for TransportStudies and Director of the Transport & EnvironmentLaboratory. Prior to joining Imperial, Marc was aresearch associate in the Centre for Sustainable RoadFreight and Energy Efficient Cities Initiative at theUniversity of Cambridge, where he also completedhis PhD.His research aims to quantify and reduce envi-ronmental impacts from transport using a range ofemissions measurement and modelling tools. Ex-amples of recent research projects include: quantifying real-world vehicleemissions; using real-world vehicle emissions data to improve emissionsmodels; evaluating economic and environmental benefits of Kinetic EnergyRecovery Systems (KERS) for road freight; and quantifying aircraft emissionsat airports. Marc is a member of the LoCITY ‘Policy, Procurement, Planningand Practice’ working group and the EQUA Air Quality Index AdvisoryBoard.