Papers

A sample of research publications coauthored by our team

Featured Publications 🡒

Optimization & AI for Modern Transit 🡒

Transit Equity, Behavior, and Outcomes 🡒

Early Work 🡒

Why Make This Public?   Because It Sets Us Apart.

Much of the microtransit industry makes bold claims without providing the data to back them up. At Modal, we believe transit agencies deserve to know the evidence behind the systems they operate.

These papers show why Modal consistently outperforms alternatives: our optimization models and algorithms are the most advanced in the field, backed by a research-to-commercial pipeline that keeps us ahead.

Of course, we protect our core IP, and many of our key algorithms remain private. But sharing some of the science allows transit providers to see the strength of our foundation and the caliber of our talent.

Featured Publications

  1. Marta Reach: Piloting an On-Demand Multimodal Transit System in Atlanta
    The MARTA Reach pilot demonstrated that On-Demand Multimodal Transit Systems (ODMTS) can effectively address the first/last mile problem by integrating on-demand shuttles with fixed rail routes, resulting in a highly valued service that shifted a significant number of trips from ride-hailing, taxis, or personal cars to public transit while achieving cost-effective operations with potential for economic sustainability.
  2. Column generation for real-time ride-sharing operations
    RTDARS applies column generation to real-time ride-sharing dispatching, minimizing average wait times to around 2.2 minutes while guaranteeing service for all customers and limiting trip deviations to 0.62 minutes, enabling transit agencies to enhance efficiency, reduce congestion, and support scalable deployment in dense urban areas like New York City.
  3. Paratransit Optimization with Constraint Programming: A Case Study in Savannah, Georgia
    This constraint programming model jointly optimizes route planning and shift scheduling for paratransit services, enabling transit agencies like Chatham Area Transit to serve 97% of requests through improved efficiency and flexible shift starts.
  4. Community-based trip sharing for urban commuting
    This research introduces an optimization approach for community-based trip sharing that clusters commuters and matches them based on schedules and locations, achieving 44% reduction in daily car usage on a mid-size city dataset, thereby alleviating traffic congestion and parking pressure for transit agencies.
  5. The impact of congestion and dedicated lanes on on-demand multimodal transit systems
    Dedicated Bus Lanes (DBLs) significantly reduce travel times on congested routes like I-85 in Metro Atlanta, enabling On-Demand Multimodal Transit Systems (ODMTS) to offer faster, more competitive trips compared to driving, which boosts system adoption by 32 percentage points for non-local riders and maintains high service levels for local users without increasing net costs.

Optimization & AI for Modern Transit

  1. Investigating large neighbourhood search for bus driver scheduling
    The proposed Large Neighbourhood Search approach enhances bus driver scheduling by integrating novel destroy operators and Column Generation for repair, outperforming existing heuristics and improving upper bounds on large real-life instances.
  2. Branch-and-check with explanations for the vehicle routing problem with time windows
    The BCE method enables transit agencies to solve VRPTW instances more efficiently by automatically generating cuts through general-purpose conflict analysis, outperforming traditional branch-and-cut in proving optimality and finding high-quality routes quickly, which can reduce operational costs and improve scheduling in real-world fleet management.
  3. T-SCORE Project M1-Multi-Modal Optimization: Development of Optimization Frameworks on On-Demand Multimodal Transit Systems
    We show that adding on-demand shuttles to existing fixed-route transit solves the first/last-mile problem while cutting both agency costs and passenger travel times. Real-city simulations in San Francisco and Salt Lake City show how to deploy it seamlessly today without changing any new rail or bus lines.
  4. Revitalizing Public Transit in Low Ridership Areas: An Exploration of On-Demand Multimodal Transit Systems
    By replacing low-ridership fixed-route bus lines with on-demand shuttles that connect to fixed transit services, transit agencies can reduce operational costs by 37.5% while significantly cutting passenger travel times - from over 40 minutes to under 17 minutes on average - through improved accessibility and dynamic ride-sharing in areas like Austin, Texas.
  5. Ridesharing and fleet sizing for on-demand multimodal transit systems
    This work optimizes ridesharing and fleet sizing in on-demand multimodal transit, enabling agencies to balance costs and service levels through integrated planning that supports scalable, efficient deployments.
  6. Optimization models for estimating transit network origin–destination flows with big transit data
    This research develops optimization models to estimate network-level origin-destination flows using AVL/APC data, enabling transit agencies to identify transfers and produce accurate O-D matrices at varying resolutions.
  7. Resiliency of on-demand multimodal transit systems during a pandemic
    Modal helps transit agencies like MARTA maintain service during pandemics by integrating high-frequency trains and buses with on-demand shuttles, reducing costs through resilient designs that handle depressed demand and social distancing while preserving convenience and accessibility.
  8. Learning Model Predictive Controllers for Real-Time Ride-Hailing Vehicle Relocation and Pricing Decisions
    This research enables transit agencies to deploy ride-hailing systems with higher spatial-temporal fidelity by learning offline the computationally intensive Model Predictive Control (MPC) for vehicle relocation and dynamic pricing, increasing riders served and improving service quality on the New York City dataset.
  9. The flexible and real-time trip sharing problems
    This research introduces robust planning methods for commute trip sharing that account for uncertainties in return trip schedules, enabling transit agencies to generate reliable driver assignments and routes using historical data and scenario sampling.
  10. The commute trip sharing problem
    This research develops optimization algorithms for a car-pooling platform that matches commuters based on spatial and temporal proximity, guaranteeing return rides and reducing vehicle usage by 57% and vehicle miles traveled by 46% while only increasing average ride time by 22%, enabling transit agencies to alleviate parking pressure and congestion through efficient, deployable ride-sharing solutions.
  11. Real-time pricing optimization for ride-hailing quality of service
    The AP-RTRS framework uses dynamic pricing and vehicle relocation to reduce waiting times during demand surges in ride-hailing systems, ensuring all riders are served within reasonable time frames while maintaining revenues and geographical fairness.
  12. Learning Model-Based Vehicle-Relocation Decisions for Real-Time Ride-Sharing: Hybridizing Learning and Optimization
    Transit agencies can deploy this hybrid learning-optimization framework to enable real-time vehicle relocation with longer planning horizons, reducing average rider waiting time by 27% compared to standard model predictive control, while maintaining computational efficiency through polynomial-time prediction and optimization steps.
  13. Branch and price for bus driver scheduling with complex break constraints
    This Branch and Price approach optimizes bus driver schedules under complex break regulations, delivering provably optimal or near-optimal solutions that reduce operational costs and improve schedule efficiency for transit agencies.
  14. Spatio-temporal point processes with attention for traffic congestion event modeling
    This research introduces an attention-based point process model that integrates traffic sensor data and police reports to predict congestion events, enabling transit agencies to anticipate and mitigate disruptions more effectively.
  15. The benefits of autonomous vehicles for community-based trip sharing
    Autonomous vehicles can reduce daily vehicle usage by 92% for community-based commute trip sharing in Ann Arbor, improving on traditional car-pooling by 34% and cutting vehicle miles traveled by 30%, enabling transit agencies to achieve significant cost savings and operational efficiency through higher vehicle utilization and centralized routing.
  16. Commuting with Autonomous Vehicles: A Branch and Cut Algorithm with Redundant Modeling
    The CTSPAV algorithm enables transit agencies to deploy autonomous vehicle fleets for commuter ride-sharing, achieving 92% reduction in daily vehicle counts and 60% decrease in peak-hour congestion through optimized mini-route scheduling, while trading off increased empty miles for enhanced efficiency and scalability in medium-sized cities.
  17. Bilevel optimization for on-demand multimodal transit systems
    This research enables transit agencies to design On-Demand Multimodal Transit Systems (ODMTS) that account for latent demand by integrating rider mode choice models into a bilevel optimization framework, leading to more cost-efficient network designs with higher ridership adoption and reduced overall costs compared to designs based solely on existing demand.
  18. Transfer-expanded graphs for on-demand multimodal transit systems
    This research provides transit agencies with a scalable optimization framework for designing On-Demand Multimodal Transit Systems (ODMTS) that integrate high-frequency buses, rail, and on-demand shuttles while limiting passenger transfers.
  19. Real-time dispatching of large-scale ride-sharing systems: Integrating optimization, machine learning, and model predictive control
    This research introduces an end-to-end framework that integrates optimization, machine learning for demand forecasting, and model predictive control to relocate idle vehicles in real-time ride-sharing systems.
  20. Optimization Models for Estimating Transit Network Origin-Destination Flows with AVL/APC Data
    This research develops optimization models to estimate network-level origin-destination flows using AVL/APC data, enabling transit agencies to identify transfers and produce accurate O-D matrices at varying resolutions.
  21. Benders decomposition for the design of a hub and shuttle public transit system
    This research applies Benders decomposition to optimize hub-and-shuttle transit networks, enabling transit agencies to reduce average travel times by a factor of two while maintaining or lowering overall system costs compared to existing setups, as demonstrated on Canberra's public transit data.
  22. Branch-And-Price-And-Check Model For The Vehicle Routing Problem With Location Resource Constraints
    This model optimizes vehicle routes under location-specific resource constraints like parking bays or forklifts, enabling transit agencies to avoid scheduling conflicts and reduce operational delays.

Transit Equity, Behavior, and Outcomes

  1. Path-Based Formulations for the Design of On-demand Multimodal Transit Systems with Adoption Awareness
    This paper reconsiders the On-Demand Multimodal Transit Systems (ODMTS) Design with Adoptions problem (ODMTS-DA) to capture the latent demand. Our new path-based optimization model, called P-Path, solves large-scale instances (35+ million constraints) optimally.
  2. Empathy and AI: Achieving Equitable Microtransit for Underserved Communities
    This research introduces an AI-driven sociotechnical system that engages users with empathy-building interventions and prosocial incentives to shift flexible trips, thereby increasing vehicle utilization and reducing wait times for critical work and medical travel in underserved small communities.
  3. Iterative Approaches for Integrating Rider Behavior into the Design of Large-Scale On-demand Multimodal Transit Systems
    This work presents iterative methods to incorporate rider behavior into on-demand multimodal transit design, helping agencies create systems that better align with user preferences and boost adoption rates.
  4. Capturing travel mode adoption in designing on-demand multimodal transit systems
    This research enables transit agencies to design On-Demand Multimodal Transit Systems (ODMTS) that account for latent demand by integrating rider mode preferences into a bilevel optimization model, resulting in high adoption rates, 53% shorter trip durations compared to existing systems, and improved access for low-income riders while operating within budget constraints.
  5. Modeling heterogeneity in mode-switching behavior under a mobility-on-demand transit system: An interpretable machine learning approach
    This research applies machine learning to predict and interpret individual mode-switching behavior in a Mobility-on-Demand (MOD) transit system, revealing that existing drivers are more sensitive to additional pickups due to privacy concerns, while current transit users are willing to share rides but reluctant to transfers.
  6. Modeling Stated Preference for Mobility-on-Demand Transit: A Comparison of Machine Learning and Logit Models
    Machine learning models, particularly random forests, outperform traditional logit models in predicting travel mode choices for integrated high-frequency transit and ridesourcing systems, enabling transit agencies to enhance forecasting accuracy and optimize service planning.
  7. Measuring transit equity of an on-demand multimodal transit system
    This study provides frameworks for assessing equity in on-demand multimodal transit, allowing agencies to ensure fair access and benefits distribution across diverse populations.
  8. Public transit for special events: Ridership prediction and train scheduling
    This research offers predictive models for special event ridership and optimized scheduling, enabling agencies to handle surges efficiently and improve service reliability.
  9. Mobility-on-demand versus fixed-route transit systems: An evaluation of traveler preferences in low-income communities
    Transit agencies can adopt mobility-on-demand systems to enhance accessibility to destinations for low-income communities, potentially improving service efficiency and user satisfaction, while addressing key concerns such as women's safety, technology inefficiency, and ride-request processes to facilitate real-world deployment.
  10. Distilling Black-Box Travel Mode Choice Model for Behavioral Interpretation
    This research applies model distillation to interpret black-box machine learning models for travel mode choice, enabling transit agencies to derive transparent decision rules from complex predictions.
  11. Prediction and Behavioral Analysis of Travel Mode Choice: A Comparison of Machine Learning and Logit Models
    Transit agencies can leverage random forest machine learning models to achieve significantly higher predictive accuracy in forecasting travel mode choices compared to traditional logit models, enabling more precise planning for emerging mobility services like high-frequency buses and on-demand shuttles.
  12. Shared E-scooters: Business, pleasure, or transit?
    The study reveals that shared e-scooters in Atlanta primarily serve as affordable last-mile solutions for business trips and leisure activities, enabling transit agencies to enhance efficiency and reduce parking needs in urban centers through potential relocation of parking spaces and integration with public transit systems.
  13. Constrained-based differential privacy for mobility services
    This research introduces Constraint-Based Differential Privacy (CBDP) to release mobility data for transit agencies, enabling the design of on-demand multimodal transit systems with minimal cost increases (e.g., 16% fleet size growth vs. 150% under standard methods) and preserved performance metrics like waiting times, while improving accuracy over existing privacy approaches by an order of magnitude for real-world deployment.

Early Work

  1. Joint vehicle and crew routing and scheduling
    This research enables transit agencies to jointly optimize vehicle and crew routes, allowing crews to interchange vehicles and reducing overall costs compared to sequential methods, while improving efficiency through synchronized scheduling and fewer crews at the expense of additional vehicles in integrated models.
  2. A multistage very large-scale neighborhood search for the vehicle routing problem with soft time windows
    This algorithm optimizes vehicle routing with soft time windows by minimizing routes, violations, and distance, enabling transit agencies to reduce fleet size and operational costs while improving service efficiency in real-world deployments.
  3. Multi-period vehicle loading with stochastic release dates
    This paper addresses multi-period vehicle loading under uncertainty, providing agencies with robust planning tools for efficient resource use in variable demand scenarios.
  4. Large neighborhood search for dial-a-ride problems
    This research introduces LNS-FFPA, a constraint-based large neighborhood search algorithm that minimizes routing costs in Dial-a-Ride Problems by efficiently reinserting customer sets, enabling transit agencies to achieve high-quality vehicle routes significantly faster than state-of-the-art methods, thus reducing operational costs and improving service efficiency in dynamic door-to-door transportation systems.
  5. Vehicle routing for the last mile of power system restoration
    This work optimizes last-mile routing for power restoration, offering adaptable techniques for transit agencies in emergency response and recovery operations to minimize downtime.
  6. A relaxation-guided approach for vehicle routing problems with black box feasibility
    This method enables transit agencies to solve complex routing problems with unknown constraints, such as loading or scheduling, by efficiently generating feasible routes using guided heuristics and column generation, leading to improved solution quality and reduced computational effort in real-world deployments.
  7. Spatial, temporal, and hybrid decompositions for large-scale vehicle routing with time windows
    This research introduces customer-based adaptive decomposition techniques that enable transit agencies to solve large-scale vehicle routing problems with time windows more efficiently by focusing on spatial, temporal, or hybrid subproblems, yielding significant improvements - such as 10.6% better solution quality over large neighborhood search after 5 minutes - while facilitating real-world deployment through faster, high-quality routing optimizations.
  8. The price of commitment in online stochastic vehicle routing
    This research develops algorithms for online stochastic vehicle routing with time windows that require immediate vehicle assignment upon request acceptance, enabling transit agencies to leverage stochastic sampling and optimization to enhance decision-making and reduce rejected customers.
  9. Large neighborhood search for the double traveling salesman problem with multiple stacks
    This research applies a large neighborhood search algorithm to optimize short-haul and long-haul pickup and delivery operations by minimizing travel time under LIFO stack constraints, enabling transit agencies to achieve cost savings and improve efficiency in real-world logistics through robust routing solutions that are consistently within 2% of best-known outcomes.
  10. Randomized adaptive spatial decoupling for large-scale vehicle routing with time windows
    The RASD scheme enables transit agencies to quickly generate high-quality vehicle routing plans for large fleets by adaptively decoupling spatial subproblems and re-optimizing them, achieving significant cost savings and efficiency gains under tight time constraints while also discovering improved solutions over extended runtimes.
  11. Waiting and relocation strategies in online stochastic vehicle routing
    This foundational work explores waiting and relocation tactics for stochastic routing, providing agencies with strategies to handle uncertainty and improve response times in on-demand services.
  12. A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows
    Transit agencies can deploy this two-stage hybrid algorithm to minimize the number of vehicles used in pickup and delivery operations with time windows, achieving 76% improvement on large benchmarks and reducing vehicle count by as much as three, while subsequently lowering total travel costs for greater operational efficiency and cost savings.
  13. Scenario-based planning for partially dynamic vehicle routing with stochastic customers
    This research develops scenario-based planning for dynamic routing under customer uncertainty, enabling agencies to build resilient schedules that adapt to real-time changes and reduce disruptions.
  14. A two-stage hybrid local search for the vehicle routing problem with time windows
    This algorithm first minimizes the number of vehicles using simulated annealing, then reduces travel costs with large neighborhood search, enabling transit agencies to achieve optimal routing plans that lower operational costs and improve fleet efficiency on standard benchmarks.

Talk with our Experts

Have questions about how these research insights apply to your agency? Modal is built by the same researchers who pioneered this work, and we're available to discuss your system, evaluate feasibility, or walk through our optimization models.

If you want to understand how on-demand transit would work in your region, our team can walk you through the modeling, tradeoffs, and pilot design.

Talk with Our Experts