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

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.