MIP-Based Joint Routing-Assignment Optimization

• MIP-based JRA optimization is an exact combinatorial approach that jointly assigns items and computes Hamiltonian cycles under complex routing and assignment constraints.
• It integrates dynamic subtour cuts, MTZ constraints, and advanced solver configurations to ensure robust and efficient global optimization.
• The methodology scales to large problems through techniques like Lagrangian relaxation and partial path reconstruction, proving effective in robotics and logistics.

Mixed-Integer-Programming (MIP)–based Joint Routing-Assignment (JRA) optimization is an exact combinatorial approach for solving problems where both assignment of items to placeholders and a closed route (Hamiltonian cycle) visiting all entities must be determined under complex constraints. MIP-based JRA methods have proven highly effective across robotic packaging, motion planning, vehicle routing, and logistics, accommodating extensions such as time frames, multi-class assignments, and capacity restrictions. This article presents the mathematical underpinnings, solver enhancements, empirical findings, and broader implications of MIP-Based JRA optimization, with technical focus on the models and algorithms developed by contemporary research groups (Qilong et al., 21 Dec 2025, Yuan, 7 Nov 2025, Bragin et al., 2022).

1. Mathematical Foundations of MIP-Based JRA

The JRA problem jointly addresses (i) assignment of $n$ items ($I = \{0, \ldots, n-1\}$) to $n_p$ placeholders ($P = \{n, \ldots, n + n_p - 1\}$) and (ii) computation of a Hamiltonian cycle over the union $V = I \cup P$, so that each node is visited exactly once. The problem is formalized as a Mixed-Integer Program:

Variables:

• Routing:

$$x_{ij} = \begin{cases} 1, & \text{if arc }(i, j) \text{ is in the cycle} \ 0, & \text{otherwise} \end{cases} \quad \forall i \neq j,\, i, j \in V$$

• Assignment (if $n_p \geq n$):

$$a_{ip} = \begin{cases} 1, & \text{if } i \in I \text{ assigned to } p \in P \ 0, & \text{otherwise} \end{cases}$$

• Placeholder selection (if $n_p > n$): $c_p \in \{0, 1\}$ for $p \in P$.
• (For time-frame variants) Sequencing: $t_i \in \{1, \ldots, n\}$ and $y_{ij} \in \{0,1\}$.

Objective:

$$\min \sum_{i \in V} \sum_{\substack{j \in V \ j \neq i}} c_{ij} x_{ij}$$

Constraints:

• Degree: $$\sum_{j \neq i} x_{ij} = 1;\;\; \sum_{j \neq i} x_{ji} = 1 \quad \forall i \in V$$
• Assignment:

For $n_p = n$:

$$\sum_{p \in P} a_{ip} = 1\ \forall i \in I;\;\; \sum_{i \in I} a_{ip} = 1\ \forall p \in P$$

For $n_p > n$:

$$\sum_{p \in P} c_p = n;\;\; \sum_{p \in P} a_{ip} = 1\ \forall i \in I;\;\; \sum_{i \in I} a_{ip} = c_p\ \forall p \in P;\;\; a_{ip} \leq c_p$$

• Link Routing to Assignment:

$x_{ij} = 0$ if $i, j \in I$ or $i, j \in P$; $x_{ip} \leq a_{ip}$, $x_{pi} \leq a_{ip}\ \forall i \in I, p \in P$

• Start/Stop: (optional fixed endpoints): $a_{n-1, 2n-1} = 1$, $x_{n-1, 2n-1} = 1$, $x_{2n-1, n-1} = 0$
• Subtour Elimination: For all $S \subset V$, $2 \leq |S| < |V|$:

$\sum_{i, j \in S,\, i \neq j} x_{ij} \leq |S| - 1$

Time-frame constraints require sequencing variables $y_{ij}$, $t_i$, with additional permutation, precedence, and linking constraints, including the Miller-Tucker-Zemlin (MTZ) subtour elimination on $y$ (Qilong et al., 21 Dec 2025).

2. MIP Model Extensions and Generalization

Several modeling extensions address practical requirements:

• Multiple Placeholder Options: For $n_p > n$, selection variables $c_p$ ensure exactly $n$ placeholders are used, allowing flexibility in packaging scenarios.
• Multi-Type Assignment: Partition $I$ and $P$ by type, with assignment and routing enforced only within types.
• Time-Frame Decomposition: Temporal grouping of items $G_1, \ldots, G_h$ with enforced within/frame permutations and inter-frame precedence.
• Vehicle Routing and Capacity: By adding capacity or time-window constraints to the routing variables $x_{ij}$, the MIP framework generalizes to capacitated VRP and time-dependent delivery (Qilong et al., 21 Dec 2025).

A key feature is the linkage between assignment and routing decisions via logical and flow constraints. Multi-type and time-frame variants illustrate the flexibility of the MIP approach for broader logistics and automation scenarios.

3. Solver Enhancements: Dynamic Cuts and Gurobi Integration

State-of-the-art scalability is achieved through advanced solver configuration:

• Subtour Cutting-Plane Callbacks: Subtours are dynamically detected by inspecting the subgraph induced by $x_{ij}=1$; for each non-trivial subtour $S$, the cut $\sum_{i,j\in S,\,i\neq j} x_{ij} \leq |S|-1$ is added on-the-fly (lazy constraint).
• Miller-Tucker-Zemlin (MTZ) Cuts: For time-frame instances, pre-added MTZ constraints on $y$ enforce global sequencing and eliminate subtours without reliance on $x$-variable cuts.
• Parameter Tuning: Gurobi is configured for balanced heuristic/cut behavior (MIPFocus=1), aggressive cut generation (Cuts=2), full parallelism (Threads=22), and large node file threshold (NodefileStart=0.5).
• Trigger Logic: The subtour callback runs only on integer incumbents to minimize overhead while maintaining guarantee of connectivity.
• Integration with Built-in Cuts: Solver-native cover, flow, and clique cuts operate jointly with custom subtour cuts, yielding significant reductions in branch-and-bound tree size (Qilong et al., 21 Dec 2025).

This configuration enables stable, near-constant solve times and robust global optimality verification for moderate instance sizes.

4. Empirical Evaluation and Comparative Performance

Empirical analysis on realistic robotic manipulation datasets provides the following core results (Qilong et al., 21 Dec 2025):

Method	Avg Time (s)	Std Dev (s)	Path Length (m)	Greedy Error (%)
MIP Solver	2.96	1.61	10.53	14.3
Shaking	34.46	71.8	10.53	–
Greedy	≪1	–	≈12.0	14.3

• MIP achieves global optima, matching the path length of the exact shaking-based solver but with order-of-magnitude lower and far more stable computation times.
• Greedy heuristics are instantaneous but consistently yield ≈14% greater path cost.
• Variance in shaking-based solver times is substantially higher due to enumeration and combinatorial explosion.
• The combination of MTZ and dynamic subtour cuts ensures effective pruning and accelerated convergence.

These outcomes establish the practical efficacy of MIP-based JRA, especially for moderate-scale problems where global optimality and constraint handling are essential (Qilong et al., 21 Dec 2025).

5. Decomposition and Large-Scale Methods: Lagrangian and Heuristics

For large instances or generic JRA forms with multi-resource constraints, direct MIP approaches may become inefficient. Surrogate "Level-Based" Lagrangian Relaxation (SLBLR) (Bragin et al., 2022) and the Partial Path Reconstruction (PPR) framework (Yuan, 7 Nov 2025) provide scalable alternatives:

• Lagrangian Relaxation: Coupling constraints are dualized, yielding a master problem in Lagrange multipliers $\lambda$, iteratively updated via Polyak-like stepsizes. The auxiliary feasibility problem detects divergence, adaptively tightens overestimates, and ensures geometric (linear) convergence in dual variables for MILP-JRA with resource allocation structure (Bragin et al., 2022).
• PPR and SLPPR (PJAR): Large JRA instances (e.g. $n=500$ or $n=1000$) are efficiently improved by extracting key nodes or spatial regions, fixing most of the cycle, and solving small MIPs on boundary nodes. Large-$\alpha$ constraints allow k-opt-like local modifications while anchoring most of the incumbent solution, providing sub-0.1% deviation from the true optimum at dramatically reduced compute time (Yuan, 7 Nov 2025).
• Polishing Loops: Iterated application of spatially localized PPR refinements (SLPPR) incrementally drives the solution toward optimality without revisiting the full $2n$-node MIP.

These methods extend MIP efficacy well into large-scale settings previously intractable for standard solvers alone, supporting near-optimal JRA tour computation (Yuan, 7 Nov 2025, Bragin et al., 2022).

6. Applications and Generalization to Logistics, Robotics, and Routing

MIP-based JRA is broadly applicable to:

• Robotic packaging, palletization, mobile manipulation—optimizing pick/place operations under assignment, sequencing, and physical arrangement constraints.
• Time-windowed warehousing, last-mile delivery, vehicle-route assignment—via time-frame or capacity extensions.
• Multi-class and multi-type logistics—through type-partitioned variables and assignment restrictions.
• Task-and-motion planning—each optimized JRA cycle can serve as an input to continuous trajectory planning modules (Qilong et al., 21 Dec 2025).

By leveraging exact combinatorial optimization (and integrating decomposition or partial-reconstruction methods for scale), the MIP-based JRA paradigm robustly addresses joint assignment-routing problems fundamental to modern automation and logistics systems.