Capturing Travel Mode Adoption in Designing On-demand Multimodal Transit Systems
CCapturing Travel Mode Adoptionin Designing On-demand Multimodal Transit Systems
Beste Basciftci
Sabancı University, Istanbul 34956, Turkey, [email protected]
Pascal Van Hentenryck
Georgia Institute of Technology, Atlanta, Georgia 30332, USA, [email protected]
This paper studies how to integrate rider mode preferences into the design of On-Demand MultimodalTransit Systems (ODMTS). It is motivated by a common worry in transit agencies that an ODMTS maybe poorly designed if the latent demand, i.e., new riders adopting the system, is not captured. The paperproposes a bilevel optimization model to address this challenge, in which the leader problem determines theODMTS design, and the follower problems identify the most cost efficient and convenient route for ridersunder the chosen design. The leader model contains a choice model for every potential rider that determineswhether the rider adopts the ODMTS given her proposed route. To solve the bilevel optimization model,the paper proposes an exact decomposition method that includes Benders optimal cuts and nogood cutsto ensure the consistency of the rider choices in the leader and follower problems. Moreover, to improvecomputational efficiency, the paper proposes upper bounds on trip durations for the follower problems andvalid inequalities that strenghten the nogood cuts.The proposed method is validated using an extensive computational study on a real data set from AAATA,the transit agency for the broader Ann Arbor and Ypsilanti region in Michigan. The study considers theimpact of a number of factors, including the price of on-demand shuttles, the number of hubs, and accessibilitycriteria. The designed ODMTS feature high adoption rates and significantly shorter trip durations comparedto the existing transit system and highlight the benefits in accessibility for low-income riders. Finally, thecomputational study demonstrates the efficiency of the decomposition method for the case study and thebenefits of computational enhancements that improve the baseline method by several orders of magnitude.
1. Introduction
This paper considers On-Demand Multimodal Transit Systems (ODMTS) (Mah´eo et al. 2019,Van Hentenryck 2019), a new type of transit systems that combine on-demand shuttles with fixedroutes served by buses or rail. ODMTS are organized around a number of hubs, on-demand shut-tles serve local demand and act as feeders to and from the hubs, and fixed routes provide high-frequency service between hubs. By dispatching in real time on-demand shuttles to pick up ridersat their origins and drop them off at their destinations, ODMTS are “door-to-door” and addressthe first/last mile problem that plagues most of the transit systems. Moreover, ODMTS addresscongestion and economy of scale by providing high-frequency services along high-density corridors.They have been shown to bring substantial convenience and cost benefits in simulation and pilotstudies in the city of Canberra, Australia (Mah´eo et al. 2019), the transit system of the University a r X i v : . [ m a t h . O C ] J a n asciftci and Van Hentenryck: Captuting Travel Mode Adoption in ODMTS of Michigan (Van Hentenryck 2019), the Ann-Arbor/Ypsilanti region in Michigan (Basciftci andVan Hentenryck 2020), and the city of Atlanta (Dalmeijer and Van Hentenryck 2020). ODMTSdiffer from micro-mobility in that they are designed and operated holistically. The ODMTS designthus becomes a variant of the hub-arc location problem (Campbell et al. 2005a,b): It is an opti-mization model that decides which bus/rail lines to open in order to maximize convenience andminimize costs (Mah´eo et al. 2019). This optimization model uses, as input, the current demand,i.e., the set of origin-destination pairs in the existing transit system.This paper is motivated by a significant worry of transit agencies: the need to capture latentdemand in the design of ODMTS . This concern, which recognizes the complex interplay betweentransit agencies and riders (Cancela et al. 2015), was also raised by Campbell and Van Woensel(2019): they articulated the potential of (1) leveraging data analytics within the planning processand (2) proposing transit systems that encourage riders to switch transportation modes. As aconsequence, rider preferences and the induced mode choices should be significant factors in thedesign of transit systems (Laporte et al. 2007). Yet, many transit agencies only consider existingriders when redesigning their network. But, as convenience improves, more riders may decide toswitch modes and adopt the transit system instead of traveling with their personal vehicles. Byignoring this latent demand, the transit system may be designed suboptimally, resulting in highercosts or poor quality of service. Basciftci and Van Hentenryck (2020) illustrated these points bycomparing the designs of an ODMTS that differ by whether they capture latent demand. Theresults highlighted the significant cost increase when latent demand is not considered as the designunder-invested in fixed routes and over-utilized on-demand shuttles. Note also that Agatz et al.(2020) highlighted the integration of stakeholder behavior in optimization models as a fundamentaltheme to address grand challenges in the next generation of transportation systems.This paper proposes a general framework to design ODMTS based on existing and latent demand .The mode preference of a rider is expressed through a choice model that, given a route in theODMTS, determines whether the rider adopts the ODMTS or continues to use her personal vehicle.At a high level, the framework is a game between the transit agency and potential riders. The transitagency proposes an ODMTS design and routes for potential riders. Riders decide whether to adoptthe transit system, inducing revenues and costs to the transit agency. This game is embodied in a bilevel optimization model, which is solved using an exact decomposition method. The method usestraditional Benders optimality cuts and nogood cuts, which are strengthened by valid inequalitiesexploiting the network structure. The approach is validated on a real case study. The contributionsof the paper can be summarized as follows:1. The paper presents a bilevel optimization approach to the design of ODMTS under rideradoption constraints. The bilevel optimization problem consists of (i) a leader problem that asciftci and Van Hentenryck: Captuting Travel Mode Adoption in ODMTS determines the transit network design and takes into account rider preferences as well asrevenues and costs from adopting riders; (ii) follower problems identify the most cost-efficientand convenient route for riders. The personalized choice models are integrated into the leaderproblem to represent the interplay between the transit agency and rider preferences. Sincethe model assumes a fixed cost for riding the transit system, the choice models capture thedesired convenience of the trips.2. The paper proposes an exact decomposition method for the bilevel optimization model. Themethod combines a Benders decomposition approach with combinatorial cuts that ensure theconsistency between rider choices and the leader decisions. Furthermore, the paper presentsvalid inequalities that significantly strengthen these combinatorial cuts, as well as prepro-cessing steps that reduce the problem dimensionality. These enhancements produce orders ofmagnitude improvements in computation times.3. The paper validates the approach using a comprehensive case study that considers the transitagency of the broad Ann Arbor/Ypsilanti region in Michigan. The case study demonstrates thebenefits of the proposed approach from adoption, convenience, cost and accessibility perspec-tives. The results highlight that the ODMTS decreases trip durations by up to 53% comparedto the existing system, induces high adoption rates for the latent demand, and operates wellinside the budget of the transit agency.The rest of the paper is organized as follows. Section 2 reviews the relevant literature. Section 3presents the problem setting and the resulting bilevel ODMTS design problem with latent demandand rider choices. Section 4 proposes theoretical results on trip durations in ODMTS. Section5 presents an exact decomposition algorithm and derives valid inequalities and problem-specificenhancements. Section 6 demonstrates the performance of the proposed approach in the case study.Section 7 concludes the paper with final remarks.
2. Related Literature
The design of transit networks organized around hubs is an emerging research area, with thegoal of ensuring reliable service and economies of scale (Farahani et al. 2013a). Campbell et al.(2005a,b) introduce a variant of this problem, the hub-arc location problem, to select the set ofarcs to open between hubs while optimizing the flow with minimum cost. Alumur et al. (2012)consider multimodal hub location and hub network design problem by taking into account bothcost and convenience aspects in satisfying demand. Mah´eo et al. (2019) examine this problem in thecontext of ODMTS, pioneering on-demand shuttles to serve all or parts of the trips, and allowingroutes that do not necessarily involve arcs between hubs. The goal is to obtain a transit networkdesign that minimizes the cost and duration of the overall trips. In these studies, user behaviour asciftci and Van Hentenryck:
Captuting Travel Mode Adoption in ODMTS is not explicitly captured within the transit network design process; instead the objective functionminimizes a weighted combination of the system cost and the travel times of the trips for existingriders of the transit system.Capturing information about rider routes into transit network design is a critical component ofensuring accessible public transit systems (Sch¨obel 2012). Guan et al. (2006) model a joint lineplanning and passenger assignment problem as a single-level mixed integer program, where ridersselect their routes during network design and the route durations are part of the objective functionalong with the costs of the transit network. Bornd¨orfer et al. (2007) study this line planning problemunder these two competing objectives by utilizing a column-generation approach as its solutionmethodology. Sch¨obel and Scholl (2006) consider identifying the routes that minimize the overalltravel time of the riders under a budget constraint on the transit network design.Another relevant line of research involving transit network design problems focuses on maximiz-ing population coverage by examining population in the neighborhood of the potential stations(Wu and Murray 2005, Matisziw et al. 2006, Curtin and Biba 2011). In these settings, travel costscan be jointly optimized with the maximization of ridership capture (Guti´errez-Jarpa et al. 2013).Mar´ın and Garc´ıa-R´odenas (2009) integrate user behavior into this planning problem by represent-ing the choices of the riders according to the network design and the cost of the resulting trip incomparison to their current mode of travel. Laporte et al. (2011a) extend this problem under thepossibility of arc failure; they aim at providing routes faster than other modes for a high proportionof the trips under a budget constraint. In these approaches, user choices are associated with thecosts of the trips to represent their mode switching behavior (Correa and Stier-Moses 2011). Dueto the complexity in modeling and solving these problems with respect to the dual perspectives oftransit agency and riders, these studies focus on single-level formulations.As should be clear at this point, the design of public transit systems involve decision-makingprocesses from multiple entities, including transit agencies and riders (Laporte et al. 2011b). Bileveloptimization is thus a key methodology to formulate these multi-player optimization problems andit has been applied to several urban transit network design problems (LeBlanc and Boyce 1986,Farahani et al. 2013b). This setting involves a leader who determines a set of decisions, and thefollowers determine their actions under these decisions. Fontaine and Minner (2014) study thediscrete network design problem where the leader designs the network to reduce congestion undera budget constraint and the riders search for the shortest path from their origin to destination. Yaoet al. (2012) and Yu et al. (2015) consider this setting over multimodal transit networks with busesand cars; they determine which bus legs are open and with which frequencies, and ensure trafficequilibrium. Bilevel optimization is also studied in toll optimization problems over multicommoditytransportation networks by maximizing the revenues obtained through tolls in the leader problem asciftci and Van Hentenryck: Captuting Travel Mode Adoption in ODMTS and obtaining the paths with minimum costs in the follower problem (Labb´e et al. 1998, Brotcorneet al. 2001). These studies are then extended to a more general problem setting when the underlyingnetwork is jointly optimized while considering the pricing aspect (Brotcorne et al. 2008). Pintoet al. (2020) also apply bilevel optimization to the joint design of multimodal transit networks andshared autonomous mobility fleets. Here, the upper-level problem is a transit network frequencysetting problem that allows for the removal of bus routes.Colson et al. (2007) provide an overview of bilevel optimization approaches with solution method-ologies and discusses traffic equilibrium constraints that may complicate the problem further whencongestion is considered. Colson et al. (2005), Sinha et al. (2018) further present possible solutionmethodologies to address bilevel optimization problems. Due to the complex nature of the bilevelproblems involving transportation networks, various studies (e.g., Bianco et al. (2009), Yao et al.(2012), Kalashnikov et al. (2016)) focus on developing heuristics as its solution methodology. Onthe other hand, Gao et al. (2005), Fontaine and Minner (2014), Yu et al. (2015) provide refor-mulation and decomposition-based solution methodologies to provide exact solutions for this classof problems. Despite this extensive literature on bilevel optimization in transportation problems,personalized rider preferences regarding transit routes have not been incorporated into the networkdesign, resulting in the omission of the latent demand during planning. To our knowledge, Bas-ciftci and Van Hentenryck (2020) provide the first study that focuses on this bilevel optimizationproblem by associating rider choices with the cost and time of those trips in the ODMTS system.The studied problem considers the specific case where the transit agency and riders subsidize thecost of the trips equally, leading rider choices to be based on a combination of these cost andconvenience. However, if pricing is not equally subsidized between these entities or rider prefer-ences solely depend on the time of the trips, then the problem becomes much more challengingto solve. To address these challenges, this paper extends this line of research and models riderpreferences that depends on trip convenience for a transit system with fixed ticket prices. Since thissetting substantially complicates exact solution methods, this paper studies an exact decompositionmethod that exploits Benders optimality cuts, combinatorial cuts, and dedicated valid inequalitiesstrengthening the combinatorial cuts. This paper also contains an extensive computational studythat includes rider adoption, cost, revenue and accessibility aspects on various instances.
3. The Bilevel Optimization Approach
This section presents a bilevel optimization approach for the ODMTS design based on a gametheoretic framework between the transit agencies and riders. The transit agency is the leader whodetermines the transit network design of the system, whereas the riders are the followers whodecide whether to adopt the transit system as their travel mode. The proposed framework aims asciftci and Van Hentenryck:
Captuting Travel Mode Adoption in ODMTS at designing the ODMTS network while taking into account both existing transit riders and thelatent demand, i.e., riders who observe the system design and performance, and decide their travelmode accordingly. Section 3.1 describes the problem setting, Section 3.2 presents the optimizationmodel, and Section 3.3 presents preprocessing steps for dimensionality reduction. This proposedproblem stays as close as possible to the original setting of the ODMTS design Mah´eo et al. (2019). The input is defined in terms of a set N of nodes associated with bus stops, a subset H ⊆ N of whichare designated as hubs. Each trip r ∈ T has an origin stop or r ∈ N , a destination stop de r ∈ N ,and a number of riders taking that trip p r ∈ Z + . The time and distance between each pair i, j ∈ N are denoted by t ij and d ij respectively. These parameters can be asymmetric but are assumed tosatisfy the triangular inequality. Since the goal is to optimize both inconvenience (e.g., travel time)and cost, the optimization problem uses a parameter θ ∈ [0 ,
1] to balance both objectives using aconvex combination. In particular, inconvenience is associated with the travel time and multipliedby θ , while travel cost is associated with the travel distance and multiplied by 1 − θ .Riders pay a fixed cost φ to use the transit system, irrespective of their routes. This fixed costper rider, ϕ = (1 − θ ) φ , becomes a revenue to the transit agency. If a leg between the hubs h, l ∈ H is open, then the transit agency incurs an investment cost β hl = (1 − θ ) b n d hl , where b is the cost ofusing a bus per mile and n is the number of buses operating in each open leg within the planninghorizon. Moreover, the transit agency incurs a service cost for each trip r ∈ T that consists ofthe weighted cost and inconvenience of using bus legs between hubs and on-demand shuttle legsbetween bus stops. More specifically, the weighted cost and inconvenience for an on-demand shuttlebetween i and j for the transit agency is given by γ rij = (1 − θ ) g d ij + θt ij , where g is the costof using a shuttle per mile. Since the operating cost of buses are already considered within theinvestment costs, each bus leg between the hubs h, l ∈ H in trip r ∈ T only incurs an inconveniencecost τ rhl = θ ( t hl + S ), where S is the average waiting time of a bus.To represent the latent demand for the transit system, the set of trips is partitioned into twogroups: riders from T (cid:48) ⊆ T currently travel with their personal vehicles, and riders from T \ T (cid:48) currently use the transit system. Riders from T (cid:48) may switch their travel mode from their personalvehicles to the ODMTS, depending on the inconvenience of the route assigned to them. Conse-quently, each trip r ∈ T (cid:48) is associated with a binary choice model C r that determines, given aproposed route, whether its rider adopts the ODMTS. More precisely, given route vectors x r , y r for trip r , C r ( x r , y r ) holds if trip r adopts the ODMTS. Since the price of the ODMTS is fixed, thispaper assumes that the choice model only depends on the trip inconvenience which is captured bythe function f r ( x r , y r ) = (cid:88) h,l ∈ H ( t hl + S ) x rhl + (cid:88) i,j ∈ N t ij y rij . asciftci and Van Hentenryck: Captuting Travel Mode Adoption in ODMTS Moreover, the paper assumes that a rider will adopt the ODMTS if her trip inconvenience in thetransit system is not more than α r times of her direct trip duration t rcur (using her personal vehicle),where α r is a parameter associated with the rider. More formally, the paper adopts the followingchoice model C r ( x r , y r ) ≡ ( f r ( x r , y r ) ≤ α r t rcur ) . (1)Although trip inconvenience is only characterized in terms of travel time in this choice model, itcan be generalized to include additional considerations. For instance, it is traditional to converttransfers into travel time to avoid trips with too many transfers. The decision variables of the optimization model are as follows: Binary variable z hl is 1 if the busleg between the hubs h, l ∈ H is open. Additionally, for each trip r ∈ T , binary variables x rhl and y rij represent whether the route selected for trip r utilizes the bus leg between the hubs h, l ∈ H , andthe shuttle leg between the stops i, j ∈ N , respectively. The optimization model also uses a binarydecision variable δ r for each trip r ∈ T (cid:48) to represent whether its rider switches her travel mode tothe ODMTS.The optimization model is given in Figure 1: it consists of a leader model and a follower problemfor each trip r . The leader problem (Equations (2a)– (2e)) determines the network design betweenthe hubs for the ODMTS, whereas the follower problem (Equations (3a)–(3f)) identifies routes foreach trip r ∈ T by utilizing the legs of the network design along with the on-demand shuttles.The leader objective (2a) minimizes the sum of (i) the investment cost of opening bus legs and(ii) the weighted cost and inconvenience of the trips of the existing riders, minus (iii) the revenues ofthose riders adopting the ODMTS. The modeling assumes that existing riders will remain loyal tothe ODMTS, given that case studies have demonstrated that ODMTS improves rider convenience.Constraint (2b) guarantees weak connectivity between the hubs by ensuring the sum of incomingand outgoing open legs to be equal to each other for each hub. Constraint (2c) captures the modechoice of the riders in T (cid:48) based on the ODMTS routes.For a given trip r , the follower problem (3) minimizes the lexicographic objective function (cid:104) d r , f r (cid:105) ,where d r represents the cost and inconvenience of trip r and f r breaks potential ties by returninga most convenient route for the rider of trip r . Observe that this choice is aligned with the travelchoice model. Constraint (3d) enforces flow conservation for the bus and shuttle legs used in trip r . Constraint (3e) ensures that the route only considers open bus legs. Proposition 1
For any z ∈ { , } | H |×| H | , a lexicographic minimizer of problem (3) exists and thelexicographic minimum is unique. asciftci and Van Hentenryck: Captuting Travel Mode Adoption in ODMTS min z hl (cid:88) h,l ∈ H β hl z hl + (cid:88) r ∈ T \ T (cid:48) p r d r + (cid:88) r ∈ T (cid:48) p r δ r ( d r − ϕ ) (2a)s.t. (cid:88) l ∈ H z hl = (cid:88) l ∈ H z lh ∀ h ∈ H (2b) δ r = C r ( x r , y r ) ∀ r ∈ T (cid:48) (2c) z hl ∈ { , } ∀ h, l ∈ H (2d) δ r ∈ { , } ∀ r ∈ T (cid:48) (2e)where ( x r , y r , d r ) are a solution to the optimization problemlex-min x rhl ,y rij ,d r (cid:104) d r , f r (cid:105) (3a)s.t. d r = (cid:88) h,l ∈ H τ rhl x rhl + (cid:88) i,j ∈ N γ rij y rij (3b) f r = (cid:88) h,l ∈ H ( t hl + S ) x rhl + (cid:88) i,j ∈ N t ij y rij (3c) (cid:88) h ∈ H if i ∈ H ( x rih − x rhi ) + (cid:88) i,j ∈ N ( y rij − y rji ) = , if i = or r − , if i = de r , otherwise ∀ i ∈ N (3d) x rhl ≤ z hl ∀ h, l ∈ H (3e) x rhl ∈ { , } ∀ h, l ∈ H, y rij ∈ { , } ∀ i, j ∈ N. (3f) Figure 1 The Bilevel Optimization Model for ODMTS Design with Travel Mode Adoption.
This proposition follows because the feasible space of a follower subproblem is not empty, sincethere is always a direct shuttle route from or r to de r . Moreover, each component of the objectiveis bounded from below.It is useful to think about the bilevel optimization problem in the following sequential way: (1)the transit agency designs the ODMTS to balance its cost and the convenience of its riders; (2)riders are then presented with the route in the ODMTS that maximizes their convenience; and(3) riders decide whether to adopt the ODMTS, to drive with their own vehicles, or to forgo thetrip. When a new rider adopts the ODMTS, the transit agency receives her fare. Note that sub-objective d r contains sub-objective f r multiplied by θ , and the lexicographic objective breaks tiesby choosing the optimal value of d r with the smallest value of f r .Observe that, once a design z is chosen, the mode choice of every rider is uniquely determined,which is important for computational reasons. Moreover, the follower problem has a totally uni-modular constraint matrix, and can be solved as a linear program using an objective of the form M d r + f r for a suitably large M . asciftci and Van Hentenryck: Captuting Travel Mode Adoption in ODMTS In the rest of the paper, a solution z ∈ { , } | H |×| H | is called an ODMTS design. Moreover, giventwo ODMTS designs z and z , z ≤ z iff z hl ≤ z hl for all h, l ∈ H . This means that z opens fewerbus legs than z . This section presents a number of preprocessing steps to simplify the bilevel optimization problem.
The objective function of the leader problem(2a) includes bilinear terms δ r d r for all trips r ∈ T (cid:48) . These terms can be linearized with an exactMcCormick reformulation since δ r is a binary variable. In particular, a bilinear term δ r d r ( r ∈ T (cid:48) )in the objective function is replaced with a new variable ν r , and the following constraints are addedto the leader problem: ν r ≤ M r δ r (4a) ν r ≤ d r (4b) ν r ≥ d r − M r (1 − δ r ) (4c) ν r ≥ , (4d)where the term M r is an upper bound on the value of d r . The following result is helpful in findingsuch a bound. Proposition 2
Let r ∈ T and ( d r ∗ , f r ∗ ) and ( d r ∗ , f r ∗ ) be the optimal objective values of the followerproblem under the ODMTS designs z and z . If z ≤ z , then d r ∗ ≥ d r ∗ .Proof: If z ≤ z , then z has at least as many bus legs as z . Hence, the feasible region of thefollower problem under z is a subset of the feasible region under z . (cid:3) For a given ODMTS design and a trip r , the follower problem (3) returns a path of least cost andinconvenience between or r and de r . As a result, by Proposition 2, the ODMTS design with nobus leg gives an upper bound on the value of d r . Similarly, the ODMTS design with all legs openreturns a lower bound that can be inserted in the leader problem to strengthen the formulation. The follower problem (3) considers all arcs between nodes i, j ∈ N for shuttle legs. However, only a subset of these arcs are needed due to the triangular inequalityon arc weights. In particular, the follower problem needs only to consider arcs i) from origin tohubs, ii) from hubs to destination, and iii) from origin to destination. This set of arcs is denoted as A r in the following. As a result, the bilevel optimization problem only uses the following decisionvariables for each trip r : y ror r h , y rhde r ∈ { , } ∀ h ∈ Hy ror r de r ∈ { , } . asciftci and Van Hentenryck: Captuting Travel Mode Adoption in ODMTS
4. Analytical Results on Trip Durations
This section presents analytical results that show how ODMTS designs impact the durations ofthe routes proposed to riders. It focuses on the general case where the trip origin and destinationare not hub locations: each such trip is of two possible forms: i) a combination of legs includingshuttle trips from origin to a hub and from a hub to destination along with bus ride(s) betweenthe hubs or ii) a direct shuttle ride from origin to destination. Section 4.1 derives upper and lowerbounds on trip durations when new arcs are added or existing arcs are removed from an ODMTSdesign. Section 4.2 identifies certain cases where a trip duration does not worsen with the additionor removal of arcs from a given design. These results are used in Section 5 in dedicated inequalitiesthat link ODMTS designs and rider choices.
This section first derives upper bounds on trip durations when new arcs are added to an ODMTSdesign. It then derives corresponding lower bounds when arcs are removed from a design.
Theorem 1
Consider transit network design z and assume that the optimal route for trip r includes shuttle trips from origin or r to hub m and from hub n to destination de r with a trip time t . For any network z ≥ z , the time t of the optimal route for trip r admits the following upperbound: t ≤ t + (1 − θ ) θ g (cid:18) d or r m + d nde r − min h,l ∈ H { d or r h + d lde r } (cid:19) = U B . (5) Proof:
Without loss of generality, assume that the optimal route of trip r under design z includes the shuttle trips from origin or r to hub h (cid:48) and from hub l (cid:48) to destination de r . Let d r ∗ = θt + (1 − θ ) g ( d or r m + d nde r ) and d r ∗ = θt + (1 − θ ) g ( d or r h (cid:48) + d l (cid:48) de r ) be the optimal objective functionvalues under designs z and z . If z ≥ z , then d r ∗ ≥ d r ∗ . It follows that: θt + (1 − θ ) g ( d or r m + d nde r ) ≥ θt + (1 − θ ) g ( d or r h (cid:48) + d l (cid:48) de r ) θt + (1 − θ ) g ( d or r m + d nde r − ( d or r h (cid:48) + d l (cid:48) de r )) ≥ θt t + (1 − θ ) θ g ( d or r m + d nde r − ( d or r h (cid:48) + d l (cid:48) de r )) ≥ t t + (1 − θ ) θ g (cid:18) d or r m + d nde r − min h,l ∈ H { d or r h + d lde r } (cid:19) ≥ t . (cid:3) Corollary 1 If m is the closest hub to origin or r and n is the closest hub to destination de r , thenthe upper bound in Theorem 1 reduces to t ≤ t . This corollary indicates that, if the route of a trip includes shuttle components from its origin anddestination to the closest hubs, then addition of arcs only makes the duration of the trip better. asciftci and Van Hentenryck:
Captuting Travel Mode Adoption in ODMTS For example, if a rider is already adopting the ODMTS under the initial design, then these riderswill keep adopting the system under the new design as the duration of the trip can only get shorter.
Theorem 2
Consider ODMTS design z and assume that the optimal route for trip r is a directshuttle trip with trip time t . For any ODMTS design z ≥ z , the time t of the optimal route fortrip r satisfies the following upper bound: t ≤ max (cid:26) t , t + (1 − θ ) θ g (cid:18) d or r de r − min h,l ∈ H { d or r h + d lde r } (cid:19)(cid:27) = U B . (6) Proof:
Under z , the optimal route for trip r involves either a direct trip from origin or r todestination de r or a combination of rides involving shuttle trips from origin or r to some hub h (cid:48) ,from some hub l (cid:48) to destination de r , and bus rides between hubs h (cid:48) , l (cid:48) . In the first case, observethat t is an upper bound on the trip duration t . In the second case, θt + (1 − θ ) gd or r de r ≥ θt + (1 − θ ) g ( d or r h (cid:48) + d l (cid:48) de r ) θt + (1 − θ ) g ( d or r de r − ( d or r h (cid:48) + d l (cid:48) de r )) ≥ θt t + (1 − θ ) θ g (cid:18) d or r de r − min h,l ∈ H { d or r h + d lde r } (cid:19) ≥ t . Depending on z , both cases are possible and the result follows. (cid:3) When z has no hub legs open, the optimal route for trip r takes time t or r de r . Therefore, for anynetwork z ≥ z , the upper bound using Theorem 2 becomes t ≤ max (cid:26) t or r de r , t or r de r + (1 − θ ) θ g (cid:18) d or r de r − min h,l ∈ H { d or r h + d lde r } (cid:19)(cid:27) . (7)If this upper bound value is duration of the direct route, then the trip must be served by anon-demand shuttle. The following corollary can thus be used as a pre-processing step to identifydirect shuttle trips. Corollary 2
For any trip r ∈ T , if min h,l ∈ H { d or r h + d lde r } ≥ d or r de r , then the trip will be servedwith on-demand shuttles only.Proof: The proof is by contradiction. Suppose that, trip r is served with on-demand shuttlesto and from hubs, and bus leg(s) between hubs under a network z where z ≥ z . Without loss ofgenerality, assume that the origin is connected to hub m and hub n is connected to the destination.Then, d or r m + d nde r ≥ min h,l ∈ H { d or r h + d lde r } ≥ d or r de r . Moreover, the time of this route is at leastthe time of the direct trip by the triangle inequality, contradicting the hypothesis by definition of d r . (cid:3) The next results derive lower bounds on trip durations. asciftci and Van Hentenryck:
Captuting Travel Mode Adoption in ODMTS Theorem 3
Consider ODMTS design z , and assume that the optimal route for trip r includesthe shuttle trips from origin or r to hub m and from hub n to destination de r with a trip time t .For any design z such z ≥ z , the time t of the optimal route for trip r has a lower bound as t ≥ t + (1 − θ ) θ g (cid:18) d or r m + d nde r − max (cid:26) max h,l ∈ H { d or r h + d lde r } , d or r de r (cid:27)(cid:19) = LB . (8) Proof:
Observe first that the optimum d r value for trip r under z is greater than or equalto the corresponding value under network design z . Without loss of generality, assume that theoptimum route of trip r under design z includes either the shuttle trips from origin or r to hub h (cid:48) and from hub l (cid:48) to destination de r , or a direct shuttle trip from origin or r to destination de r . Inthe first case, θt + (1 − θ ) g ( d or r m + d nde r ) ≤ θt + (1 − θ ) g ( d or r h (cid:48) + d l (cid:48) de r ) t + (1 − θ ) θ g ( d or r m + d nde r − ( d or r h (cid:48) + d l (cid:48) de r )) ≤ t t + (1 − θ ) θ g (cid:18) d or r m + d nde r − max h,l ∈ H { d or r h + d lde r } (cid:19) ≤ t . In the second case, t + (1 − θ ) θ g ( d or r m + d nde r − d or r de r ) ≤ t , completing the proof. (cid:3) Theorem 4
Consider ODMTS design z , and assume that the optimal route for trip r is a directshuttle trip from origin or r to destination de r with a trip time t . For any network z , z ≥ z , thetime t of the optimum route for trip r will be t = t = LB .Proof: As the feasible solutions under z is a subset of the feasible solutions under z , theoptimum route of trip r with respect to the follower problem will remain as a direct shuttle tripfrom origin or r to destination de r . (cid:3) This section presents two specific but important cases where the duration of the studied trip cannotbecome worse when more bus legs are added. The first case considers a trip route where shuttlesconnect the origin and destination to hubs and where additional arcs do not make closer hubsavailable. Given ODMTS design z , define the set of active hubs H ( z ) = { h ∈ H : (cid:80) l ∈ H z hl > } .Due to the weak connectivity constraint (2b), (cid:80) l ∈ H z hl > (cid:80) l ∈ H z lh > h ∈ H .Define the following minimum distances from/to node i ∈ N to/from any active hub under z as −→ d min i ( z ) := min h ∈H ( z ) { d ih } and ←− d min i ( z ) := min h ∈H ( z ) { d hi } . Finally, define −→ W i ( z ) = { h ∈ H \ H ( z ) : d ih < −→ d min i ( z ) } and ←− W i ( z ) = { h ∈ H \ H ( z ) : d hi < ←− d min i ( z ) } as the set of non-active hubs that are asciftci and Van Hentenryck: Captuting Travel Mode Adoption in ODMTS closer to the origin and destination than the active hubs respectively. The next theorem showsthat, if the non-active hubs closer to the origin and destination of a trip r in the current designremain inactive in a larger design, the duration of trip r can only improve. Theorem 5
Consider ODMTS design z , and assume that the optimal route for trip r includesthe shuttle trips from origin or r to hub m , and from hub n to destination de r , with a trip time t .If m and n are the closest active hubs to the origin and destination, i.e., d or r m = −→ d min or r ( z ) and d nde r = ←− d min de r ( z ) , then for any network design z satisfying z ∈ { z ∈ { , } | H |×| H | : z hl = 1 ∀ ( h, l ) s.t. z hl = 1 , (cid:88) l ∈ H z hl = 0 ∀ h ∈ −→ W or r ( z ) , (cid:88) l ∈ H z hl = 0 ∀ h ∈ ←− W de r ( z ) } , then the time t of the optimal route for trip r in z satisfies t ≤ t .Proof: By definition of z , −→ d min or r ( z ) = −→ d min or r ( z ) and ←− d min de r ( z ) = ←− d min de r ( z ). This implies that d or r h ≥ −→ d min or r ( z ) and d hde r ≥ ←− d min de r ( z ) for all hubs h ∈ H ( z ). Since the cost only depends on thedistance of the shuttle rides, the cost of the optimal route under z is g ( −→ d min or r ( z ) + ←− d min de r ( z )), andthe corresponding cost under z become g ( d (cid:48) + d (cid:48) ), where d (cid:48) ≥ −→ d min or r ( z ) and d (cid:48) ≥ ←− d min de r ( z ). Sincethe latter cost is greater than or equal to the former one, and z ≥ z , it must be the case that t ≤ t . (cid:3) The next result identifies the set of arcs whose removal from the transit design do not impactthe duration of the associated trip.
Theorem 6
Consider design z , and assume that the optimal route of trip r takes time t . Ifdesign z is obtained from z by removing some arcs that are not used on the optimal route for r ,then the trip duration for r under z remains t .
5. Solution Methodology
This section proposes a solution methodology that decompose the bilevel problem (2) into a masterproblem and subproblems. The approach combines a traditional Benders decomposition to generateoptimality cuts with combinatorial Benders cuts to reconcile rider choices in the master problemwith those induced by the optimal routes in the follower subproblems. More specifically, the masterproblem consists of the leader problem with variables ( { z hl } h,l ∈ H , { δ r } r ∈ T (cid:48) , { d r } r ∈ T ) where the riderchoice constraint (2c) is relaxed. In each iteration, the follower subproblems are solved to generateoptimality cuts on variables d r . In addition, combinatorial cuts are introduced to guarantee the asciftci and Van Hentenryck: Captuting Travel Mode Adoption in ODMTS consistency between C r ( x r , y r ) and the master variable δ r . These “basic” combinatorial cuts arefurther improved using the results of Section 4. The proposed decomposition algorithm convergeswhen the lower bound obtained by the master problem, and the upper bound constructed from thefeasible solutions of the subproblems are close enough. The rest of this section formally introducesthe decomposition algorithm along with the several enhancements. Section 5.1 and Section 5.2present the master problem and the Benders subproblems. Section 5.3 proposes the cut generationprocedure for the optimality cut and the combinatorial cuts for coupling the choice model and thenetwork design. Section 5.4 proposes valid inequalities that enforce the relationship between theODMTS designs and the rider choices. Section 5.5 summarizes the decomposition algorithm, andproves its finite convergence. Finally, Section 5.6 discusses Pareto-optimal cut generation procedurefor enhancing the performance of the solution methodology. The initial master problem (9) of the solution algorithm can be formulated as a relaxation of thebilevel problem (2): min (cid:88) h,l ∈ H β hl z hl + (cid:88) r ∈ T \ T (cid:48) p r d r + (cid:88) r ∈ T (cid:48) p r ( ν r − ϕ ) (9a)s.t. (2b) , (2d) , (2e) , (4) . At each iteration of the algorithm, the relaxed master problem (9) determines an ODMTS designto be evaluated by the subproblems. Then, Benders cuts and combinatorial cuts are added to thisproblem following the procedure proposed in Section 5.3 to ensure the optimality and consistencybetween the rider choices in the master problem and the follower routes.
Given a transit network design solution { ¯ z hl } h,l ∈ H obtained by the master problem, the subproblemfor each trip r can be formulated using the follower problem (3) over the objective function ˆ d r = M d r + f r and its associated coefficients ˆ τ hl and ˆ γ ij . The resulting problem can be formulated asfollows: min (cid:88) h,l ∈ H ˆ τ rhl x rhl + (cid:88) i,j ∈ A r ˆ γ rij y rij (10a)s.t. (cid:88) h ∈ H if i ∈ H ( x rih − x rhi ) + (cid:88) i,j ∈ A r ( y rij − y rji ) = , if i = or r − , if i = de r , otherwise ∀ i ∈ N (10b) x rhl ≤ ¯ z hl ∀ h, l ∈ H (10c)0 ≤ x rhl ≤ , ∀ h, l ∈ H, ≤ y rij ≤ ∀ i, j ∈ A r . (10d) asciftci and Van Hentenryck: Captuting Travel Mode Adoption in ODMTS The model exploits the totally unimodular property of the follower problem under a given binarysolution { ¯ z hl } h,l ∈ H and uses the arc set A r , eliminating the unnecessary arcs for the on-demandshuttles. The dual of subproblem (10) is expressed in terms of the dual variables u ri and v rhl thatcorrespond to constraints (10b) and (10c):max ( u ror r − u rde r ) − (cid:88) h,l ∈ H ¯ z hl v rhl (11a)s.t. u rh − u rl − v rhl ≤ ˆ τ rhl ∀ h, l ∈ H (11b) u ri − u rj ≤ ˆ γ rij ∀ i, j ∈ A r (11c) u ri ≥ ∀ i ∈ N, v rhl ≥ ∀ h, l ∈ H. (11d)Note the primal subproblem (10) is always feasible and bounded as each trip can be served by adirect shuttle trip. Therefore, the dual subproblem (11) is feasible and bounded as well. Bendersoptimality cuts in the form d r ≥ (¯ u ror r − ¯ u rde r ) − (cid:88) h,l ∈ H z hl ¯ v rhl (12)are thus added to the master problem at each iteration using the optimal solution (¯ u r , ¯ v r ) of thedual subproblem. This section presents how to achieve the consistency of the rider choices in the master problemand those induced by the subproblems.
Definition 1 (Choice Consistency)
For a given trip r , the solution values { ¯ z hl } h,l ∈ H and ¯ δ r ofthe master problem are consistent with an optimal solution ( ¯x r , ¯y r , ¯ d r ) of the follower problem (3) under the design { ¯ z hl } h,l ∈ H if ¯ δ r = C r ( ¯x r , ¯y r ) . To ensure choice consistency, two possible cases must be considered:1. Solution values { ¯ z hl } h,l ∈ H and ¯ δ r are inconsistent with C r ( ¯x r , ¯y r ) when(a) ¯ δ r = 1 and C r ( ¯x r , ¯y r ) = 0;(b) ¯ δ r = 0 and C r ( ¯x r , ¯y r ) = 1.2. Solution values { ¯ z hl } h,l ∈ H and ¯ δ r are consistent with C r ( ¯x r , ¯y r ).By Proposition 1, the lexicographic minimum of problem (3) is unique and hence the routes ofthe lexicographic minimizers have the same cost and inconvenience under a given ODMTS design.Therefore, it is sufficient to relate the rider choices with the ODMTS design to ensure the consis-tency in these decisions. In particular, the first inconsistency (case 1(a)) can be eliminated withthe combinatorial cut (cid:88) ( h,l ):¯ z hl =0 z hl + (cid:88) ( h,l ):¯ z hl =1 (1 − z hl ) ≥ δ r . (13) asciftci and Van Hentenryck: Captuting Travel Mode Adoption in ODMTS Proposition 3
Constraint (13) removes inconsistency 1(a).
The second inconsistency (case 1(b)) can be eliminated with the cut (cid:88) ( h,l ):¯ z hl =0 z hl + (cid:88) ( h,l ):¯ z hl =1 (1 − z hl ) + δ r ≥ . (14) Proposition 4
Constraint (14) removes inconsistency 1(b).
Combinatorial cuts (13) and (14) guarantee the consistency between the rider choice variables andthe choices induced by ¯ z . We can further strengthen these cuts by exploiting the properties of thechoice model (1). Based on the analyses in Section 4, it is possible to add new valid inequalities tothe master problem at each iteration. This section proposes valid inequalities for the studied problem (2) to strengthen the relationshipbetween transit network design and rider choice variables. The first result utilizes the upper boundvalues on the duration of the trips.
Proposition 5
For ODMTS design z , consider the upper bound U B in Theorems 1 and 2. If arider of trip r adopts the transit system under z , and U B ≤ α r t rcur , then the rider also adopts theODMTS under any design z such that z ≤ z , Proposition 5 allows for obtaining combinatorial cuts.
Proposition 6
For a given transit network design z , if the condition in Proposition 5 holds fortrip r , then the consistency cut becomes (cid:88) ( h,l ): z hl =1 (1 − z hl ) + δ r ≥ . (15)The second result exploits the lower bound values on the duration of the trips. Proposition 7
For design z , consider the lower bound LB on trip duration as derived in Theo-rems 3 and 4. If the rider of trip r do not adopt the ODMTS under z , and LB ≥ α r t rcur , then therider will not adopt the ODMTS under any network design z such that z ≥ z . Proposition 7 enables deriving combinatorial cuts as follows.
Proposition 8
For a given design z , if the condition in Proposition 7 holds for trip r , thenconsistency cut becomes (cid:88) ( h,l ): z hl =0 z hl ≥ δ r . (16) asciftci and Van Hentenryck: Captuting Travel Mode Adoption in ODMTS By leveraging the lifted network designs introduced in Section 4.2, additional valid inequalitiesare proposed to enhance the consistency cuts as follows.
Proposition 9
For a given transit network design z , if the condition in Theorem 5 holds and therider of trip r adopts the ODMTS under z , then the consistency cut becomes: (cid:88) h ∈−→ W orr ( z ) ∪←− W der ( z ) ,l ∈ H z hl + (cid:88) ( h,l ): z hl =1 (1 − z hl ) + δ r ≥ Proof:
For any design z in the form described in Theorem 5, t ≤ t . Therefore, if the riderof trip r adopts the ODMTS under z , then t ≤ t ≤ α r t rcur . This result implies adoption of theODMTS for trip r by setting δ r to 1, under any design z . (cid:3) For a given transit network design z , if the arc(s) satisfying the condition in Theorem 6 areremoved from z , then the rider choices remain the same. Proposition 10
If the rider of trip r adopts the ODMTS under design z , then the followinginequality is valid: (cid:88) h ∈A r ( z ) (1 − z hl ) + (cid:88) ( h,l ): z hl =0 z hl + δ r ≥ On the other hand, if the rider of trip r does not adopt the ODMTS under z , then the followinginequality is valid: (cid:88) h ∈A r ( z ) (1 − z hl ) + (cid:88) ( h,l ): z hl =0 z hl ≥ δ r (19) Algorithm 1 summarizes the decomposition algorithm.
Proposition 11
Algorithm 1 converges in finitely many iterations.Proof:
First observe that there are finitely many combinatorial cuts (13) and (14) that can beadded to ensure the relationship between network design and rider preferences as both variablesare binary. As the inclusion of all consistency cuts converts the decomposition algorithm into aBenders decomposition algorithm, Algorithm 1 has finite convergence.
To further accelerate the solution methodology, the decomposition algorithm generate Pareto-optimal cuts (Magnanti and Wong 1981). Each subproblem is first solved to identify its optimal asciftci and Van Hentenryck:
Captuting Travel Mode Adoption in ODMTS Algorithm 1
Decomposition Algorithm Set LB = −∞ , U B = ∞ , z ∗ = ∅ . while U B > LB + (cid:15) do Solve the relaxed master problem (9) and obtain the solution ( { ¯ z hl } h,l ∈ H , { ¯ δ r } r ∈ T (cid:48) , { ¯ d r } r ∈ T ). Update LB . for all r ∈ T do Solve the subproblem (11) under { ¯ z hl } h,l ∈ H and obtain ( d r ∗ , f r ∗ ). Add optimality cut in the form (12) to the relaxed master problem (9). for all r ∈ T (cid:48) do if { ¯ z hl } h,l ∈ H and ¯ δ r are inconsistent with C r ( ¯x r , ¯y r ) then Add cuts in the form (13) or (14) to the relaxed master problem.
Add cuts discussed in Section 5.4 if the necessary conditions are satisfied. if C r ( x r , y r ) is 1 then Set ˆ δ r = 1. else Set ˆ δ r = 0. (cid:100) U B = (cid:80) h,l ∈ H β hl ¯ z hl + (cid:80) r ∈ T \ T (cid:48) p r d r ∗ + (cid:80) r ∈ T (cid:48) p r ˆ δ r ( d r ∗ − ϕ ). if (cid:100) U B < U B then
Update
U B as (cid:100) U B , z ∗ = ¯ z .objective function value, i.e., Υ r (¯ z ) for trip r and design ¯ z . The second step solves the Paretosubproblem max ( u ror r − u rde r ) − (cid:88) h,l ∈ H z hl v rhl (20a)s.t. u rh − u rl − v rhl ≤ ˆ τ rhl ∀ h, l ∈ H (20b) u ri − u rj ≤ ˆ γ rij ∀ i, j ∈ A r (20c)( u ror r − u rde r ) − (cid:88) h,l ∈ H ¯ z hl v rhl = Υ r (¯ z ) (20d) u ri ≥ ∀ i ∈ N, v rhl ≥ ∀ h, l ∈ H, (20e)where constraint (20d) is added and the objective function (20a) use a core point z that srictlysatisfies the weak connectivity constraint (2b). For a given η ∈ (0 , z hl = η for all h, l ∈ H . asciftci and Van Hentenryck: Captuting Travel Mode Adoption in ODMTS
6. Computational Study
This section presents a case study using a real data set from AAATA, the transit agency serving thebroader Ann Arbor and Ypsilanti area of Michigan. Section 6.1 introduces the experimental setting.Section 6.2 presents the ODMTS design under different configurations, and provides a detailedanalysis in comparison to the current transit system. Section 6.3 discusses the computationalperformance of the proposed solution approach.
The case study is based on the AAATA transit system that operates over 1,267 bus stops, some ofwhich are designated as hubs. It uses all the trips from 6 pm to 10 pm, i.e., which consists primarilyfrom commuting trips from work to home. There are 1,503 trips, each associated with an originand a destination bus stop, for a total of 5,792 riders. As the time and distance between bus stoppairs do not satisfy triangular inequality, a preprocessing step is applied to ensure this property.The experimental settings define different rider preferences depending on income levels. Morespecifically, as the income level of the riders increases, they become less tolerant to increases in tripduration. To this end, the experiments categorize the trips into three groups: high-income, middle-income, and low-income trips. This categorization in income levels is based on the destination stopof each trip, which is used as a proxy for the residential address of riders of that trip. Out of the1,503 trips, there are 476 low-income, 819 middle-income, and 208 high-income trips with 1,754,3,316, and 7,22 riders respectively. The experimental settings also assume that all low-income ridersmust use the transit system, whereas a certain percentage of riders from middle-income and high-income levels have the option to switch to the ODMTS from their current mode of travel withpersonal vehicles. In particular, 100%, 75%, and 50% of the trips from the low-income, middle-income and high-income categories must utilize the transit system, while the remaining ones have amode decision to make. Consequently, the value of the parameter α r in choice function (1) becomessmaller as the income level of the riders increases. In particular, α r is set to 1.5 and 2 for the tripsassociated with high-income and middle-income riders respectively.The bus cost per mile, b , is set to $ g , is set to $ φ of using the ODMTS $ n = 16 buses within the four-hour planning horizon for each openleg between the hubs with an average waiting time S of 7.5 minutes. The cost and inconvenienceparameter θ is 0.001 in the case study. Computational experiments are conducted using Gurobi9.0 as the solver on an Intel i5-3470T 2.90 GHz machine with 8 GB RAM. asciftci and Van Hentenryck: Captuting Travel Mode Adoption in ODMTS Figure 2 Network Design for the ODMTS with 10 Hubs.
Riders adopting ODMTS Existing riders Riders not adopting ODMTSIncome ODMTS direct AAATA ODMTS direct AAATA ODMTS direct AAATAlow NA 16.05 6.90 25.63 NAmedium 4.21 3.61 14.64 11.27 5.03 21.53 25.91 7.73 31.88high 4.61 4.61 15.42 9.84 5.31 21.06 19.96 8.37 29.77
Table 1 Trip Duration Analysis under 10 Hubs Design.
This section studies the ODMTS designs under different assumptions. Section 6.2.1 presents thebaseline ODMTS design and analyses its trip duration, adoption rates, and accessibility. Thefollowing sections examine how the baseline design changes under various assumptions. Sections6.2.2–6.2.5 examines configurations where (1) the cost of operating on-demand shuttles becomeshigher, (2) ridership increases, (3) travel choices are associated with riders who cannot affordpersonal vehicles, and (4) the number of hubs is increased and the ridership also grows. Finally,Section 6.2.6 compares the baseline with the four configurations with respoect to adoption rates,costs, and revenues obtained.
The baseline ODMTS design is depicted in Figure2 and it opens 7 hubs. Bus stops are colored by income level: red in low-income regions, grayin middle-income regions, and green in high-income regions. 89% of middle-income and 70% ofhigh-income riders adopt the ODMTS. asciftci and Van Hentenryck:
Captuting Travel Mode Adoption in ODMTS Riders adopting ODMTS Existing riders Riders not adopting ODMTSIncome ODMTS direct AAATA ODMTS direct AAATA ODMTS direct AAATAlow NA 18.39 6.91 25.63 NAmedium 3.21 2.82 12.19 14.16 5.03 21.53 27.38 7.23 29.14high 4.47 4.47 14.42 10.41 5.36 21.06 21.09 8.37 29.99
Table 2 Trip Duration Analysis under 10 Hubs Design with Increased On-Demand Shuttle Cost.
Table 1 reports various statistics on trip durations per income level for existing riders, ridersadopting the designed ODMTS, and those not adopting it. More precisely, the table uses thefollowing classification: i) riders who choose to adopt the ODMTS, ii) existing riders of the transitsystem who have no mode choice and thus necessarily adopt the ODMTS, and iii) riders withchoice who do not adopt the designed ODMTS. For each rider type and each income level, thetable reports three trip duration: the duration in the designed ODMTS, the duration of the direct trip, and the duration in the existing
AAATA transit system.The table highlights that the ODMTS routes are significantly shorter than those of the existingtransit system. For existing riders, the trip durations reduced by 37%, 48%, and 53% for low-income, middle-income, and high-income riders. This is critical since many of these riders may nothave an alternative transportation mean, and the ODMTS should not increase their travel time. Itis interesting to examine low-income riders whose trips take longer than 40 minutes in the existingtransit system. These trips, called low-income long transit (LILT) trips , constitute 28% of thelow-income rides and have an average transit time of 51.39 minutes. Under the baseline ODMTSdesign their average trip duration decreased to 32.21 minutes, a 37% reduction in transit time.For riders with mode choice, the durations of the existing transit routes are also significantlyreduced under the baseline ODMTS design. Interestingly, riders who adopt the ODMTS haveroutes almost as short as direct trips. The reduction in average trip duration is 71% and 70%for middle-income and high-income riders who adopt the ODMTS design, making the proposedODMTS substantially more attractive. The riders who do not adopt ODMTS have longer directtrip times: although the baseline ODMTS improves over the existing system, the reduction intransit time is not enough to induce a mode change.
Consider the case where thecost of on-demand shuttles increases by 50%. Figure 3 depicts the resulting ODMTS design whichnow opens all hubs and significantly increases their connectivity. The resulting ODMTS thus reliesmore on the bus network and less on the on-demand shuttles to serve the trips. The overall adoptionrates decreased slightly, as 85% of the middle-income and 69% of the high-income riders adoptthe system. This reduction in adoption is obviously directly linked to longer transit times. Table 2reports the average trip durations corresponding to each rider class under this setting. asciftci and Van Hentenryck:
Captuting Travel Mode Adoption in ODMTS Figure 3 Network Design for the ODMTS with 10 Hubs with Increased On-Demand Shuttle Cost.
Riders adopting ODMTS Existing riders Riders not adopting ODMTSIncome ODMTS direct AAATA ODMTS direct AAATA ODMTS direct AAATAlow NA 17.33 6.90 25.63 NAmedium 3.71 3.17 13.69 12.06 5.03 21.53 24.71 7.30 29.31high 4.53 4.53 14.39 10.09 5.31 21.06 20.85 8.38 30.17
Table 3 Trip Duration Analysis under 10 Hubs Design with Doubled Ridership.
This section examines the effect of increased rider-ship and studies the ODMTS design when the number of riders doubles. The resulting ODMTSdesign is illustrated in Figure 4. Again, all of the hubs are open and most of the bus legs from thebaseline design also operate in the new design. Furthermore, the design increases connectivity tothe lower-income communities by opening new bus legs in the corresponding regions. On the otherhand, adoption ratios in terms of the trips decreased marginally: 86% of middle-income and 69%of high-income riders utilize the resulting system.Table 3 presents the average trip durations for this design. Similar to the base case, the ODMTSperforms better than the current transit system. The trip durations for existing riders becomeslightly longer in the new design as more bus legs are utilized.
The next results concern accessibility, a critical met-ric for transit systems. As mentioned earlier, it is critical to ensure that low-income riders with nopersonal vehicles can be served by the transit system within reasonable transit times. Otherwise, asciftci and Van Hentenryck:
Captuting Travel Mode Adoption in ODMTS Figure 4 Network Design for the ODMTS with 10 Hubs with Doubled Ridership.
Riders adopting ODMTS Existing riders Riders not adopting ODMTSIncome ODMTS direct AAATA ODMTS direct AAATA ODMTS direct AAATAlow 32.40 11.99 51.50 13.01 5.65 19.07 49.24 10.05 50.46medium 3.71 3.17 13.69 12.06 5.03 21.53 24.71 7.30 29.31high 4.53 4.53 14.39 10.09 5.31 21.06 20.85 8.38 30.17
Table 4 Trip Duration Analysis under 10 Hubs Design with Doubled Ridership and Rider Choices for LILT trips. they may lose accessibility to jobs, education, health-care, and other amenities, since the trip dura-tion may become impractical. Consider again the LILT trips discussed in Section 6.2.1. To studyaccessibility, these trip riders are associated with a choice model with α r parameter set to 4. If atrip duration becomes longer than four times than the direct trip time, these riders will not ableto utilize the system anymore and lose accessibility to major opportunities. Out of 476 low-incometrips, there are 132 such LILT trips. The results are presented for the case of doubled ridership.Under this model, 96% of low-income trips utilize the ODMTS system and almost all of theLILT riders adopt the ODMTS, demonstrating the system ability to meet accessibility needs. TheODMTS design is the same as in Figure 4.Table 4 presents the trip duration results with this choice model and doubled ridership. As thedesign remains the same, the middle-income and high-income trips have the same adoption ratesand trip durations as in Table 3. LILT riders who adopt the ODMTS have an average trip durationless than three times that of the direct trip duration, and significantly shorter than the averagetrip duration by the existing transit system. On the other hand, LILT riders who do not adopt the asciftci and Van Hentenryck: Captuting Travel Mode Adoption in ODMTS Figure 5 Visualization of Sample LILT Trips Not Adopting ODMTS.
ODMTS have much longer trip durations, although they have shorter trips on average comparedto the current system. Figure 5 visualizes two of them, which are representative of trips for whichriders do not adopt the ODMTS. The trips share the same destination (denoted by “de”) but havedifferent origins (denoted by “or1” and “or2”). Their routes are illustrated with orange dashedroutes from origins to destination. More specifically, the trip with origin “or1” uses an on-demandshuttle to reach the closest open hubs, but results in a long trip due to many transfers betweenhubs. On the other hand, the trip with origin “or2” utilizes the on-demand shuttles for longertrip segments, but it involves a transfer to the city center, increasing the trip duration. In general,however, all the LILT trips with destination points in the vicinity of the eastern-most hub adoptthe ODMTS even when their origins are in the city center.
It is also interesting to study the effect of increasingthe number of hubs as ridership increases. Figure 6 presents the ODMTS design for 20 hubs anddoubled ridership. The resulting design opens 14 hubs and the bus network has a significantlybroader geographical coverage. In this setting, 85% of middle-income and 68% of high-income ridersadopt the ODMTS respectively. Table 5 reports the average trip duration: the more expansive busnetwork induces increases of 11%, 18%, 1% in average trip durations for low-income, middle-income,and high-income riders respectively. asciftci and Van Hentenryck:
Captuting Travel Mode Adoption in ODMTS Figure 6 Network Design for the ODMTS with 20 Hubs with Doubled Ridership.
Riders adopting ODMTS Existing riders Riders not adopting ODMTSIncome ODMTS direct AAATA ODMTS direct AAATA ODMTS direct AAATAlow NA 19.21 6.90 25.63 NAmedium 3.05 2.64 11.22 14.19 5.03 21.53 24.21 7.12 28.94high 4.02 4.02 14.02 10.17 5.31 21.06 20.26 8.41 29.54
Table 5 Trip Duration Analysis under 20 Hubs Design with Doubled Ridership.
Table 6 presents a detailed comparison of the ODMTSdesigns considered in this section with respect to the adoption, cost, and revenue. The revenue isassumed to be $ refers to the baseline design from Section 6.2.1, to the10 hub design with increased on-demand shuttle costs from Section 6.2.2, to the 10 hubdesign with doubled ridership from Section 6.2.3, to the 10 hub design with doubledridership and accessibility considerations from Section 6.2.4, and to the 20 hub designwith doubled ridership from Section 6.2.5. Columns “MI” and “HI” represent the percentage ofthe middle and high income riders who adopt the ODMTS. No low-income riders have a choicemodel, except in . Column “ asciftci and Van Hentenryck: Captuting Travel Mode Adoption in ODMTS Adoption Revenue & CostsMI (%) HI (%)
89 70 5792 (5402) 13505.00 2440.80 13553.31 0.46
85 69 5792 (5326) 13315.00 3564.59 17516.07 1.46
86 69 11584 (10700) 26750.00 4073.14 23847.84 0.11
86 69 11584 (10620) 26550.00 4073.14 23642.55 0.11
85 68 11584 (10608) 26520.00 4959.34 20285.19 -0.12
Table 6 Adoption, Cost and Revenue Comparison under Different ODMTS Settings.
10 hubs 20 hubsIncome
Table 7 Direct Trip Identification Analysis. per rider: it is obtained by deducting the revenue from the sum of the investment and travel costsand dividing by the number of ODMTS riders.The first interesting result is that the baseline design would be profitable for a price of $ $ $ $ $ , , and respectively. This section reports a number of computational results on the bilevel optimization model, includingthe impact of the preprocessing steps and the valid inequalities. Table 7 reports on the abilityto detect direct trips for instances with 10 and 20 hubs. 32% and 25% of the trips are identifiedas direct in the 10 hubs and 20 hubs instances. The percentage decreases for 20 hubs since thebus network is more expansive. In 10 hubs setting, the highest percentage of direct trips are high-income, as the hub locations are further away from the origin and destination of these trips. This asciftci and Van Hentenryck:
Captuting Travel Mode Adoption in ODMTS (a) 10 Hubs Instance. seconds) O p t i m a li t y ga p ( % ) Base caseEnhanced case (b) 20 Hubs Instance. seconds) O p t i m a li t y ga p ( % ) Base caseEnhanced case
Figure 7 Impact of the Enhancements on Computational Performance. percentage reduces substantially for 20 hubs for high-income class, especially in comparison toother rider classes, demonstrating the importance of hub locations and the number of hubs for thisanalysis.Figures 7a and 7b examine the benefits of the upper bound on the follower problem presentedin Section 3.3.1 in combination with the valid inequalities proposed in Section 5.4. The figuresuse the baseline instance with 10 Hubs studied in Section 6.2.1 and the 20 Hubs instance studiedin Section 6.2.5. They report the optimality gap and the runtime at each iteration of the algo-rithm. A time limit of 2 hours and 12 hours are specified for the instances with 10 Hubs and 20Hubs. The results demonstrate the significant computational impact of the upper bounds and validinequalities: the proposed decomposition algorithm is capable of producing high-quality solutionsquickly for this real case study and brings improvements of several orders of magnitude comparedto a decomposition algorithm that only relies on Benders and nogood cuts. Observe also that theperformance remains the same when doubling the ridership as the number of O-D pairs does notchange.
7. Conclusion
This paper studied how to integrate rider mode preferences into the design of ODMTS. Thisfunctionality was motivated by the desire to capture the impact of latent demand, a key worryof transit agencies. The paper proposed a bilevel optimization model to address this challenge,in which the leader problem determines the ODMTS design, and the follower problems identifythe most cost efficient and convenient route for riders under the chosen design. The leader modelcontains a choice model for every potential rider that determines whether the rider adopts theODMTS given her proposed route. asciftci and Van Hentenryck:
Captuting Travel Mode Adoption in ODMTS To solve the bilevel optimization model, the paper proposed a decomposition method thatincludes Benders optimal cuts and nogood cuts to ensure the consistency of the rider choices in theleader and follower problems. Moreover, to improve the computational efficiency of the method,the paper proposed upper and lower bounds on trip durations for the follower problems and validinequalities that strenghten the nogood cuts using the problem structure.The paper also presented an extensive computational study on a real data set from AAATA, thetransit agency for the broader Ann Arbor and Ypsilanti region in Michigan. The study consideredthe impact of a number of factors, including the price of on-demand shuttles, the number of hubs,and accessibility criteria. It analyzed the adoption rate of the ODTMS for various class of riders(low-income, middle-income, and high-income). The designed ODMTS feature high adoption ratesand significantly shorter trip durations compared to the existing transit system both for existingriders and riders who adopted the ODMTS. Under increased ridership and/or the availability ofmore hubs, trip durations may increase as they use more bus legs between hubs and less on-demandshuttles; however, adoption rates are not impacted much and the net profit of the transit agencyincreases significantly through economies of scale. The results further highlighted the benefits inaccessibility for low-income riders as their rip durations decrease and remain reasonable. Finally,the computational study demonstrated the efficiency of the decomposition method for the casestudy and the benefits of computational enhancements.Future work will consider more complex choice models (e.g., involving the cost of transfers)and/or restrictions on acceptable routes. Scaling the approach to large metropolitan areas is alsoa priority.
Acknowledgments
Many thanks to Julia Roberts at AAATA for sharing the transit data and for many interesting discus-sions. This research is partly supported by NSF Leap HI proposal NSF-1854684, and Department of EnergyResearch Award 7F-30154.
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