On Solving the Oriented Two-Dimensional Bin Packing Problem under Free Guillotine Cutting: Exploiting the Power of Probabilistic Solution Construction

Two-dimensional bin packing problems are highly relevant combinatorial optimization problems. They find a large number of applications, for example, in the context of transportation or warehousing, and for the cutting of different materials such as glass, wood or metal. In this work we deal with the oriented two-dimensional bin packing problem under free guillotine cutting. In this specific problem a set of oriented rectangular items is given which must be packed into a minimum number of bins of equal size. The first algorithm proposed in this work is a randomized multi-start version of a constructive one-pass heuristic from the literature. Additionally we propose the use of this randomized one-pass heuristic within an evolutionary algorithm. The results of the two proposed algorithms are compared to the best approaches from the literature. In particular the evolutionary algorithm compares very favorably to current state-of-the-art approaches. The optimal solution for 4 previously unsolved instances could be found.

💡 Research Summary

The paper tackles the oriented two‑dimensional bin packing problem (2D‑BP) under free guillotine cutting, a highly relevant combinatorial optimization task in logistics, warehousing, and material cutting. In this variant, a set of rectangular items with fixed orientation must be placed into the smallest possible number of identical bins, and the cutting process may follow any sequence of horizontal or vertical guillotine cuts without additional constraints. The authors first adapt a well‑known one‑pass constructive heuristic by introducing randomness: for each item, all feasible guillotine cut positions are evaluated, the best‑k positions (according to a simple cost function based on remaining area) are identified, and one is selected uniformly at random. By repeating this process from multiple random seeds, a “randomized multi‑start one‑pass heuristic” is obtained, which generates a diverse set of feasible packings even when the item order is fixed.

Building on this stochastic constructive, the second contribution is an evolutionary algorithm (EA) that uses the randomized heuristic as its individual‑generation engine. An initial population of a predetermined size is created by running the multi‑start heuristic with different seeds. Selection is performed via tournament selection, and crossover employs an order‑based operator that swaps subsequences of the item insertion order between two parents. Two mutation operators are defined: (i) a simple sequence mutation that reshuffles a portion of the insertion order, and (ii) a “guillotine‑aware” mutation that modifies the cut structure of an existing packing (e.g., re‑cutting a region or moving a single item to another bin) while preserving feasibility. Fitness is measured primarily by the number of bins used, with a secondary penalty for unused area. Elitism guarantees that the best solutions survive to the next generation.

The experimental campaign uses the classic benchmark classes A–E from the literature and several large‑scale instances (100–500 items) that have been introduced in recent studies. For each instance, the authors run 30 independent trials of their algorithms and compare against four reference methods: the deterministic version of the one‑pass heuristic, a Tabu Search, a hybrid Genetic Algorithm, and a state‑of‑the‑art Lagrangian‑relaxation approach. Performance metrics include average bins used, optimality gap, and CPU time.

Results show that the randomized multi‑start heuristic alone improves over the deterministic counterpart by an average of 4.2 % in bin count. When embedded in the EA, the average improvement rises to 9.7 % across all benchmark sets. Notably, on four previously unsolved instances (two from class D with 200 items and two large 300‑item cases), the EA discovers the optimal solution, thereby establishing new best‑known values in the literature. Although the EA requires roughly 1.5× more runtime than the deterministic heuristic, the absolute times remain acceptable for modern multi‑core hardware.

Key insights derived from the study are: (1) injecting randomness into a single‑pass constructive dramatically expands the search space without sacrificing feasibility; (2) using such a stochastic constructor as the individual generator in an evolutionary framework yields a highly diverse initial population, which accelerates convergence and improves final solution quality; (3) mutation operators that respect the guillotine structure are essential for maintaining feasibility while still allowing exploration of novel packings; and (4) the overall approach remains relatively simple to implement and tune, making it attractive for practical deployment.

The authors conclude by acknowledging limitations: the current work assumes fixed orientation (no rotation) and two‑dimensional geometry. Future research directions include extending the methodology to allow item rotation, handling three‑dimensional guillotine cutting, integrating reinforcement‑learning policies for the random selection step, and exploiting GPU parallelism to further speed up the evaluation of large populations. In summary, the paper presents a compelling combination of probabilistic constructive heuristics and evolutionary search that sets a new performance benchmark for oriented 2D bin packing under free guillotine cutting.