Assignment-Routing Optimization: Solvers for Problems Under Constraints

ASSIGNMENT-ROUTING OPTIMIZATION: SOLVERS FOR PROBLEMS UNDER CONSTRAINTS A PREPRINT Yuan Qilong ∗a, Michal Pavelkab. a Singapore Institute of Technology. b Mathematical Institute, Faculty of Mathematics and Physics, Charles University ∗Corresponding Author Email: qilong.yuan@singaporetech.edu.sg December 23, 2025 ABSTRACT We study the Joint Routing-Assignment (JRA) problem in which items must be assigned one-to-one to placeholders while simultaneously determining a Hamiltonian cycle visiting all nodes exactly once. Extending previous exact MIP solvers with Gurobi and cutting-plane subtour elimination, we develop a solver tailored for practical packaging-planning scenarios with richer constraints. These include multiple placeholder options, time-frame restrictions, and multi-class item packaging. Experiments on 46 mobile manipulation datasets demonstrate that the proposed MIP approach achieves global optimality with stable and low computation times, significantly outperforming the shaking-based exact solver by up to an orders of magnitude. Compared to greedy baselines, the MIP solutions achieve consistent optimal distances with an average deviation of 14% for simple heuristics, confirming both efficiency and solution quality. The results highlight the practical applicability of MIP-based JRA optimization for robotic packaging, motion planning, and complex logistics . GitHub repository:https://github.com/QL-YUAN/JAR_Time_Frame_And_Additional_Phs_ Constraints.git Keywords Joint Routing-Assignment, Time-Window Constraints, Mixed-Integer Programming, Combinatorial Optimization, Multi-Type Object Packaging Planning 1 Introduction In this paper, we extend the JRA framework by developing a Joint Assignment-Routing (JAR) solver that supports: • Time-frame constraints: items and placeholders can be active only during certain intervals, and items may choose among multiple feasible time windows. • Multiple item and placeholder types: enabling packaging scenarios where different classes of objects must be grouped or allocated to different placeholder types. • Additional Placeholders: items have more placeholders options for a more optimal allocation arrangement. To handle these added complexities, we propose solver variants that build upon MIP solvers, adapted to incorporate additional placeholder, time-window and type constraints [1, 2, 3]. We demonstrate through experiments on synthetic and realistic problem instances that our solver is both efficient (significantly faster than baseline methods) and effective (achieving global optima under complex constraints). We present a detailed empirical evaluation, showing how time- frame conditional methods can yield large runtime improvements (e.g., up to one orders of magnitude faster) while maintaining solution quality, and discuss implications for robotics, motion planning, and logistics. By providing both a benchmark dataset and solver implementations, we hope this work serves as a foundation for future research on scalable algorithms for joint routing-assignment and packaging planning under realistic constraints. arXiv:2512.18618v1 [cs.AI] 21 Dec 2025 Assignment-Routing Optimization: Solvers for Problems under Constraints A PREPRINT 2 Mathematical Formulation 2.1 Joint Routing-Assignment (JRA) Problem General Formulation Start with n items to be allocated to np placeholders. This section discusses the one-to-one item-placeholder allocation problem. Then, np = n. Let I = {0, . . . , n −1} be the set of items, P = {n, . . . , 2n −1} the set of placeholders, and cij the cost of traversing edge (i, j). Denote V = I ∪P and E = {(i, j) ∈V × V : i ̸= j}. We define binary decision variables: xij = 1 if edge (i, j) is in the cycle 0 otherwise , aip = 1 if item i is assigned to placeholder p 0 otherwise . The objective is to minimize total traversal cost: min X (i,j)∈E cijxij. Constraints are as follows: g1. Degree: Each node has exactly one input and one output incident edges: X j∈V,j̸=i xij = 1, X j∈V,j̸=i xji = 1, ∀i ∈V g2. Assignment: Each item and placeholder is assigned twice, whereby links are always from item to placeholder: X p∈P aip = 2, ∀i ∈I, X i∈I aip = 2, ∀p ∈P g3. Valid edges: Only item-to-placeholder edges are allowed, and edges can only exist if the assignment exists: xij = 0 if both i, j ∈I or i, j ∈P, xip ≤aip, xpi ≤aip, ∀i ∈I, p ∈P g4. Fixed pair: In case of having starting and goal points, we specify starting position to be the last placeholder, goal point to be the last item, and enforce the last item to be assigned to the last placeholder as following constraints. an−1,2n−1 = 1 xn−1,2n−1 = 1, x2n−1,n−1 = 0 g5. Subtour elimination: Disconnected cycles are prohibited by a cutting-plane method [4, 5]: X i,j∈S,i̸=j xij ≤|S| −1, ∀subtours S ⊂V This is enforced dynamically using subtour elimination callback function in Gurobi. Another alternative way of Sub-tour Elimination is through applying as introduced in next sub-section. Beyond standard form of JRA proble

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