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A Branch-and-Cut Algorithm for the Optimal Design of Parking Lots with One-way and Two-way Lanes

Helen Thomas, Tarun Rambha

TL;DR

This paper tackles optimizing parking-lot designs to maximize perpendicular stalls under both two-way and one-way driving lanes by rasterizing plots into grids and formulating a generalized MIP. It advances beyond flow-based models by introducing connectivity cuts and a branch-and-cut framework that dramatically improves solution times, while extending the model to handle one-way traffic with directionality constraints. The approach is validated on 325 NYC parking lots, showing substantial speedups and higher stall counts for one-way configurations, with meaningful improvements even under longer time budgets. Practical extensions address turning constraints, multiple entrances/exits, and turn-restriction considerations, highlighting the method’s potential for real-world parking facility design and optimization.

Abstract

We address the problem of maximizing the number of stalls in parking lots where vehicles park perpendicular to the driveways. Building on recent research on two-way driving lanes, we first formulate a mixed integer program to maximize the number of parking stalls using a flow-based approach. Parking lots are rasterized into a grid, and the proposed MIP model optimizes them in a generic manner, adapting to the grid resolution and stall size without requiring custom formulations. The constraints ensure the connectivity of parking stalls and driveways to the entrance/exit. This formulation is then extended to the case of one-way driving lanes. We then propose valid inequalities and a branch-and-cut algorithm for the one-way and two-way lane configurations. This approach eliminates flow variables, big-M type constraints, and improves solution times for medium-sized instances. The effectiveness of the suggested models is showcased on 325 parking lots from New York City. For instances in which the flow version could be solved in 15 minutes, the branch-and-cut algorithm improved the median runtimes by 87.43% for the one-way case and by 79.36% for the two-way case and resulted in better optimality gaps for the other instances, compared to the baseline flow-based formulation. Similar advantages were observed when run with a time budget of two hours. One-way configurations accommodated, on average, 18.63% more vehicles on average than their two-way counterparts across all instances. Modifications to the proposed formulations that consider the turning characteristics of vehicles and the presence of multiple entrances and exits are also examined.

A Branch-and-Cut Algorithm for the Optimal Design of Parking Lots with One-way and Two-way Lanes

TL;DR

This paper tackles optimizing parking-lot designs to maximize perpendicular stalls under both two-way and one-way driving lanes by rasterizing plots into grids and formulating a generalized MIP. It advances beyond flow-based models by introducing connectivity cuts and a branch-and-cut framework that dramatically improves solution times, while extending the model to handle one-way traffic with directionality constraints. The approach is validated on 325 NYC parking lots, showing substantial speedups and higher stall counts for one-way configurations, with meaningful improvements even under longer time budgets. Practical extensions address turning constraints, multiple entrances/exits, and turn-restriction considerations, highlighting the method’s potential for real-world parking facility design and optimization.

Abstract

We address the problem of maximizing the number of stalls in parking lots where vehicles park perpendicular to the driveways. Building on recent research on two-way driving lanes, we first formulate a mixed integer program to maximize the number of parking stalls using a flow-based approach. Parking lots are rasterized into a grid, and the proposed MIP model optimizes them in a generic manner, adapting to the grid resolution and stall size without requiring custom formulations. The constraints ensure the connectivity of parking stalls and driveways to the entrance/exit. This formulation is then extended to the case of one-way driving lanes. We then propose valid inequalities and a branch-and-cut algorithm for the one-way and two-way lane configurations. This approach eliminates flow variables, big-M type constraints, and improves solution times for medium-sized instances. The effectiveness of the suggested models is showcased on 325 parking lots from New York City. For instances in which the flow version could be solved in 15 minutes, the branch-and-cut algorithm improved the median runtimes by 87.43% for the one-way case and by 79.36% for the two-way case and resulted in better optimality gaps for the other instances, compared to the baseline flow-based formulation. Similar advantages were observed when run with a time budget of two hours. One-way configurations accommodated, on average, 18.63% more vehicles on average than their two-way counterparts across all instances. Modifications to the proposed formulations that consider the turning characteristics of vehicles and the presence of multiple entrances and exits are also examined.

Paper Structure

This paper contains 25 sections, 6 theorems, 41 equations, 31 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Gaps and contributions
  4. A generalized MIP formulation
  5. Problem setup
  6. Single-purpose constraints
  7. Accessibility constraints
  8. Driveway connectivity constraints
  9. One-way parking lot configuration
  10. Exit connectivity constraints
  11. Directionality constraints and dead ends
  12. A branch-and-cut algorithm
  13. Two-way driveways
  14. Strengthened valid inequalities
  15. Avoiding infeasible solutions -- separation procedure
  16. ...and 10 more sections

Key Result

Proposition 1

Let $V \in \mathcal{V}_{pq}$ be a vertex separator and let the partitions created in $G$ after removing it be $V_{p q}$ and $V_{p q}'$. Then the inequality where $c_{kl} = |D^{-1}_{kl} \cap V_{p q}'|$ is valid for the feasible region of the formulation $\mathcal{F}^{\textsc{flow}}_{\textsc{2W}}$.

Figures (31)

  • Figure 1: Processing a parking lot instance
  • Figure 2: Optimal parking lot configurations for two-way and one-way driving lane configurations
  • Figure 3: Decision variables of the integer program. The example here assumes parking fields with $\omega = 2$ and $\ell=3$ and square driving fields with $\delta=4$. Blocked cells are shown in red. The entrance field is anchored at $(0,5)$.
  • Figure 4: Sets of parking and driving fields containing the yellow cell $(6,7)$
  • Figure 5: Driving field neighbors of parking field at $(6,6)$ (in left and center panels) and the need for additional constraints on the driving fields (right).
  • ...and 26 more figures

Theorems & Definitions (13)

  • Remark
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Theorem 1
  • ...and 3 more