Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Efficient and Almost Optimal Solver for the Joint Routing-Assignment Problem via Partial JRA and Large-α Optimization

Qilong Yuan

TL;DR

The paper addresses the Joint Routing–Assignment (JRA) problem, which couples item-to-placeholder assignment with a Hamiltonian tour and is challenging for large-scale instances. It introduces Partial Path Reconstruction (PPR) and Spatially Localized PPR (SLPPR) within the PJAR framework, together with Large-$\alpha$ constraints, to obtain near-optimal solutions efficiently. Empirical results on datasets with $n=300,500,1000$ show average deviations as low as $0.00\%$ after polishing, with substantial time savings compared to exact MIP approaches; the method also highlights limitations of traditional $k$-opt approaches for large-scale JRA. The framework is applicable beyond JRA to TSP-like problems and related routing/assignment contexts, offering a scalable, high-accuracy alternative for robotics, logistics, and manufacturing task planning.

Abstract

The Joint Routing-Assignment (JRA) optimization problem simultaneously determines the assignment of items to placeholders and a Hamiltonian cycle that visits each node pair exactly once, with the objective of minimizing total travel cost. Previous studies introduced an exact mixed-integer programming (MIP) solver, along with datasets and a Gurobi implementation, showing that while the exact approach guarantees optimality, it becomes computationally inefficient for large-scale instances. To overcome this limitation, heuristic methods based on merging algorithms and shaking procedures were proposed, achieving solutions within approximately 1% deviation from the optimum. This work presents a novel and more efficient approach that attains high-accuracy, near-optimal solutions for large-scale JRA problems. The proposed method introduces a Partial Path Reconstructon (PPR) solver that first identifies key item-placeholder pairs to form a reduced subproblem, which is solved efficiently to refine the global solution. Using this PJAR framework, the initial heuristic merging solutions can be further improved, reducing the deviation by half. Moreover, the solution can be iteratively polished with PPR based solver along the optimization path to yield highly accurate tours. Additionally, a global Large-α constraint is incorporated into the JRA model to further enhance solution optimality. Experimental evaluations on benchmark datasets with n = 300, 500, and 1000 demonstrate that the proposed method consistently delivers almost optimal solutions, achieving an average deviation of 0.00% from the ground truth while maintaining high computational efficiency. Beyond the JRA problem, the proposed framework and methodologies exhibit strong potential for broader applications. The Framework can be applied to TSP and related optimization problems.

An Efficient and Almost Optimal Solver for the Joint Routing-Assignment Problem via Partial JRA and Large-α Optimization

TL;DR

The paper addresses the Joint Routing–Assignment (JRA) problem, which couples item-to-placeholder assignment with a Hamiltonian tour and is challenging for large-scale instances. It introduces Partial Path Reconstruction (PPR) and Spatially Localized PPR (SLPPR) within the PJAR framework, together with Large- constraints, to obtain near-optimal solutions efficiently. Empirical results on datasets with show average deviations as low as after polishing, with substantial time savings compared to exact MIP approaches; the method also highlights limitations of traditional -opt approaches for large-scale JRA. The framework is applicable beyond JRA to TSP-like problems and related routing/assignment contexts, offering a scalable, high-accuracy alternative for robotics, logistics, and manufacturing task planning.

Abstract

The Joint Routing-Assignment (JRA) optimization problem simultaneously determines the assignment of items to placeholders and a Hamiltonian cycle that visits each node pair exactly once, with the objective of minimizing total travel cost. Previous studies introduced an exact mixed-integer programming (MIP) solver, along with datasets and a Gurobi implementation, showing that while the exact approach guarantees optimality, it becomes computationally inefficient for large-scale instances. To overcome this limitation, heuristic methods based on merging algorithms and shaking procedures were proposed, achieving solutions within approximately 1% deviation from the optimum. This work presents a novel and more efficient approach that attains high-accuracy, near-optimal solutions for large-scale JRA problems. The proposed method introduces a Partial Path Reconstructon (PPR) solver that first identifies key item-placeholder pairs to form a reduced subproblem, which is solved efficiently to refine the global solution. Using this PJAR framework, the initial heuristic merging solutions can be further improved, reducing the deviation by half. Moreover, the solution can be iteratively polished with PPR based solver along the optimization path to yield highly accurate tours. Additionally, a global Large-α constraint is incorporated into the JRA model to further enhance solution optimality. Experimental evaluations on benchmark datasets with n = 300, 500, and 1000 demonstrate that the proposed method consistently delivers almost optimal solutions, achieving an average deviation of 0.00% from the ground truth while maintaining high computational efficiency. Beyond the JRA problem, the proposed framework and methodologies exhibit strong potential for broader applications. The Framework can be applied to TSP and related optimization problems.

Paper Structure

This paper contains 47 sections, 41 equations, 4 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Connections to existing problem classes
  3. Pickup-and-delivery and the stacker-crane problem.
  4. Assignment - routing integration.
  5. Purposes of the Article
  6. Mathematical Formulation
  7. Simplified Model Constraints
  8. Path Improvement: Partial Path Reconstruction (PPR) Method
  9. Breaking and Decomposition of the Cycle.
  10. Identification of Segments.
  11. Reserved Edge Set.
  12. Summary.
  13. Solution Recovery and Reconstruction of the Complete Tour
  14. Removal of Temporary Linkages.
  15. Reintegration of Reserved Linkages.
  16. ...and 32 more sections

Figures (4)

  • Figure 1: Illustration of Workflow of The Proposed Hungarian algorithm
  • Figure 2: Three figures displayed side by side.
  • Figure 3: Polishing Path Solution with SLPPR:Examples with Five Circles. (a)Before polishing. (b).Breaking connections around the circles. (c).Optimized reconnection with SLPPR. (d). Reconstructed tour with less distance cost.
  • Figure 4: Sample of Final Optimal Path for n=300 (in blue) (Merged Initial Guess is in Red).