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Solving the Line-Based Dial-a-Ride Problem by Generating Stopping Patterns

Antonio Lauerbach, Sven Mallach, Kendra Reiter, Marie Schmidt, Michael Stiglmayr

Abstract

In the line-based dial-a-ride problem (liDARP), vehicles operate along a predefined bus line, with the possibility of skipping stations and turning when empty. Motivated by the practical observation that tight passenger time windows often limit pooling in on-demand services, we introduce a new variant of this transportation system by removing all temporal constraints, which we call the liDARP without TWs. We introduce a new MILP formulation for the liDARP without TWs, which constructs feasible tours as sequences of stopping patterns; first, we consider a fundamental single-vehicle, single-pass special case. Based on our insights, we develop a branch-and-price algorithm where the pricing problem generates profitable stopping patterns. For practical applications, we additionally propose a root node heuristic, using the stopping patterns generated at the root node. Computational experiments show that our branch-and-price algorithm is competitive, finding solutions with a MIP gap of less than 5% for large instances in 60 minutes. Further, the root node heuristic scales to instances with up to 100 requests, outperforming the state-of-the-art and reaching optimality gaps of less than 5% within 15 minutes. This method is highly effective in generating solutions for practical applications, where solving large problems quickly is more valuable than reaching optimality.

Solving the Line-Based Dial-a-Ride Problem by Generating Stopping Patterns

Abstract

In the line-based dial-a-ride problem (liDARP), vehicles operate along a predefined bus line, with the possibility of skipping stations and turning when empty. Motivated by the practical observation that tight passenger time windows often limit pooling in on-demand services, we introduce a new variant of this transportation system by removing all temporal constraints, which we call the liDARP without TWs. We introduce a new MILP formulation for the liDARP without TWs, which constructs feasible tours as sequences of stopping patterns; first, we consider a fundamental single-vehicle, single-pass special case. Based on our insights, we develop a branch-and-price algorithm where the pricing problem generates profitable stopping patterns. For practical applications, we additionally propose a root node heuristic, using the stopping patterns generated at the root node. Computational experiments show that our branch-and-price algorithm is competitive, finding solutions with a MIP gap of less than 5% for large instances in 60 minutes. Further, the root node heuristic scales to instances with up to 100 requests, outperforming the state-of-the-art and reaching optimality gaps of less than 5% within 15 minutes. This method is highly effective in generating solutions for practical applications, where solving large problems quickly is more valuable than reaching optimality.
Paper Structure (40 sections, 2 theorems, 7 equations, 8 figures, 5 tables)

This paper contains 40 sections, 2 theorems, 7 equations, 8 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Definition
  4. Modeling and Mathematical Formulation
  5. Modeling Stopping Patterns
  6. Mixed-Integer Linear Programming Model
  7. Positions
  8. Overlapping Requests
  9. Parameters
  10. Variables
  11. Model
  12. Generating Stopping Patterns
  13. Complexity
  14. Solving the most profitable stopping pattern Problem
  15. ILP for the most profitable stopping pattern Problem
  16. ...and 25 more sections

Key Result

Theorem 1

The decision version of the liDARP without TWs is $\mathsf{NP}$-complete even in the case where there is only one vehicle of capacity one and when the distances are Euclidean.

Figures (8)

  • Figure 1: Example instance and solution for the liDARP without TWs.
  • Figure 2: Line with requests (origins as colored circles, destinations as colored squares) and the two stopping patterns used in \ref{['fig:example-solution']}. Filled markers are visited, unfilled markers are skipped stations.
  • Figure 3: (a) An instance of clique where $b=3$, with the vertices $1$, $2$, and $4$ forming a $3$-clique (highlighted in red). (b) The most profitable uncapacitated stopping pattern instance constructed from the clique instance in (a). An optimal stopping pattern includes the vertex nodes corresponding to the $3$-clique in (a) with six of the (red) edge requests being served by this stopping pattern. The base, particularity, and consistency requests are omitted for clarity.
  • Figure 4: Flow of the Branch-and-Price Algorithm for the liDARP without TWs.
  • Figure 5: Best bound (\ref{['fig:model-comparison:eb_bound']}\ref{['fig:model-comparison:bnp_bound']}) and best incumbent (\ref{['fig:model-comparison:eb_incumbent']}\ref{['fig:model-comparison:bnp_incumbent']}) found for the ALAEB model and branch-and-price algorithm, after a given number of seconds of runtime. On the bottom of each figure, the instances for which no incumbent could be found with the respective approach are indicated.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • proof
  • proof