Multimodal Mobility Systems: Joint Optimization of Transit Network Design and Pricing
MMultimodal Mobility Systems: Joint Optimization ofTransit Network Design and Pricing
Qi Luo
Clemson UniversityClemson, SC, USA [email protected]
Samitha Samaranayake
Cornell UniversityIthaca, NY, USA [email protected]
Siddhartha Banerjee
Cornell UniversityIthaca, NY, USA [email protected]
Abstract
The performance of multimodal mobility systems relies onthe seamless integration of conventional mass transit ser-vices and the advent of Mobility-on-Demand (MoD) services.Prior work is limited to individually improving various trans-port networks’ operations or linking a new mode to an exist-ing system. In this work, we attempt to solve transit networkdesign and pricing problems of multimodal mobility sys-tems en masse. An operator (public transit agency or privatetransit operator) determines frequency settings of the masstransit system, flows of the MoD service, and prices for eachtrip to optimize the overall welfare. A primal-dual approach,inspired by the market design literature, yields a compactmixed integer linear programming (MILP) formulation. How-ever, a key computational challenge remains in allocating anexponential number of hybrid modes accessible to travelers.We provide a tractable solution approach through a decom-position scheme and approximation algorithm that acceler-ates the computation and enables optimization of large-scaleproblem instances. Using a case study in Nashville, Ten-nessee, we demonstrate the value of the proposed model. Wealso show that our algorithm reduces the average runtime by60% compared to advanced MILP solvers. This result seeks toestablish a generic and simple-to-implement way of revamp-ing and redesigning regional mobility systems in order tomeet the increase in travel demand and integrate traditionalfixed-line mass transit systems with new demand-responsiveservices.
CCS Concepts: • Applied computing → Transportation ;• Theory of computation → Market equilibria ; •
Math-ematics of computing → Combinatorial optimization . Keywords:
Multimodal mobility systems, Market design,Mixed Interger Programming
ACM Reference Format:
Qi Luo, Samitha Samaranayake, and Siddhartha Banerjee. 2021.Multimodal Mobility Systems: Joint Optimization of Transit Net-work Design and Pricing. In
Nashville ’21: ACM/IEEE InternationalConference on Cyber-Physical Systems, May 19–21, 2021, Nashville,TN .
ACM, New York, NY, USA, 11 pages. https://doi.org/
Nashville ’21, May 19–21, 2021, Nashville, TN © 2021 Association for Computing Machinery. https://doi.org/
Emerging Mobility-on-Demand (MoD) solutions, from ride-hailing platforms (e.g., Uber, Lyft) to on-demand bus ser-vices [1], provide responsive and reliable mobility options tourban commuters. Nevertheless, there are growing concernsthat MoD would compete with or substitute for conventionaltransit, increase traffic congestion, or adopt discriminatoryprice mechanisms [5, 7, 9]. This work proposes a new frame-work for building a multimodal mobility system as a firststep in the full integration of Mobility-on-Demand (MoD)services and regional mass transport networks. Mass transit(MT) operates as an affordable top-down system where usershave to adapt to available routes and service times. On theother hand, MoD platforms provide flexible door-to-doorservice but are not affordable for every traveler. This workaims to explore the opportunity to get the best of both sys-tems by tightly integrating the two systems at the systemdesign level, thereby understanding the true potential of afully integrated multimodal mobility system.A primary motive for establishing a multimodal mobilitysystem is to ensure that MT and MoD services are seam-lessly integrated into an efficient transportation network.The system’s success hinges upon the confluence of twomodes and meeting the latent travel demand given user pref-erences. MT provides low-cost services along fixed routesto a prearranged timetable. In contrast, MoD can provideaccess to underserved communities by MT (due to variousreasons) at a higher expense. On a network design level,integrating MoD helps extend MT’s reach through first- orlast-mile connections; On an operational level, redirectingtravelers by MoD can harness underused MT routes andreduce the overall operational costs. This work aims to pro-vide a tractable optimization framework that simultaneouslydesign networks and determines service prices in favor ofthe public good.
Multimodal mobility systems are an emerging service thatcuts across traditional transit boundaries and emerging MoDapplications. Transportation agencies are playing a promi-nent role in shaping such services by experimenting withnew service models and rules for governing integration withon-demand mobility service providers [13]. For example,the Chicago Regional Transportation Authority and Lyft a r X i v : . [ c s . C E ] F e b ashville ’21, May 19–21, 2021, Nashville, TN Luo, et al. launched an incentive program that connects commutersfrom nearby metro-stations to office buildings [14]. The localtransportation agency is responsible for allocating resourcesbetween improving MT services and subsidizing MoD ser-vices as first- or last-mile connections. MoD, either ownedby the public sector or a private company, can benefit fromreaching a broader basis of customers [2].A vast stream of literature has studied the centralized plan-ning, tactical, and operational decisions in transit or MoDsystems. Planning decisions include choosing service regionsor bus stop locations. Tactical decisions include setting busroutes’ frequency. Operational decisions include timetablingof MT and routing of MoD. Interested readers are referredto the reviews on MT [8] and this stream of work on MoD[16]. More recently, joint decision-making on different lev-els has been studied thoroughly taking both short-term andlong-term travelers’ behavior into consideration. Zhang etal. [18] investigated the joint optimization of MT servicefrequency and the fare to maximize a weighted sum of profitand consumers’ surplus. Sun and Szeto [15] evaluated theeffectiveness of sectional fares in MT by solving a bilevelprogram that jointly determines the fares and frequency set-ting. These approaches do not apply to a multimodal settingdue to the exponential blowup in computation time whenincluding multiple modes.Modeling travelers’ mode choice is a critical technical chal-lenge of optimizing multimodal networks. The latent traveldemand is realized only when the prices and the qualityof service (QoS) are both satisfying. In general, this choicemodel is treated as a lower-level optimization in centralizeddecisions. Beheshtian et al. [3] studied the multimodal mar-ketplace in which service providers bids for the use of roadsegments and travelers pay the roadway tolls by the clearingprices. In addition, transportation agencies may incentivizetravelers to use one mode of transport over the others. Brandset al. [4] designed a tradable permit scheme to regulate roadtransport externalities and empirically tested the existence ofdynamic equilibrium on the mobility market. Characterizingthose user equilibria is a computationally heavy task, regard-less of the recourse under various tactical and operationalplans. A simplified framework is proposed by Wischik [17].It studied the multimodal mobility market’s knock-on effecton travelers’ route and mode choices by combining a discretechoice model with a multipath resource allocation model.By solving a single optimization problem, a transportationagency can infer the choice models and then control trafficflows externalized by travelers’ prices. Leveraging this idea,this work focuses on solving a joint pricing and networkdesign problem for multimodal mobility systems. To the best of our knowledge, this work is the first attemptto propose a scalable framework for solving the joint pric-ing and network design problem in multimodal mobility networks using exact methods. This problem is notoriouslydifficult to solve even in single-mode transportation net-works, and exacerbated by the complexities introduced inthe multimodal setting. Two critical challenges in the mul-timodal setting are the growth of feasible combinations ofmodes and modeling user preferences. Our work computesthe optimal prices for hybrid trips by a primal-dual approachdeveloped in the market design literature. The computedprices ensure that each traveler is willing to pay for theirutility-maximizing option. The optimal design of the mar-ketplace achieves maximal social welfare. Besides, the multi-modal mobility network needs to determine the frequency ofMT routes and relocate MoD vehicle fleets to meet travel de-mand. Since the number of possible modes is enormous, wecombine approximation algorithms, Benders decomposition,and cutting-plane approaches into a unified solver to obtainimplementable algorithm for optimal pricing and operationalplanning. This approach’s performance in facilitating theconvergence of the two modes is experimentally evaluatedusing a real-world case study in Nashville, Tennessee.The outline of this work is as follows. Section 2 describesthe formulation of the joint pricing and network design prob-lem. Section 3 presents a decomposition scheme that takesadvantage of the special structure of the subproblems. Sec-tion 4 conducts a case study applying the developed approachto the city of Nashville. Finally, we draw some conclusionsin Section 5.
We study a joint pricing and network design problem in amultimodal mobility system. With a welfare maximizationgoal in its mind, a transportation agency is responsible forpricing multimodal mobility services and determining theroutes and frequency setting of MT to meet demand foraffordable and efficient mobility services.
The following example (Figure 1) shows how to coordinatethe operations of MT and MoD subject to travelers’ behaviorto maximize overall welfare.
Figure 1.
Pricing and network design problem in a simplemultimodal networkThe network has two one-way MT lines, 𝐼 and 𝐼𝐼 , and MoDis available on each edge. As MoD serves trip requests in ultimodal Mobility Systems: Joint Optimization of Transit Network Design and Pricing Nashville ’21, May 19–21, 2021, Nashville, TN an on-demand fashion, it needs to rebalance idle vehicles tohigh demand locations [12]. The rebalancing flows betweeneach pair of vertices 𝑖, 𝑗 at steady-state are 𝑟 𝑖 𝑗 . There are twotypes of travelers 𝜃 and 𝜃 with different utility functions.Travel demand from vertex 1 to vertex 3 is split to 𝜆 , ( 𝜃 ) and 𝜆 , ( 𝜃 ) , respectively. Travelers of type 𝜃 are price-sensitiveand travelers of type 𝜃 are QoS-sensitive. Figure 2.
Travelers’ choice model in a multimodal networkTravelers have unified access to use a mix of two mo-bility services through a trip planner. For example, eachtraveler from vertex 1 to vertex 3 are provided with threeoptions (called “mode” throughout this work): (a) taking low-frequency MT Line I, (b) taking MoD, or (c) a multimodalmode, i.e., taking MoD to vertex 2 and transfer to MT Line 𝐼 or Line 𝐼𝐼 , which collectively arrive at a high frequency(Figure 2). We define the time between MT vehicles past agiven point as the headway, and the MT service frequencyin unit of the average number of vehicles per unit time isthe inverse of the headway. Assume that Line 𝐼 and Line 𝐼𝐼 operate at frequencies of 𝑓 𝐼 and 𝑓 𝐼𝐼 , respectively. If a travelerchooses mode (c), she will take the first arriving MT vehicleat the vertex 2. We model this option by creating a bundle oftwo lines from which each traveler chooses Line 𝑖 ∈ { 𝐼, 𝐼𝐼 } by a probability of 𝑓 𝑖 /( 𝑓 𝐼 + 𝑓 𝐼𝐼 ) . Since mode (b) is expensiveand mode (a) may not meet travelers’ QoS requirements, im-proving the competitiveness of mode (c) when appropriateis the primary goal of this work.After the system presents the aforementioned modes andtheir corresponding prices and QoS, each traveler choosesthe mode 𝑚 that maximizes the ( 𝑣 𝑚 − 𝑝 𝑚 ) , 𝑚 ∈ { 𝑚 , 𝑚 , 𝑚 } ,where 𝑣 𝑚 is the valuation and 𝑝 𝑚 is the charged price (in-cluding both MT fares and MoD fares). Without integratedmultimodal access, 𝜃 travelers tend to take Line I and 𝜃 trav-elers tend to choose MoD. As a result, Line I is overloaded,and the MoD vehicle fleet stagnates at vertex 3.From the transportation agency’s perspective, the goal isto create a multimodal market in which the market clear-ance prices for all modes, including these hybrid modes, leadto maximal social welfare. This work focuses on frequencysetting and rebalancing decisions in addition to pricing isbecause frequency setting is the most fundamental tacticaldecision in MT and rebalancing is the unique and central task in MoD operations. In this market design procedure,an obvious trade-off is the resource allocation between MTand MoD. Providing high-frequency MT services on Line 𝐼 enhances the QoS while increasing its operating costs. Inthis example, it is potentially beneficial to subsidize sometravelers to use the multimodal mode (c) to gain access tothe low-cost Line 𝐼𝐼 . On the other hand, they should not beover-subsidized to choose the MoD mode (b). Since the sys-tem’s welfare is measured by the difference between utilitiesreceived from serving travel demand and the incurred op-erating costs, this approach can redistribute realized trafficstreams to improve MT’s utilization while reducing the costof operating the MoD vehicle fleet. In a general setting, the multimodal mobility network isrepresented by a directed graph 𝐺 = ( 𝑉 , 𝐸 ) consisting of 𝑛 vertices corresponding to pickup/dropoff locations. Thereare 𝐿 transit lines where each line is a sequence of edges { 𝑒 𝑖 𝑗 } ,which joins a sequence of distinct vertices. MoD is assumedto be available on each edge. A traveler from the startingvertex 𝑠 to the terminal vertex 𝑡 is provided with a menu ofavailable hybrid modes consisting of MT and MoD denotedas M 𝑠𝑡 . She employs a mixed strategy, i.e., the trip plannerprovides her a randomized option from the chosen modesto maximize her expected utility. Let [ 𝐾 ] : = { , , . . . , 𝐾 } throughout this work and [ 𝐾 ] denote the Cartesian productof sets. The notation used throughout this work is summa-rized in the nomenclature Table 4 in Appendix A. The ser-vice frequency on each line is a vector 𝑓𝑓𝑓 ∈ Z | 𝐿 | . There existsa natural lower bound such that, when setting the frequencyat this bound, MT becomes noncompetitive for all travel-ers compared with other options. We rescale the frequencysuch 𝑓 ℓ ∈ [ 𝐹 ] for all ℓ ∈ [ 𝐿 ] . The transit network’s config-uration is fixed throughout the frequency setting decision.Let 𝑧 ℓ,𝑓 be a binary variable corresponding to line ℓ beingset to frequency 𝑓 , i.e., 𝑧 ℓ𝑓 = 𝑓 is turned on and 𝑧 ℓ𝑓 = off . Line ℓ ’s vehicle ca-pacity is V ℓ . The unit setup cost incurred for raising thefrequency of a transit line is 𝑜 ℓ so the total setup cost of MTis (cid:205) ℓ ∈[ 𝐿 ] ,𝑓 ∈[ 𝐹 ] 𝑜 ℓ 𝑧 ℓ,𝑓 .When a set of MT lines 𝐿 𝑖 𝑗 passes through the same edge 𝑒 𝑖 𝑗 , the system creates a set of bundled services B 𝑖 𝑗 : = comb ( 𝐿 𝑖 𝑗 ) .Each bundled service 𝑏 𝑖 𝑗 ∈ B 𝑖 𝑗 includes one or more MTlines. The probability of a traveler who chooses 𝑏 𝑖 𝑗 takingline ℓ ∈ 𝑏 𝑖 𝑗 is proportional to its frequency 𝑓 ℓ / (cid:205) ℓ ′ ∈ 𝑏 𝑖𝑗 𝑓 ℓ ′ , i.e.,on a first-come-first-serve (FCFS) basis. The on/off decisionfor each bundle is denoted a binary variable 𝑧 𝑏 𝑖𝑗 𝜓 where 𝜓 ℓ isthe corresponding frequency in 𝜓 at its ℓ ∈ 𝑏 𝑖 𝑗 entry. In exam-ple of Figure 1, 𝑏 = { 𝐼, 𝐼𝐼 } and 𝜓 = { 𝑓 𝐼 , 𝑓 𝐼𝐼 } . 𝑧 𝑏 𝑖𝑗 𝜓 = 𝑏 𝑖 𝑗 to all modes ashville ’21, May 19–21, 2021, Nashville, TN Luo, et al. passing the edge 𝑒 𝑖 𝑗 . As each traveler is allowed to transferat most once, a preprocess truncates the set of modes M 𝑠𝑡 when it is expanded to include bundled services at varyingfrequencies. We consider a large market, fluidscaling of the system. Each vertex 𝑠 has exogenous, non-atomic flows { 𝜆 𝑠𝑡 , 𝑡 ∈ 𝑉 /{ 𝑠 }} . Heterogeneous travelers oftype 𝜃 ∈ Θ have valuations drawn from a discrete distri-bution 𝑞 . Thus the flows are split as 𝜆 𝑠𝑡 ( 𝜃 ) : = 𝜆 𝑠𝑡 𝑞 𝑠𝑡 ( 𝜃 ) .Travelers from a starting vertex 𝑠 to a terminal vertex 𝑡 plantheir trips from options (called “hybrid modes”) 𝑚 ∈ M 𝑠𝑡 defined by a sequence of edges, each specifying the mode oftransport as {( 𝑠, 𝑖 ) , ( 𝑖 , 𝑖 ) , ... ( 𝑖 𝑘 , 𝑡 )} .Observing existing services and their corresponding QoSand prices, travelers make their trip plans (possibly usinga multimodal option). A traveler from vertex 𝑠 to vertex 𝑡 chooses the mode 𝑚 ∈ M 𝑠𝑡 that maximizes their utility 𝑣 𝜃𝑚 − 𝑝 𝜃𝑚 . The valuation for trip ( 𝑠, 𝑡 ) is type- and mode-dependent, and denoted as 𝑣𝑣𝑣 ( 𝜃 ) ∈ R | 𝑀 𝑠𝑡 | . 𝑥 𝜃𝑚 ∈ [ , ] de-notes the probability that a traveler of type 𝜃 traveling from 𝑠 to 𝑡 chooses mode 𝑚 ∈ M 𝑠𝑡 and (cid:205) 𝑚 ∈M 𝑠𝑡 𝑥 𝜃𝑚 = 𝑚 ∈ M 𝑠𝑡 . The total operational cost of hybrid mode 𝑚 is given by 𝑐 𝑚 = (cid:205) 𝑒 𝑖𝑗 ∈ 𝐸 𝑚 𝑐 𝑖 𝑗 + (cid:205) ℓ ∈ 𝐿 : ℓ ∼ 𝑚 𝑐 ℓ , where ℓ ∼ 𝑚 means that mode 𝑚 uses Line ℓ . Payments for MoD aredistance-based, and we let 𝑐 𝑖 𝑗 denotes the operating cost ofcarrying passengers by MoD from 𝑖 and 𝑗 . Payments for MTare made per-use. The chosen Line ℓ ’s fares depend on theod pair 𝑠 − 𝑡 and the operational cost of Line ℓ is 𝑐 ℓ . This section proposes a genericMILP formulation for the joint pricing and network designproblem in (1). The transit network design problem deter-mines the MT frequency 𝑓 for all ℓ ∈ [ 𝐿 ] ( 𝑧𝑧𝑧 ) and rebalancingflows of MoD ( 𝑟𝑟𝑟 ). The objective is maximizing the overallwelfare collected from operating the multimodal mobilitysystem. The welfare measures the difference between the util-ity of serving all travel demand, operational costs of hybridmodes, MoD’s rebalancing costs , and MT’s setup costs.The constraints are categorized as follows:1. (1a) are the capacity constraints on each edge. Thisconstraint guarantees that the total MT flow passingthrough edge 𝑒 𝑖 𝑗 (left-hand side) is no larger than thetotal capacity carried by MT lines (right-hand side).The flow distribution for each MT line is satisfied byenforcing a FCFS rule in trip assignment.2. (1b) - (1c) describe the relationship between MT’s linesetups and bundled service setups. A bundle 𝑏 𝑖 𝑗 isaccessible if and only if all lines included ℓ ∈ 𝑏 𝑖 𝑗 areoperated at the frequency levels of 𝜓 .3. (1d) - (1e) are the ranking constraints for MT frequencysetting. When the system determines Line ℓ ’s frequency to be at level ℓ , low-frequency options are turned onas travelers tend to choose a high-QoS service whenpaying the same price.4. (1f) is the budget constraint for MT. The cycle time ofLine ℓ is 𝑇 ℓ and the total number of MT vehicles is R .5. (1g) ensures the flow-balance (via rebalancing) of theMoD fleet.6. (1h) is a linkage constraint in mode generation. A mode 𝑚 is available if each ℓ ∼ 𝑚 is open. It can be relaxedto 𝐶𝑥 𝜃𝑚 ≤ 𝑧 ℓ,𝑓 with arbitrary 𝐶 > 𝑧,𝑥,𝑟𝑧,𝑥,𝑟𝑧,𝑥,𝑟 ∑︁ ( 𝑠,𝑡 ) ∈[ 𝑛 ] 𝜃 ∈ Θ 𝑠𝑡 𝜆 𝑠𝑡 ( 𝜃 ) ∑︁ 𝑚 ∈M 𝑠𝑡 ( 𝑣 𝜃𝑚 − 𝑐 𝑚 ) 𝑥 𝜃𝑚 − ∑︁ 𝑖,𝑗 𝑐 𝑖 𝑗 𝑟 𝑖 𝑗 − ∑︁ ℓ ∈[ 𝐿 ] 𝑓 ∈[ 𝐹 ] 𝑜 ℓ 𝑧 ℓ,𝑓 (1) 𝑠.𝑡 . ∑︁ ( 𝑠,𝑡 ) : 𝑒 𝑖𝑗 ∈( 𝑠,𝑡 ) 𝜃 ∈ Θ ,𝑚 ∈M 𝑠𝑡 𝜆 𝑠𝑡 ( 𝜃 ) 𝑥 𝜃𝑚 ≤ ∑︁ ( ℓ,𝑓 ) ∈( 𝑏 𝑖𝑗 ,𝜓 ) , ∀ 𝜓 ∈[ 𝐹 ] | 𝑏𝑖𝑗 | , ∀ 𝑏 𝑖𝑗 ∈B 𝑖𝑗 V ℓ 𝑧 ℓ,𝑓 , ∀ 𝑒 𝑖 𝑗 ∈ 𝐸 (1a) 𝑧 𝑏 𝑖𝑗 𝜓 ≤ 𝑧 ℓ,𝑓 , ∀( ℓ, 𝑓 ) ∈ ( 𝑏 𝑖 𝑗 ,𝜓 ) , ∀ 𝜓 ∈ [ 𝐹 ] | 𝑏 𝑖𝑗 | , ∀ 𝑏 𝑖 𝑗 ∈ B 𝑖 𝑗 , ∀ 𝑒 𝑖 𝑗 ∈ 𝐸 (1b) 𝑧 𝑏 𝑖𝑗 𝜓 ≥ ∑︁ ( ℓ,𝑓 ) ∈( 𝑏 𝑖𝑗 ,𝜓 ) 𝑧 ℓ,𝑓 − | 𝑏 𝑖 𝑗 | + , ∀ 𝜓 ∈ [ 𝐹 ] | 𝑏 𝑖𝑗 | , ∀ 𝑏 𝑖 𝑗 ∈ B 𝑖 𝑗 , ∀ 𝑒 𝑖 𝑗 ∈ 𝐸 (1c) 𝑧 ℓ,𝑓 ≤ 𝑧 ℓ,𝑓 ′ , ∀ 𝑓 ≥ 𝑓 ′ , ∀ ℓ ∈ [ 𝐿 ] (1d) 𝑧 ℓ,𝑓 ≥ 𝑧 ℓ,𝑓 ′′ , ∀ 𝑓 ≤ 𝑓 ′′ , ∀ ℓ ∈ [ 𝐿 ] (1e) ∑︁ ℓ ∈[ 𝐿 ] ∑︁ 𝑓 ∈[ 𝐹 ] 𝑇 ℓ 𝑧 ℓ,𝑓 ≤ R (1f) ∑︁ 𝑗 ∈[ 𝑛 ] (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) 𝑟 𝑗𝑖 + ∑︁ 𝜃 ∈ Θ 𝑚 : ( 𝑗,𝑖 ) ∈ 𝐸 𝑚 𝜆 𝑠𝑡 ( 𝜃 ) 𝑥 𝜃𝑚 (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) = ∑︁ 𝑘 ∈[ 𝑛 ] (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) 𝑟 𝑖𝑘 + ∑︁ 𝜃 ∈ Θ 𝑚 : ( 𝑖,𝑘 ) ∈ 𝐸 𝑚 𝜆 𝑠𝑡 ( 𝜃 ) 𝑥 𝜃𝑚 (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) , ∀ 𝑖 ∈ [ 𝑛 ] (1g) 𝑥 𝜃𝑚 ≤ 𝑧 ℓ,𝑓 , ∀( ℓ, 𝑓 ) ∼ 𝑚, ∀ 𝜃 ∈ Θ , ∀ ℓ ∈ 𝐿, ∀ 𝑓 ∈ [ 𝐹 ] (1h) ∑︁ 𝑚 ∈M 𝑀𝑠𝑡 𝑥 𝜃𝑚 ≤ 𝑞 𝑠𝑡 ( 𝜃 ) , ∀ 𝜃 ∈ Θ , ∀( 𝑠, 𝑡 ) ∈ [ 𝑛 ] (1i) 𝑧 𝑏 𝑖𝑗 𝜓 ∈ { , } , ∀ 𝜓 ∈ [ 𝐹 ] | 𝑏 𝑖𝑗 | , ∀ 𝑏 𝑖 𝑗 ∈ B 𝑖 𝑗 , ∀ 𝑒 𝑖 𝑗 ∈ 𝐸 (1j) 𝑧 ℓ,𝑓 ∈ { , } , ∀ 𝑓 ∈ [ 𝐹 ] , ∀ ℓ ∈ [ 𝐿 ] (1k)0 ≤ 𝑥 𝜃𝑚 ≤ , ∀ 𝜃 ∈ Θ , 𝑚 ∈ M 𝑠𝑡 , ( 𝑠, 𝑡 ) ∈ [ 𝑛 ] (1l) 𝑟 𝑖 𝑗 ≥ , ∀ 𝑒 𝑖 𝑗 ∈ 𝐸 (1m)This formulation overcomes two main technical challengesin prior work. First, the transit network’s design subject to ultimodal Mobility Systems: Joint Optimization of Transit Network Design and Pricing Nashville ’21, May 19–21, 2021, Nashville, TN travelers’ mode choice is modeled by a primal-dual approach.This approach conserves the user equilibrium and avoids thecomputational tractability issue in the conventional bilevelprogramming approach [15]. Second, considering 𝑝 𝑚 as pri-mal variables under the demand elasticity leads to a nonlin-ear mathematical program. We convert the pricing decisionsinto a dual problem (explained in Section 2.2.4) and sim-plify the formulation by linearization tricks. This model canbe extended to including the MoD platform as a separatedecision-maker [2]. The privately-owned MoD company isbetter off by participating in this multimodal network to gaina broader basis of customers. The LP-relaxationof (1) computes the dual prices 𝑝 𝜃𝑚 for each mode 𝑚 ∈ M 𝑠𝑡 .The dual variables for the MILP formulations are as follows:1. Edge-base price: 𝑝 𝑖 𝑗 for each 𝑒 𝑖 𝑗 ∈ 𝐸 corresponding to(1a).2. Bundled price: 𝛽 𝑏 𝑖𝑗 𝜓 and 𝜂 𝑏 𝑖𝑗 𝜓 for ( 𝑏 𝑖 𝑗 ,𝜓 ) ∈ (B 𝑖 𝑗 , | 𝐹 | | 𝑏 𝑖𝑗 | ) and 𝑒 𝑖 𝑗 ∈ 𝐸 corresponding to (1b) and (1c).3. Line price: 𝑤 ℓ,𝑓 ,𝑓 ′ with 𝑓 ≤ 𝑓 ′ , 𝑓 , 𝑓 ′ ∈ [ 𝐹 ] correspond-ing to (1d) and (1e).4. MT vehicle setup price: 𝑢 corresponding to (1f).5. MoD connection price: 𝛾 𝑖 for 𝑖 ∈ [ 𝑛 ] corresponding to(1g).6. Line setup cost: 𝜁 𝜃𝑚ℓ𝑓 for all ( ℓ, 𝑓 ) ∼ 𝑚, 𝜃 ∈ Θ , ℓ ∈ 𝐿, 𝑓 ∈ [ 𝐹 ] corresponding to (1h).7. Path-base utility: 𝜈 𝜃,𝑠𝑡 for all 𝜃 ∈ Θ , 𝑚 ∈ M 𝑠𝑡 , ( 𝑠, 𝑡 ) ∈[ 𝑛 ] corresponding to (1i).Traveler of type 𝜃 traveling from 𝑠 to 𝑡 is charged a price 𝑝 𝜃𝑚 computed x as follows: 𝑝 𝜃𝑚 = ∑︁ 𝑒 𝑖𝑗 ∈ 𝐸 𝑚 (cid:104) 𝑝 𝑖 𝑗 + 𝛽 𝑏 𝑖𝑗 𝜓 + 𝜂 𝑏 𝑖𝑗 𝜓 + 𝛾 𝑖 (cid:105) + 𝑤 ℓ,𝑓 ,𝑓 ′ + 𝑢 + 𝜁 𝜃𝑚ℓ𝑓 ,𝑚 ∼ ( 𝑏 𝑖 𝑗 ,𝜓 ) , (2)where 𝑚 ∼ ( 𝑏 𝑖 𝑗 ,𝜓 ) means that the bundled service on edge 𝑒 𝑖 𝑗 is included in the hybrid mode 𝑚 ; 𝐸 𝑚 and 𝑉 𝑚 are edgesand vertices trespassed by this mode.It is worth mentioning that the dual prices are not well-defined in the original MILP (1). Section 3 justifies the defini-tion of dual prices with a decomposition scheme. After fixingthe frequency setting of MT, the travelers’ mode choices areformulated as an LP [17] and the pricing problem regardingthe realized network flow is precisely defined. Solving thesesubproblems will return part of the dual price and feasibil-ity/optimality cuts. The algorithm then resolves the masterproblem to approach a feasible frequency-setting plan forMT such that the total welfare is maximized. In the finalstep, extra setup costs for MTs are added evenly for all users,reflecting the equal-sharing practice in public transportationsectors. This decomposition is also computationally efficientas the subproblems are solvable in polynomial time. Proposition 2.1.
Under the optimal dual prices 𝑝 𝜃 , travelersof 𝜃 from vertex 𝑠 to vertex 𝑡 in graph 𝐺 will adopt a mixedstrategy 𝑥 ∗ 𝜃𝑚 over available modes 𝑚 ∈ M 𝑠𝑡 such that 𝑥 𝜃𝑚 ∈ arg max ∑︁ 𝜃 ∈ Θ ,𝑚 ∈M 𝑠𝑡 ( 𝑣 𝜃𝑚 − 𝑝 𝜃𝑚 ) 𝑥 𝜃𝑚 , ∀( 𝑠, 𝑡 ) ∈ [ 𝑛 ] . This mixed strategy for choosing modes can be imple-mented by a randomized assignment policy as follows. Ob-serving that each optimal mode 𝑚 gives type 𝜃 travelers thesame utility, the system can randomly provide mode 𝑚 witha probability of 𝑥 ∗ 𝜃𝑚 and keep track of the past assignment. Inthe following assignment, the system matches the empiricaldistribution of mode assignments with these 𝑥 ∗ 𝜃𝑚 . The joint optimization of transit network design and pricingformulated above is notoriously hard to solve directly regard-ing the large-scale urban networks. This section starts witha negative result for the integrality gap of the original MILPformulation. To overcome the computational challenge, wedesign a bilevel decomposition scheme such that the dualprices of modes are computed in recourse. Finally, we shedlight on the computational complexity of the proposed algo-rithm.
The dual prices can be computed from the constraints (1a)-(1f) in the LP-relaxation of (1). Nevertheless, since both thefrequency setting 𝑧 and mode choice 𝑥 are integer variables,we show that the integrality gap of the original formulationis arbitrarily large.In a simple network of two vertices, 𝐿 =
1, and 𝐹 equals aconstant ˆ 𝐹 , we consider only MT in a round trip. Without lossof generality, we assume the optimal frequency is 𝑓 ∗ suchthat (cid:205) 𝑓 ∈[ 𝑓 ∗ ] 𝑇 𝐼 𝑧 𝐼,𝑓 ≤ R and (cid:205) 𝑓 ∈[ 𝑓 ∗ + ] 𝑇 𝐼 𝑧 𝐼,𝑓 > R . Hence, 𝑧 𝐼, = · · · = 𝑧 𝐼,𝑓 ∗ = 𝑧 𝐼,𝑓 ∗ + = . . . 𝑧 𝐼,𝐹 =
0. We candenote the mode choice variable 𝑥 𝑓 in this case as the modesonly differ in frequency. The setting up cost is 𝑓 ∗ 𝑜 ℓ and thetotal capacity is 𝑓 ∗ 𝑉 𝐼 . The constraint (1a) determines 𝑥 𝑓 as 𝑥 𝑓 > 𝑣 𝜃 𝑓 − 𝑐 𝑓 for 𝑓 ∈ [ 𝑓 ∗ ] and equals0 for others. We denote the objective value as 𝑣 ∗ . We canfind fractional solutions for its LP-relaxation counterpart asfollows. We set 𝑧 𝐼,𝑓 = 𝑓 ∗ / ˆ 𝐹 for all 𝑓 ∈ [ ˆ 𝐹 ] which is feasiblefor the constraints in (1). The total capacity and the setupcosts are equal to the original problem while all modes areopen. 𝑥 𝑓 is expanded to 𝑓 ∈ [ ˆ 𝐹 ] and the integrality gapmax { 𝑣 𝜃 𝑓 − 𝑐 𝑓 } 𝑓 ∈[ 𝑓 ∗ + , ˆ 𝐹 ] / 𝑣 ∗ is unbounded.The large integrality gap motivates the development ofa decomposition framework in Figure 3. The original MILPis computationally intractable as the dimension of binary ashville ’21, May 19–21, 2021, Nashville, TN Luo, et al. Figure 3.
Decomposition framework for joint pricing andtransit network design problemvariables 𝑧 and 𝑥 are in the scale of 2 𝐿 × 𝐹 ×| 𝐸 | and the resultingnumber of constraints explodes.We decompose the MILP formulation to two levels ofsubproblems that solve 𝑧 , 𝑥 , and 𝑟 sequentially. We start with sub-problems for any transit network design, which are solvedthrough fully polynomial time approximation schemes. Let 𝑦 𝜃𝑚 = 𝜆 𝑠𝑡 ( 𝜃 ) 𝑥 𝜃𝑚 and the values from the subproblem ˆ 𝑦 𝜃𝑚 = 𝜆 𝑠𝑡 ( 𝜃 ) ˆ 𝑥 𝜃𝑚 . Given a MT network design 𝑧𝑧𝑧 and travelers’ choice 𝑦𝑦𝑦 , the sub-subproblem 𝑆𝑈 𝐵 ( 𝑧, 𝑦𝑧, 𝑦𝑧, 𝑦 ) (the MoD rebalancing prob-lem) is a min-cost flow problem: 𝑆𝑈 𝐵 ( 𝑧, 𝑦𝑧, 𝑦𝑧, 𝑦 ) = min 𝑟𝑟𝑟 ∑︁ 𝑖,𝑗 𝑐 𝑖 𝑗 𝑟 𝑖 𝑗 (3) 𝑠.𝑡 . ∑︁ 𝑗 ∈[ 𝑛 ] (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) 𝑟 𝑗𝑖 + ∑︁ 𝜃 ∈ Θ 𝑚 : ( 𝑗,𝑖 )∈ 𝐸 𝑚 ˆ 𝑦 𝜃𝑚 (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = ∑︁ 𝑘 ∈[ 𝑛 ] (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) 𝑟 𝑖𝑘 + ∑︁ 𝜃 ∈ Θ 𝑚 : ( 𝑖,𝑘 )∈ 𝐸 𝑚 ˆ 𝑦 𝜃𝑚 (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) , ∀ 𝑖 ∈ [ 𝑛 ] (3a) 𝑟 𝑖 𝑗 ≥ , ∀ 𝑒 𝑖 𝑗 ∈ 𝐸 (3b)The dual of the MoD rebalancing problem ismax 𝛾𝛾𝛾 ∑︁ 𝑖 ∈[ 𝑛 ] (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) ∑︁ 𝜃 ∈ Θ 𝑚 : ( 𝑖,𝑘 ) ∈ 𝐸 𝑚 ˆ 𝑦 𝜃𝑚 − ∑︁ 𝜃 ∈ Θ 𝑚 : ( 𝑗,𝑖 ) ∈ 𝐸 𝑚 ˆ 𝑦 𝜃𝑚 (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) 𝛾 𝑖 (4) 𝑠.𝑡 . 𝛾 𝑖 − 𝛾 𝑗 ≤ 𝑐 𝑖 𝑗 , ∀ 𝑒 𝑖 𝑗 ∈ 𝐸𝛾 𝑖 unrestricted , ∀ 𝑖 ∈ [ 𝑛 ] The feasibile set of the dual problem is independent of 𝑥𝑥𝑥 . Solving the subsubproblem adds either feasibile cuts or optimal cuts to the subproblem 𝑃 ( 𝑧𝑧𝑧 ) in (5): 𝑃 ( 𝑧𝑧𝑧 ) = max 𝑦𝑦𝑦 ∑︁ ( 𝑠,𝑡 ) ∈[ 𝑛 ] 𝜃 ∈ Θ 𝑠𝑡 𝑚 ∈M 𝑠𝑡 ( 𝑣 𝜃𝑚 − 𝑐 𝑚 ) 𝑦 𝜃𝑚 + 𝜙 (5) 𝑠.𝑡 . (1a) , (1h) , (1i) (5a) ∑︁ 𝑖 ∈[ 𝑛 ] 𝑒 (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) ∑︁ 𝜃 ∈ Θ 𝑚 : ( 𝑖,𝑘 ) ∈ 𝐸 𝑚 𝑦 𝜃𝑚 − ∑︁ 𝜃 ∈ Θ 𝑚 : ( 𝑗,𝑖 ) ∈ 𝐸 𝑚 𝑦 𝜃𝑚 (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) 𝛾 𝑒𝑖 ≥ 𝜙 (5b) ∑︁ 𝑖 ∈[ 𝑛 ] 𝑢 ] (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) ∑︁ 𝜃 ∈ Θ 𝑚 : ( 𝑖,𝑘 ) ∈ 𝐸 𝑚 𝑦 𝜃𝑚 − ∑︁ 𝜃 ∈ Θ 𝑚 : ( 𝑗,𝑖 ) ∈ 𝐸 𝑚 𝑦 𝜃𝑚 (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) 𝛾 𝑢𝑖 ≥ ≤ 𝑦 𝜃𝑚 ≤ 𝜆 𝑠𝑡 , ∀ 𝜃 ∈ Θ , 𝑚 ∈ M 𝑠𝑡 , ( 𝑠, 𝑡 ) ∈ [ 𝑛 ] (5d)where 𝛾 𝑒 are extreme points of (4) and 𝛾 𝑢 are unboundeddirections of (4).The traveler choice subproblem (5) is a variation of theweighted multicommodity flow problem (WMFP). If eachtraveler adopts a deterministic mode choice strategy, it isreduced to an integral MFP problem, which is a well-knownNP-complete problem (MAX SNP-hard). Compared to a di-rect LP approach, the subproblem computation’s main bottle-neck is the number of commodities | Θ | · |M 𝑠𝑡 | in multimodalmobility networks. A primal-dual approximation algorithm(Algorithm 1) is used to simultaneously compute travelers’routes and their corresponding prices. The essence of thealgorithm is repeated solve shortest-path problems on thedual solutions to WMFP such that the approximation ratiois independent of the number of commodities [6].The dual problem of the fractional subproblem (5) is asfollows: 𝐷 ( 𝑧𝑧𝑧 ) = min ∑︁ ( ℓ,𝑓 ) ∈( 𝑏 𝑖𝑗 ,𝜓 ) , ∀ 𝜓 ∈[ 𝐹 ] | 𝑏𝑖𝑗 | , ∀ 𝑏 𝑖𝑗 ∈B 𝑖𝑗 𝑉 ℓ ˆ 𝑧 ℓ,𝑓 𝑝 𝑖 𝑗 + ∑︁ 𝜃 ∈ Θ 𝑚 ∈M 𝑠𝑡 ( ℓ,𝑓 )∼ 𝑚 ˆ 𝑧 ℓ,𝑓 𝜁 𝜃𝑚ℓ𝑓 ∑︁ 𝜃 ∈ Θ ( 𝑠,𝑡 ) ∈[ 𝑛 ] 𝜆 𝑠𝑡 ( 𝜃 )( 𝜆 𝑠𝑡 ( 𝜃 ) 𝜈 𝜃,𝑠𝑡 + 𝛼 𝜃𝑚 ) (6) 𝑠.𝑡 . ∑︁ 𝑒 𝑖𝑗 ∈ 𝐸 𝑝 𝑖 𝑗 + ∑︁ ( ℓ,𝑓 )∼ 𝑚 𝜁 𝜃𝑚ℓ𝑓 + 𝛼 𝜃𝑚 + 𝜈 𝜃,𝑠𝑡 ≥ 𝑣 𝜃𝑚 − 𝑐 𝑚 , ∀ 𝜃 ∈ Θ , ∀ 𝑚 ∈ M 𝑠𝑡 , ∀( 𝑠, 𝑡 ) ∈ [ 𝑛 ] (6a) 𝐴 ⊺ 𝜙 𝛽𝛽𝛽 ≥ 𝑝𝑝𝑝, 𝜁𝜁𝜁 ,𝜈𝜈𝜈, 𝛼𝛼𝛼, 𝛽𝛽𝛽 ≥ 𝜌, 𝜁 , 𝜈, 𝛼, 𝛽𝜌, 𝜁 , 𝜈, 𝛼, 𝛽𝜌, 𝜁 , 𝜈, 𝛼, 𝛽 are dual variables corresponding to con-straints (1a), (1h), (1i), (5d), (5b) - (5c), respectively. 𝐴 𝜙 is thecoefficient matrix of (5b) and (5c). We denote the objectivefunction of the dual problem as 𝑔 ( 𝑝𝑝𝑝, 𝜁𝜁𝜁 ,𝜈𝜈𝜈, 𝛼𝛼𝛼, 𝛽𝛽𝛽 ) .The approximation algorithm for WMCF (Algorithm 1)provides a ( − 𝜖 ) approximation, where 𝜖 is the error toler-ance. The number of constraints in (5) is upper-bounded by ultimodal Mobility Systems: Joint Optimization of Transit Network Design and Pricing Nashville ’21, May 19–21, 2021, Nashville, TN Algorithm 1:
Approximation subroutine for travelerchoice problem
Result:
Dual prices related to traveler choice andbenders cuts (7b), (7c).Initialization: frequency setting ˆ 𝑧 in a network; 𝑦𝑦𝑦 = 𝑝𝑝𝑝, 𝜁𝜁𝜁 ,𝜈𝜈𝜈, 𝛼𝛼𝛼, 𝛽𝛽𝛽 = 𝛿 ; while new cuts are generated from (4) dofor ( 𝑠, 𝑡 ) ∈ [ 𝑛 ] , 𝑖 ∈ [ log + 𝜖 + 𝜖𝛿 ] , 𝑗 ∈ [ 𝐾 ] do 𝑃 ← min 𝐷 ( ˆ 𝑧 ) P 𝑗 ; while 𝑝 (P 𝑠𝑡𝑖 ) < min { , 𝛿 ( + 𝜖 ) 𝑖 } do 𝑉 ∗ ← min 𝑒 𝑖𝑗 ∈ 𝑃 𝑉 ℓ ˆ 𝑧 ℓ,𝑓 ; 𝑦 𝜃𝑚 ← 𝑦 𝜃𝑚 + 𝑉 ∗ ; ∀ 𝑒 𝑖 𝑗 ∈ 𝑃 , 𝑝 𝑖 𝑗 ← 𝑝 𝑖 𝑗 ( + 𝜖𝑉 ∗ 𝑉 ℓ ˆ 𝑧 ℓ,𝑓 ) Update 𝑃 . endend Solving subsubproblem
𝑆𝑈 𝐵 ( 𝑧𝑧𝑧,𝑦𝑦𝑦 ) with ˆ 𝑦𝑦𝑦 and addeither feasibility or optimality cuts. end ¯ 𝑚 : = 𝑛 | Θ | 𝐿𝐹 . Let 𝛿 = 𝜖 / ¯ 𝑚 be the initial value of dual vari-ables in (6). The number of commodities is 𝐾 = |M 𝑠𝑡 | · | Θ | .The paths of commodity 𝑖 for each ( 𝑠, 𝑡 ) is denoted as P 𝑠𝑡𝑖 .We assume that, using the dual values in (6), there existsa shortest-path oracle that finds the path with minimumcost for P 𝑠𝑡𝑖 in polynomial time as min 𝐷 ( ˆ 𝑧 ) P 𝑠𝑡𝑖 . The shortestpath computed using the dual values is 𝑝 (P 𝑠𝑡𝑖 ) . Dual pricesfor each commodity are derivatives from the subroutine inAlgorithm 1. The top levelof the original formulation (1) is a frequency setting problemwith bundled services:max 𝑧𝑧𝑧 𝜏 − ∑︁ ℓ ∈[ 𝐿 ] 𝑓 ∈[ 𝐹 ] 𝑜 ℓ 𝑧 ℓ,𝑓 (7) 𝑠.𝑡 . (1b) − (1f) , (1j) − (1k) (7a) 𝑔 ( 𝑝𝑝𝑝 𝑒 , 𝜁𝜁𝜁 𝑒 ,𝜈𝜈𝜈 𝑒 , 𝛼𝛼𝛼 𝑒 , 𝛽𝛽𝛽 𝑒 ) ≥ 𝜏 (7b) 𝑔 ( 𝑝𝑝𝑝 𝑢 , 𝜁𝜁𝜁 𝑢 ,𝜈𝜈𝜈 𝑢 , 𝛼𝛼𝛼 𝑢 , 𝛽𝛽𝛽 𝑢 ) ≥ ( 𝑝𝑝𝑝 𝑒 , 𝜁𝜁𝜁 𝑒 ,𝜈𝜈𝜈 𝑒 , 𝛼𝛼𝛼 𝑒 , 𝛽𝛽𝛽 𝑒 ) are extreme points of the dual subproblem(6) and ( 𝑝𝑝𝑝 𝑢 , 𝜁𝜁𝜁 𝑢 ,𝜈𝜈𝜈 𝑢 , 𝛼𝛼𝛼 𝑢 , 𝛽𝛽𝛽 𝑢 ) represent the unbounded direc-tions.We use the lift-and-project cutting-plane method for solv-ing the binary optimization in (7). For completeness, wedescribe this method in Appendix B. The final algorithm issummarized in Algorithm 2.We make the following claim regarding the computationalefficiency of the algorithm: Theorem 3.1.
The total iterations of the proposed algorithmthat computes the near-optimal prices and transit networkdesign is bounded by 𝑂 ( 𝐿 log ( 𝐹 )) . Algorithm 2:
Algorithm for joint pricing and transitnetwork design problem
Result:
Frequency setting for MT, rebalancing flowsfor MoD, and prices.Initialization: 𝑧𝑧𝑧 = 𝑧 ℓ,𝑓 and 𝑧 𝑏 𝑖𝑗 𝜓 ; for ℓ ∈ [ 𝐿 ] do Update 𝑍 ℓ = [ 𝑧 ℓ,𝑓 ] 𝑓 ∈[ 𝐹 ] such that, if 𝑧 ℓ,𝑓 >
0, set 𝑧 ℓ,𝑓 ′ = 𝑓 ′ < 𝑓 and 𝑧 ℓ,𝑓 ′ = 𝑓 ′ > 𝑓 . for 𝑓 ∈ [ 𝐹 ] do Solve the LP-relaxation of (7) to obtain 𝑍 ∗ ;Generate lift-and-project cuts with 𝑍 ∗ for 𝑍 ∈ R 𝐿 × 𝐹 for (7) (Appendix A) ;Update all bundled services 𝑧 𝑏 𝑖𝑗 𝜓 . end Solve subproblem (5) by Algorithm 1 andgenerate either feasibility or optimality cuts; end
The sketch proof for this main theorem is as follows. First,the lift-and-project cuts’ rank is bounded by the numberof binary variables (i.e.,
𝐿 𝐹 ). Note that we do not need torecompute variables for bundled services as they are pro-vided if and only if their contained lines are all on. Second,the consecutive variables for the frequencies on each linefollows (1d) and (1e). It is equivalent to a bisection search forthe optimal frequency on each line. Furthermore, the totalnumber of iterations is reduced to 𝑂 ( 𝐿 log ( 𝐹 )) for the mas-ter problem. Finally, the subproblem is LP and the feasibleregion of the dual problem is independent of 𝑧𝑧𝑧 . Thus, thealgorithm is solvable in polynomial time. We conduct a numerical experiment that retrofits the cur-rent MTA bus system in Nashville, Tennessee with MoD asa first or last-mile connection. Residents in Nashville, likemost metropolitan areas in the United States, have unequalaccess to transit systems. Therefore, determining the efficacyof extending the MT structure with the assistance of MoDconnections is a meaningful activity.Table 1 summarizes the data used in the experiment. Thetrip demand data estimates the origin-destination matrixat the census tract level of daily commuting trips, whichis used for 𝜆 𝑠𝑡 . There are two types of travelers 𝜃 ∈ { , } corresponding to price-sensitive and QoS-sensitive travelers.We can include more types of travelers by expanding set Θ in other applications. For all o-d pairs, 𝑞 𝑠𝑡 ( ) = 𝑞 𝑠𝑡 ( ) = . 𝑣 𝜃𝑚 = (cid:40) 𝑝 𝜃𝑚 + (cid:205) 𝑒 𝑖𝑗 ∈ 𝐸 𝑚 ( 𝑐 𝑓 · 𝑓 𝑒 − 𝑐 𝑙 · 𝑙 𝑒 ) 𝜃 = 𝑝 𝜃𝑚 + (cid:205) 𝑒 𝑖𝑗 ∈ 𝐸 𝑚 ( 𝑐 𝑓 · 𝑓 𝑒 − 𝑐 𝑙 · 𝑙 𝑒 ) 𝜃 = . ashville ’21, May 19–21, 2021, Nashville, TN Luo, et al. Each candidate mode set M 𝑠𝑡 includes all combinations ofbus routes with no more than two transfers and MoD as first-or last-mile connections. We truncated M 𝑠𝑡 in long-distancetrips if there exist more than five hybrid combinations ofMoD and MT, because travelers are not capable of evaluat-ing a large size of alternative options. For each combination,travelers consider all possible frequencies of MT as in Propo-sition 2.1. Table 1.
Summary of numerical experiment data sourcesItem Datasource DescriptionTrip demand Access Nashville [10] Origin-destinationmovement matrixBus route Nashville MTA [11] GIS and GTFSCostparameters Operational cost [2] MT cycling time,setup cost,Rebalancingcost of MoD
Figure 4.
Nashville MTA bus system (green lines) and origin-destination flow (outflow) data (chloropleth map)The MTA bus routes and trip demand data are shown inFigure 4. The size of optimization in (1) depends on fixednetwork properties such as the scale of MT network and thedimension of trip demand od matrix, as well as user-specificdesign parameters such as the MT frequency levels 𝐹 orthe total number of MT vehicles R . Table 2 summarizes thebenchmark model used throughout the remaining analysis. The optimal frequency setting of the MTA bus system isshown in Figure 5. The lines connecting suburban regionsto the downtown area are expected to run more frequentlythan the downtown area. MoD carries travelers to these linesand aggregate flow via these main corridors. This result, onthe other hand, has limitations since o-d trip demand data
Table 2.
Summary of parameters in Nashville MTA casestudyItem Notation ValueBus lines 𝐿 𝐹 | 𝑉 | 𝜆 𝑠𝑡 > ( 𝑠, 𝑡 ) |M 𝑠𝑡 | 𝑧 ℓ,𝑓 𝑧 𝑏 𝑖𝑗 𝜓 𝑥 𝜃𝑚 , 𝑟 𝑖 𝑗 R V ℓ Figure 5.
Optimal frequency setting for Nashville MTA bussystemThe computational efficiency of the proposed algorithmis shown in Table 3. Two benchmarks that we comparedwith are state-of-the-art MIP solvers (Gurobi 9.0 on a 1.1GHz Dual-Core processor). The runtime of the proposedalgorithm is significantly shorter for both cases. When thesize of the network or the frequency level increases, ourapproach outperforms the MIP solver as the proposed algo-rithm only solves a sequence of LP. Besides, the MIP solverscannot return the dual prices directly because it is not well-defined. In contrast, as the subproblems are multicommodityflow problems with rebalancing constraints, dual prices arewell-defined for each ( 𝑠, 𝑡 ) path in the multimodal network. ultimodal Mobility Systems: Joint Optimization of Transit Network Design and Pricing Nashville ’21, May 19–21, 2021, Nashville, TN Table 3.
Computational results for the Nashville MTA casestudy
Method Runtime(second)
UB-LBgap (%)
Objective
Inst. 1 Inst. 2 Inst. 1 Inst. 2 Inst. 1 Inst. 2MIP solver withconservative cutgeneration ∗ .
74% 0 .
0% 4246 5932
MIP solver withautomatic cutgeneration ∗ .
93% 0 .
0% 4050 5569
Benders decom-position andapproximationalgorithm • Instance 1 is from Table 2. • Instance 2 changes the frequency level 𝐹 = and types of customers | Θ | = . • Runtime with ∗ means the solver hits the time limits and stops early. • 𝑈 𝐵 − 𝐿𝐵 gap is the difference between the upper bound and lower bound ofthe solutions. Since several operational parameters of the status-quo sys-tem are unknown, we conduct a sensitivity analysis in Figure6. The most critical design parameters for MT are the totalnumber of buses R and vehicular capacities 𝑉 ℓ . We test theoptimal design’s sensitivity with regard to these two param-eters.As the MT system’s available resources increase, the over-all welfare increases significantly as more travelers can savemoney by using MT facilities. Note that we do not considerthe purchase costs of additional MT buses, but assume thecurrent system’s bus fleet size may vary from Table 2. Onthe other hand, it is not inherently a profitably practice touse large-capacity vehicles as we can raise the frequencieson specific routes to offset the reduced per-vehicle capacity.This coincides with the recent development in operatingsmall-size flexible MT. As a building block for any multimodal mobility ecosystem,this work studies a joint pricing and transit network designproblem. A unified MILP framework solves the frequencysetting of MT in the master problem and solves the price andMoD rebalancing in subproblems sequentially. This primal-dual approach is substantially easier to implement and moreefficient in computation compared to the traditional bilevelprogram with equilibrium constraints. With the advent ofa shared, on-demand, and diverse mobility economy, a casestudy in Nashville, TN, highlights the ability of using thisapproach in solving real-world challenges.The limitations of this work include the type of choicemodel for travelers that can be accommodated by this lin-ear model, even though it admits the MNL, which is themost common model of discrete choice in transportation.Furthermore, we presume that information in the system is (a)
Sensitivity on total vehicle number (b)
Sensitivity on vehicle capacity
Figure 6.
Sensitivity analysis of the optimal value regardingMT parameterspublicized such that each traveler is completely rational inmaking their travel plans, which might not be the case inreality. We leave these questions for future work.
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A Summary of notation
Table 4.
Summary of notationNotation DefinitionDecision variables: 𝑥 𝜃𝑚 Type 𝜃 traveler’s mode choice strategyfor mode 𝑚𝑧 ℓ,𝑓 Binary variable for setting MT route ℓ ’sfrequency at level 𝑓𝑧 𝑏 𝑖𝑗 𝜓 Binary variable for setting bundled MTservices 𝑏 𝑖 𝑗 ’s frequency at level 𝜓𝑟 𝑖 𝑗 MoD’s rebalancing flows on edge 𝑒 𝑖 𝑗 𝑦 𝜃𝑚 Intermediate variable 𝑦 𝜃𝑚 = 𝜆 𝑠𝑡 ( 𝜃 ) 𝑥 𝜃𝑚 𝑝 𝜃𝑚 Prices for mode 𝑚 charged for type 𝜃 travelers 𝑝, 𝜁 , 𝑣, 𝛼, 𝛽𝑝, 𝜁 , 𝑣, 𝛼, 𝛽𝑝, 𝜁 , 𝑣, 𝛼, 𝛽 Dual variables of subproblem 𝜏, 𝜙
Intermediate variables for benders’decompositionParameters: 𝐺 = ( 𝑉 , 𝐸 ) Graph of multimodal mobility networkconsists of vertices 𝑉 and edges 𝐸𝑛 Cardinality of vertices 𝑉 Θ Type space of travelers 𝜆 𝑠𝑡 Travel demand from vertex 𝑠 to vertex 𝑡𝑞 𝑠𝑡 ( 𝜃 ) Probability density of type 𝜃 for 𝑠 − 𝑡 trip 𝐿 Total number of MT lines 𝐹 Maximum frequency level for each line M 𝑠𝑡 Set of hybrid modes available for 𝑠 − 𝑡 trip 𝑣 𝜃𝑚 Valuation of mode 𝑚 to type 𝜃 travelers 𝐸 𝑚 Edges trespassed by mode 𝑚𝑉 𝑚 Vertices trespassed by mode 𝑚 B 𝑖 𝑗 Set of bundled MT services on edge 𝑒 𝑖 𝑗 𝑜 ℓ Unit setup cost for line ℓ𝑐 𝑚 Operational cost of mode 𝑚 R Total number of available MT vehicles 𝑇 ℓ Cycling time of line ℓ P 𝑠𝑡𝑖 Set of paths for 𝑖 𝑡ℎ commodity’s 𝑠 − 𝑡 trip in subproblem’s approximationalgorithm 𝐾 Number of commodities in subproblem 𝜖, 𝛿
Error tolerance in approximationalgorithm V ℓ Vehicle capacity on line ℓ Acronyms:MoD Mobility-on-DemandMNL Multinomial Logit model ultimodal Mobility Systems: Joint Optimization of Transit Network Design and Pricing Nashville ’21, May 19–21, 2021, Nashville, TN
B Lift-and-project cuts for master problem
We first apply the convexification procedure to all constraintsin (7). For example, we can denote the left-hand coefficientmatrix of (1b) as 𝐴 . Since 𝑧 ℓ,𝑓 ≥
0, we can rewrite theconstraint as 𝑧 ℓ,𝑓 ( 𝑍 𝑏 𝑖𝑗 − 𝐴 𝑏 𝑖𝑗 𝑍 ℓ ′ ) ≤ , ∀ 𝑏 𝑖 𝑗 ∈ B 𝑖 𝑗 , ∀ 𝑒 𝑖 𝑗 ∈ 𝐸 ( − 𝑧 ℓ,𝑓 )( 𝑍 𝑏 𝑖𝑗 − 𝐴 𝑏 𝑖𝑗 𝑍 ℓ ′ ) ≤ , ∀ 𝑏 𝑖 𝑗 ∈ B 𝑖 𝑗 , ∀ 𝑒 𝑖 𝑗 ∈ 𝐸 and linearize the constraints as (cid:40) 𝑧 ℓ,𝑓 ← 𝑧 ℓ,𝑓 𝑧 ℓ,𝑓 𝑧 ℓ ′ ,𝑓 ′ ← 𝑤 ℓ,𝑓 , ℓ ≠ ℓ or 𝑓 ′ ≠ 𝑓 . Denote the generic form of constraints in the LP-relaxationas 𝐴𝑍 ≤ 𝑏 . Given the fractional solution 𝑍 ∗ , the lift-and-project cuts are generated by the following cut generating LP: maximize 𝜅𝑍 ∗ − 𝜅 (8) 𝑠.𝑡 . 𝜅 ≤ 𝑢𝐴 + 𝑢 𝑒𝑒𝑒 ℓ,𝑓 𝜅 ≤ 𝑢𝐴 − 𝑣 𝑒𝑒𝑒 ℓ,𝑓 𝜅 ≥ 𝑢𝑏𝜅 ≥ 𝑣𝑏 − 𝑣 ⊺ 𝑢 + 𝑢 + ⊺ 𝑣 + 𝑣 = 𝑢, 𝑣, 𝑢 , 𝑣 ≥ 𝑒𝑒𝑒 ℓ,𝑓 is an all-zero vector but one at the ℓ, 𝑓 entry.Let 𝐴 ℓ,𝑓 be the column index of 𝐴 . We add the strengthenedlift-and-project cuts for 𝑧 ℓ ′ ,𝑓 ′ as follows: ∑︁ ℓ ∈[ 𝐿 ] ,𝑓 ∈[ 𝐹 ] 𝛼 ℓ,𝑓 𝑧 ℓ,𝑓 ≤ 𝛽, where 𝛼 ℓ,𝑓 = max { 𝑢 ⊺ 𝐴 ℓ,𝑓 + 𝑢 ⌈ 𝑚 ℓ,𝑓 ⌉ , 𝑣 ⊺ 𝐴 ℓ,𝑓 − 𝑣 ⌊ 𝑚 ℓ,𝑓 ⌋} , ( ℓ ′ , 𝑓 ′ ) ≠ ( ℓ, 𝑓 ) max { 𝑢 ⊺ 𝐴 ℓ,𝑓 − 𝑢 , 𝑣 ⊺ 𝐴 ℓ,𝑓 + 𝑣 } , ( ℓ ′ , 𝑓 ′ ) = ( ℓ, 𝑓 ) max { 𝑢 ⊺ 𝐴 ℓ,𝑓 , 𝑣 ⊺ 𝐴 ℓ,𝑓 } , ℓ = 𝐿, 𝑓 = 𝐹𝛽 = min { 𝑢 ⊺ 𝑏, 𝑣 ⊺ 𝑏 + 𝑢 } 𝑚 ℓ,𝑓 = 𝑣 ⊺ 𝐴 ℓ,𝑓 − 𝑢 ⊺ 𝐴 ℓ,𝑓 𝑢0
0, we can rewrite theconstraint as 𝑧 ℓ,𝑓 ( 𝑍 𝑏 𝑖𝑗 − 𝐴 𝑏 𝑖𝑗 𝑍 ℓ ′ ) ≤ , ∀ 𝑏 𝑖 𝑗 ∈ B 𝑖 𝑗 , ∀ 𝑒 𝑖 𝑗 ∈ 𝐸 ( − 𝑧 ℓ,𝑓 )( 𝑍 𝑏 𝑖𝑗 − 𝐴 𝑏 𝑖𝑗 𝑍 ℓ ′ ) ≤ , ∀ 𝑏 𝑖 𝑗 ∈ B 𝑖 𝑗 , ∀ 𝑒 𝑖 𝑗 ∈ 𝐸 and linearize the constraints as (cid:40) 𝑧 ℓ,𝑓 ← 𝑧 ℓ,𝑓 𝑧 ℓ,𝑓 𝑧 ℓ ′ ,𝑓 ′ ← 𝑤 ℓ,𝑓 , ℓ ≠ ℓ or 𝑓 ′ ≠ 𝑓 . Denote the generic form of constraints in the LP-relaxationas 𝐴𝑍 ≤ 𝑏 . Given the fractional solution 𝑍 ∗ , the lift-and-project cuts are generated by the following cut generating LP: maximize 𝜅𝑍 ∗ − 𝜅 (8) 𝑠.𝑡 . 𝜅 ≤ 𝑢𝐴 + 𝑢 𝑒𝑒𝑒 ℓ,𝑓 𝜅 ≤ 𝑢𝐴 − 𝑣 𝑒𝑒𝑒 ℓ,𝑓 𝜅 ≥ 𝑢𝑏𝜅 ≥ 𝑣𝑏 − 𝑣 ⊺ 𝑢 + 𝑢 + ⊺ 𝑣 + 𝑣 = 𝑢, 𝑣, 𝑢 , 𝑣 ≥ 𝑒𝑒𝑒 ℓ,𝑓 is an all-zero vector but one at the ℓ, 𝑓 entry.Let 𝐴 ℓ,𝑓 be the column index of 𝐴 . We add the strengthenedlift-and-project cuts for 𝑧 ℓ ′ ,𝑓 ′ as follows: ∑︁ ℓ ∈[ 𝐿 ] ,𝑓 ∈[ 𝐹 ] 𝛼 ℓ,𝑓 𝑧 ℓ,𝑓 ≤ 𝛽, where 𝛼 ℓ,𝑓 = max { 𝑢 ⊺ 𝐴 ℓ,𝑓 + 𝑢 ⌈ 𝑚 ℓ,𝑓 ⌉ , 𝑣 ⊺ 𝐴 ℓ,𝑓 − 𝑣 ⌊ 𝑚 ℓ,𝑓 ⌋} , ( ℓ ′ , 𝑓 ′ ) ≠ ( ℓ, 𝑓 ) max { 𝑢 ⊺ 𝐴 ℓ,𝑓 − 𝑢 , 𝑣 ⊺ 𝐴 ℓ,𝑓 + 𝑣 } , ( ℓ ′ , 𝑓 ′ ) = ( ℓ, 𝑓 ) max { 𝑢 ⊺ 𝐴 ℓ,𝑓 , 𝑣 ⊺ 𝐴 ℓ,𝑓 } , ℓ = 𝐿, 𝑓 = 𝐹𝛽 = min { 𝑢 ⊺ 𝑏, 𝑣 ⊺ 𝑏 + 𝑢 } 𝑚 ℓ,𝑓 = 𝑣 ⊺ 𝐴 ℓ,𝑓 − 𝑢 ⊺ 𝐴 ℓ,𝑓 𝑢0 + 𝑣0