QMC-VSBPP: Quadratic Multi-Constraint Bin Packing

• The paper introduces QMC-VSBPP, a bin packing generalization that incorporates multi-dimensional capacities, heterogeneous bin types, and quadratic separation penalties.
• It presents both a quadratic integer programming formulation and its linearized MILP version, enabling the use of exact solvers like Gurobi and a specialized RKO-ACO metaheuristic.
• Experimental results demonstrate that while exact methods face scalability issues on large instances, RKO-ACO consistently finds superior or improved solutions across benchmark cases.

The Quadratic Multiple Constraints Variable-Sized Bin Packing Problem (QMC-VSBPP) is a combinatorial optimization problem that generalizes the classical bin packing paradigm by incorporating multiple capacity dimensions per bin, heterogeneous bin types, and quadratic interaction costs between item pairs. The problem involves assigning a set of items to bins of various types with distinct capacities across several resource dimensions, while minimizing total bin usage costs and penalties arising from separating specific item pairs. This formalism is motivated by practical settings in logistics, cloud computing, and resource allocation where diverse constraints and inter-item dependencies are inherent (Santos et al., 15 Nov 2025).

1. Formal Problem Definition

QMC-VSBPP considers a set of items $I = \{1,\ldots,n\}$ and a set of bin types $M = \{1,\ldots,m\}$, each characterized by resource capacities $Q_{mr}$ across dimensions $D = \{1,\ldots,d\}$, and fixed costs $c_m$. The index set of potential bins is $B = \{1,\ldots,n\}$ (one per item in the worst case). Each item $i$ has weight $w_{ir}$ on dimension $r\in D$, and the penalty $c_{is}$ is incurred if items $i$ and $s$ are placed in different bins.

Two binary decision variables govern assignments: $x_{ij} = 1$ if item $i$ is assigned to bin $j$, and $y_{jm} = 1$ if bin $j$ is opened as type $m$. The quadratic interaction cost manifests as $c_{is} x_{ij}(1 - x_{sj})$ in the objective, penalizing separation of item pairs.

The full quadratic integer programming formulation is:

$$\min \sum_{j \in B} \sum_{m \in M} c_m y_{jm} + \sum_{j \in B} \sum_{i \in I} \sum_{s \in I} c_{is}[x_{ij}(1-x_{sj})]$$

subject to: $\sum_{j \in B} x_{ij} = 1 \quad \forall i \in I$

$\sum_{m \in M} y_{jm} \leq 1 \quad \forall j \in B$

$$\sum_{i \in I} w_{ir} x_{ij} \leq \sum_{m \in M} Q_{mr} y_{jm} \quad \forall j \in B, r \in D$$

$x_{ij}, y_{jm} \in \{0,1\}$

The objective combines fixed bin usage costs and quadratic separation penalties. Each item is assigned to exactly one bin, bins use at most one type, and multi-dimensional resource constraints are enforced. This formulation encapsulates the classical bin packing problem as a special case.

2. Quadratic-to-Linear Reformulation

Quadratic terms in the objective complicate direct application of standard MIP solvers. A linearized model introduces auxiliary binary variables $z_{ijs}$ to represent $x_{ij}(1-x_{sj})$. The revised problem formulation is:

$$\min \sum_{j\in B} \sum_{m\in M} c_m y_{jm} + \sum_{j\in B} \sum_{i\in I} \sum_{s\in I} c_{is} z_{ijs}$$

Subject to the original packing (assignment and capacity) constraints, plus: $z_{ijs} \leq x_{ij}$

$z_{ijs} \leq 1 - x_{sj}$

$z_{ijs} \geq x_{ij} - x_{sj}$

$z_{ijs} \in \{0,1\}$

This linearization yields a mixed-integer linear program (MILP), enabling the use of exact branch-and-bound solvers such as Gurobi. The auxiliary variable construction ensures correct encoding of interaction cost logic.

3. Exact and Heuristic Solution Approaches

Two principal solution strategies have been reported for QMC-VSBPP: (1) exact (quadratic and linearized) MIP models solved with Gurobi, and (2) a metaheuristic grounded in Random-Key Optimization (RKO) and Ant Colony Optimization (ACO), termed RKO-ACO (Santos et al., 15 Nov 2025).

3.1. Exact Solvers (Gurobi)

Both the original quadratic and linearized MIP models were solved with Gurobi 12.0.2, typically with a 3600 s time limit per instance. Empirical results over 96 benchmark instances indicated:

• The original model often yielded better feasible (upper bound) solutions in 62% of cases, especially for larger $n$.
• The linearized formulation generated strictly tighter lower bounds in 67% of instances and found proven optimal solutions for several small cases ($n=25$).
• For large instances ($n=200$), the optimality gap remained large (≈90% on average), indicating severe computational hardness for both models.

3.2. RKO-ACO Metaheuristic

The RKO-ACO method encodes candidate solutions as vectors $\chi \in [0,1)^{n+3}$. The first $n$ components determine item ordering for assignment, with the remainder encoding high-level solution strategies (allocation heuristics, number of pre-opened bins, local search parameters). A decoder constructs feasible solutions from this representation.

RKO-ACO maintains an archive of elite solutions, each assigned a weight $\omega_l$ for selection; new "ants" are sampled from the truncated normal distributions parametrized by archive members, with adaptive exploration scale. Key metaheuristic parameters—including archive size, ant count, exploration (via $\xi$), and selection pressure ($q$)—are adaptively tuned by online Q-learning. Local post-processing includes bin-type replacement, bin merging, and item relocation, further improving solution quality.

4. Experimental Evaluation and Results

Experiments utilized 96 benchmark instances from Meng et al. (2022), systematically varying items ($n$), bin types ($m$), dimensions ($d$), and cost functions (B1–B4: linear, convex, concave, mixed).

4.1. Metrics

Primary performance metrics were:

• Lower-bound gap: $(\mbox{BestSolution} - LB)/\mbox{BestSolution} \cdot 100\%$
• Best Relative Percentage Deviation (BRPD) to best-known solution (BKS)
• Average Relative Percentage Deviation (ARPD) over 30 runs

4.2. Computational Findings

Method	Strengths	Weaknesses
Quadratic model	Better feasible upper bounds quickly	Loose lower bounds
Linearized model	Tighter lower bounds	Poorer feasible bounds
RKO-ACO	Best or new best solutions, scalable	No provable optimality

• Linearized model yielded tighter lower bounds in 67% of instances; for small $n$ some cases were solved to proven optimality.
• Both MIP models retained large average optimality gaps ($\sim$90%) for $n=200$ after 1 hour.
• RKO-ACO matched or improved BKS in all 96 instances: for $n=50$ and $n=100$ all BKS were improved; ARPD $\leq 0$ in 93% of all runs. RKO-ACO exceeded the performance of Meng et al.'s VNS by factors of 2–7 in speed for moderate $n$.
• Statistical tests (ANOVA, Wilcoxon) indicated significant superiority of RKO-ACO over VNS ($p<10^{-17}$) and over Gurobi ($p<10^{-14}$).
• Multi-thread scaling was near-linear up to 8 threads for moderate instance sizes.
• For large $n$, lower bounds remain loose for all methods, reflecting inherent complexity.

5. Algorithmic Innovations: RKO-ACO Metaheuristic

The RKO-ACO approach integrates random-key encoding with a continuous ACO search and online Q-learning for hyperparameter control:

• Each solution vector encodes both item assignment order and bin/strategy meta-parameters.
• Archive-based elitist sampling where new solutions (ants) are generated by sampling each variable from a truncated normal with mean and SD determined by archive leaders and dispersion parameter $\xi$.
• Adaptive Q-learning controls archive size, number of ants, selection pressure $q$, and exploration $\xi$ by optimizing over a discrete set of parameter choices in response to observed improvement/reward dynamics.
• Embedded local search improvements operate both at the bin-type (switching/merging) and item relocation levels based on cost reduction.

This architecture enables RKO-ACO to scale efficiently, maintain solution diversity, and robustly escape local minima in the expanded QMC-VSBPP search space.

6. Key Insights and Strategy Selection

QMC-VSBPP is notably resistant to exact resolution for large-scale instances due to the combination of multi-dimensional bin packing, bin heterogeneity, and quadratic separation penalties. Empirical evidence indicates:

• For small to medium-size instances, the linearized MIP furnishes strong provable lower bounds for optimality gap reporting, sometimes yielding global optima.
• For large instances or when rapid high-quality feasible solutions are required, RKO-ACO supplies best-known or empirically optimal assignments within seconds to minutes, scaling well with available hardware.
• The original quadratic model is most effective for quickly generating high-quality feasible upper bound solutions in smaller instances; however, its lower bounds are looser than the linearized version.
• Both exact approaches struggle to close the optimality gap as instance size grows, a typical symptom of quadratic combinatorial explosion.

7. Benchmarking, Limitations, and Reference Values

Current state-of-the-art methods and benchmarks for QMC-VSBPP are as follows:

• For benchmarking and reporting, use the linearized Gurobi MIP to establish lower bounds.
• For obtaining feasible solutions under time constraints or for large-scale problems, RKO-ACO is recommended owing to its superior practical performance and scalability.
• All key reference values (lower and upper bounds, benchmark protocol) are provided in the cited work and supporting repository (Santos et al., 15 Nov 2025).

These results establish new upper and lower bounds for QMC-VSBPP and set a baseline for future algorithmic development. Both approaches highlight the trade-off between provable optimality and empirically driven solution quality in complex quadratic packing problems.