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Operational route planning under uncertainty for Demand Adaptive Systems

Benedikt Lienkamp, Mike Hewitt, Axel Parmentier, Maximilian Schiffer

TL;DR

This work addresses real-time operational route planning for Demand Adaptive Systems by formulating a Markov decision process with rolling-horizon look-ahead and solving a two-stage stochastic program at each decision point. A sample-based deterministic decomposition enables parallel solution of full-information subproblems, and a consensus-based heuristic along with a myopic baseline provide scalable alternatives. Extensive Munich-based data experiments show that the exact decomposition yields the best profits and passenger service levels within seconds, while the heuristic substantially speeds up computation with minimal loss in quality; even a limited number of scenarios suffices to approach the full-information benchmark, especially on highly flexible or highly fixed routes. Practically, converting fixed lines to DAS can significantly boost revenue and served passengers, with substantial gains achieved by increasing route flexibility and/or encouraging modest passenger walking distances. The study also reveals that a targeted mix of flexibility and traveler behavior can dramatically reduce cost per passenger, highlighting actionable levers for transit operators.

Abstract

With an increasing need for more flexible mobility services, we consider an operational problem arising in the planning of Demand Adaptive Systems (DAS). Motivated by the decision of whether to accept or reject passenger requests in real time in a DAS, we introduce the operational route planning problem of DASs. To this end, we propose an algorithmic framework that allows an operator to plan which passengers to serve in a DAS in real-time. To do so, we model the operational route planning problem as a Markov decision process (MDP) and utilize a rolling horizon approach to approximate the MDP via a two-stage stochastic program in each timestep to decide on the next action. Furthermore, we determine the deterministic equivalent of our approximation through sample-based approximation. This allows us to decompose the deterministic equivalent of our two-stage stochastic program into several full information planning problems, which can be solved in parallel efficiently. Additionally, we propose a consensus-based heuristic and a myopic approach. We perform extensive numerical studies based on real-world data provided to us by the public transportation provider of Munich, Germany. We show that our exact decomposition yields the best results in under five seconds, and our heuristic approach reduces the serial computation time by 17 - 57% compared to our exact decomposition, with a solution quality decline of less than one percent. From a managerial perspective, we show that by switching a fixed-line bus route to a DAS, an operator can increase profit by up to 49% and the number of served passengers by up to 35% while only increasing the travel distance of the bus by 14%. Furthermore, we show that an operator can reduce their cost per passenger by 43 - 51% by increasing route flexibility and that incentivizing passengers to walk slightly longer distances reduces the cost per passenger by 83-85%.

Operational route planning under uncertainty for Demand Adaptive Systems

TL;DR

This work addresses real-time operational route planning for Demand Adaptive Systems by formulating a Markov decision process with rolling-horizon look-ahead and solving a two-stage stochastic program at each decision point. A sample-based deterministic decomposition enables parallel solution of full-information subproblems, and a consensus-based heuristic along with a myopic baseline provide scalable alternatives. Extensive Munich-based data experiments show that the exact decomposition yields the best profits and passenger service levels within seconds, while the heuristic substantially speeds up computation with minimal loss in quality; even a limited number of scenarios suffices to approach the full-information benchmark, especially on highly flexible or highly fixed routes. Practically, converting fixed lines to DAS can significantly boost revenue and served passengers, with substantial gains achieved by increasing route flexibility and/or encouraging modest passenger walking distances. The study also reveals that a targeted mix of flexibility and traveler behavior can dramatically reduce cost per passenger, highlighting actionable levers for transit operators.

Abstract

With an increasing need for more flexible mobility services, we consider an operational problem arising in the planning of Demand Adaptive Systems (DAS). Motivated by the decision of whether to accept or reject passenger requests in real time in a DAS, we introduce the operational route planning problem of DASs. To this end, we propose an algorithmic framework that allows an operator to plan which passengers to serve in a DAS in real-time. To do so, we model the operational route planning problem as a Markov decision process (MDP) and utilize a rolling horizon approach to approximate the MDP via a two-stage stochastic program in each timestep to decide on the next action. Furthermore, we determine the deterministic equivalent of our approximation through sample-based approximation. This allows us to decompose the deterministic equivalent of our two-stage stochastic program into several full information planning problems, which can be solved in parallel efficiently. Additionally, we propose a consensus-based heuristic and a myopic approach. We perform extensive numerical studies based on real-world data provided to us by the public transportation provider of Munich, Germany. We show that our exact decomposition yields the best results in under five seconds, and our heuristic approach reduces the serial computation time by 17 - 57% compared to our exact decomposition, with a solution quality decline of less than one percent. From a managerial perspective, we show that by switching a fixed-line bus route to a DAS, an operator can increase profit by up to 49% and the number of served passengers by up to 35% while only increasing the travel distance of the bus by 14%. Furthermore, we show that an operator can reduce their cost per passenger by 43 - 51% by increasing route flexibility and that incentivizing passengers to walk slightly longer distances reduces the cost per passenger by 83-85%.

Paper Structure

This paper contains 20 sections, 1 theorem, 17 equations, 18 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Problem description and mathematical formulations
  4. Full information problem setting
  5. Dynamic and stochastic problem setting
  6. Methodology
  7. Full information approximation
  8. Decomposition of sample-based approximation
  9. Consensus-based heuristic approach
  10. Myopic approach
  11. Computational analysis
  12. Computational effectiveness of the proposed methods
  13. Computational effectiveness for varying CSF
  14. Distribution of computation times for varying CSF
  15. Computational effectiveness for varying numbers of scenarios
  16. ...and 5 more sections

Key Result

Theorem 4.1

Problem prob: two stage Q decomp is infeasible for $\sigma \in \Omega$ with $y_{r_{\theta}} = 1$ if and only if Problem prob: two stage Q decomp is infeasible for $S_{\theta} \cup r_{\theta}$ with $y_{r_{\theta}} = 1$.

Figures (18)

  • Figure 1: A line with compulsory stops (red) and optional stops (blue)
  • Figure 2: Planned bus route with compulsory and optional stops before requests received
  • Figure 3: Different bus routes for different accepted passenger requests
  • Figure 4: Visualization of our sample-based approximation where we consider possible future request scenarios at the same time
  • Figure 5: Visualization of our sample-based approximation decomposition into different subproblems where we independently solve different scenarios $\sigma_i$ with acceptance (A) or rejection (R) of request $r_{\theta}$
  • ...and 13 more figures

Theorems & Definitions (2)

  • Theorem 4.1
  • proof