Mixed-integer linear programming models for 3D irregular strip packing problems

Improving neural-based heuristics for 2D strip packing through local search in heuristic space

In practical CNC machining for aerospace and automotive components, machine availability is constrained by tool changes due to tool wear, while each machine's continuous processing time is limited by tool life. Furthermore, total real-time power consumption must not exceed the predetermined peak power threshold at any time within the planning horizon. This study investigates an identical parallel machine scheduling problem with tool changes under the peak power consumption constraint, aiming to minimise the makespan. We propose two Mixed Integer Linear Programming (MILP) models: one based on the traditional scheduling procedure and the other inspired by the two-dimensional Strip Packing (SP) problem, and develop a Greedy heuristic to efficiently solve large-scale instances. Experimental results show that both models yield optimal solutions only for small-scale instances. The proposed Greedy heuristic is capable of obtaining near-optimal solutions for small-scale instances and outperforms three adapted benchmark algorithms (LJM-LMM, MSA and HGA) in both efficiency and solution quality for large-scale problems. Finally, we derive managerial insights to support industrial decision-making, offering a systematic approach to simultaneously meet peak power requirements, optimise tool utilisation, and enhance production effectiveness in real-world manufacturing settings.

A dynamic clustering search algorithm for the rectangular strip packing problem

To evaluate the surgical outcomes of maxillary medication-related osteonecrosis of the jaw (MRONJ) at different disease stages and to analyze the comparative efficacy of different surgical techniques on the prognosis of stage Ⅲ patients. A detailed retrospective analysis was conducted on the clinical data of 136 patients with maxillary MRONJ who underwent surgical treatment in the Department of Oral and Maxillofacial Surgery of Peking University School and Hospital of Stomatology from April 2014 to February 2024. All patients were rigorously classified according to the 2022 American Association of Oral and Maxillofacial Surgeons (AAOMS) staging criteria: Stage Ⅰ (n=8), stage Ⅱ (n=30), and stage Ⅲ (n=98). The surgical interventions included local lesion resection with primary direct closure, buccal fat pad packing, and iodoform gauze packing. The patients were systematically followed up for a period of 1 year postoperatively to comprehensively assess several key outcome measures: Complete mucosal healing, resolution of pain, effective infection control, and radiological improvement of maxillary sinus inflammation based on serial computed tomography scans. Statistical analysis was performed using SPSS version 20.0. Continuous variables were expressed as mean±standard deviation and compared using the t-test, while categorical variables were expressed as numbers and percentages and compared using the χ2 test or Fisher' s exact test as appropriate. A P-value < 0.05 was considered statistically significant for all analyses. The overall short-term (3 months) cure rate was 91.2% (124/136), which improved to a long-term (1 year) cure rate of 94.9% (129/136). A stage-stratified analysis revealed excellent long-term cure rates: 100.0% (8/8) for stage Ⅰ, 96.7% (29/30) for stage Ⅱ, and 93.9% (92/98) for stage Ⅲ, with no statistically significant difference in outcomes across the different stages (P=0.611). Among the 98 stage Ⅲ patients, 34 were treated with buccal fat pad transfer (BFPT group) and 64 with iodine strip packing (ISP group), with no significant differences in baseline demographic or clinical characteristics between the two groups, ensuring comparability. The BFPT group demonstrated a statistically significant superior performance in achieving oroantral fistula closure both at the short-term (79.4% vs. 23.4%, P < 0.001) and long-term (85.3% vs. 54.7%, P=0.002) follow-up assessments. In contrast, the ISP group showed a markedly greater degree of improvement in maxillary sinus inflammation, as quantified by a standardized radiographic scoring system, with significantly greater reductions in inflammation scores at both the 3-month (P=0.029) and 12-month (P=0.014) follow-up intervals. Surgical management of maxillary MRONJ results in high rates of success with a favorable complication profile. For advanced (stage Ⅲ) disease, the choice of surgical technique entails a strategic trade-off: The buccal fat pad procedure is more conducive to achieving reliable soft tissue closure and oroantral fistula resolution, whereas iodoform gauze packing provides superior management and resolution of concomitant maxillary sinusitis. Consequently, the selection of surgical technique should be individualized, based on a careful consideration of the patient's specific anatomical defect, the extent of sinus involvement, and their overall clinical condition.

In the strip packing problem, the objective is to pack a set of two-dimensional items into a strip of fixed width such that the total height of the packing is minimized. The current state-of-the-art exact approach for the problem uses a decomposition framework in which the main problem (MP) fixes the item abscissas and the strip height, whereas the subproblem (SP) determines whether a set of item ordinates resulting in a feasible packing exists. Even though this decomposition framework has already been used several times in the literature, implementation details were often obfuscated, limiting the outreach of the approach. We address this issue by thoroughly describing and testing various builds for this framework, investigating important features such as the way to forbid an infeasible solution in the MP (e.g., by rejecting them or through a no-good cut) and the techniques used to solve the MP and the SP. One of our findings is that a minor implementation tweak such as changing the random seed between two MP iterations can bring the same level of improvement as a more involved feature such as strengthening the no-good cuts. From our extensive experiments, we identify two versions of the framework that produce complementary results: one where the main problem is solved with integer linear programming and the other where it is solved with constraint programming. We then train a reinforcement learning agent to find the best hybridization of these two algorithms and show that the resulting approach obtains state-of-the-art results on benchmark instances.

Transformer-based placement heuristic for online 2D strip packing problem

The next generation of radar systems will include advanced digital front-end technology in the apertures allowing for spatially subdividing radar tasks over the array, the so-called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">split-aperture phased array</i> (SAPA) concept. The goal of this article is to introduce radar resource management for the SAPA concept and to demonstrate the added benefit of the SAPA concept for active tracking tasks. To do so, the radar resource management problem is formulated and solved by employing the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">quality of service based resource allocation model</i> framework. As active tracking tasks may be scheduled simultaneously, the resource allocation of tasks becomes dependent on the other tasks. The solution to the resource allocation problem is obtained by introducing the adaptive fast traversal algorithm combined with a 3-D strip packing algorithm to handle task dependencies. It will be demonstrated by a simulation example that the SAPA concept can significantly increase the number of active tracks of a multifunction radar system compared to scheduling tasks sequentially.

In the Strip Packing problem (SP), we are given a vertical half-strip \([0,W]\times[0,\infty)\) and a set of \(n\) axis-aligned rectangles of width at most \(W\) . The goal is to find a non-overlapping packing of all rectangles into the strip such that the height of the packing is minimized. A well-studied and frequently used practical constraint is to allow only those packings that are guillotine separable, i.e., every rectangle in the packing can be obtained by recursively applying a sequence of edge-to-edge axis-parallel cuts (guillotine cuts) that do not intersect any item of the solution. In this paper, we study approximation algorithms for the Guillotine Strip Packing problem (GSP), i.e., the Strip Packing problem where we require additionally that the packing needs to be guillotine separable. This problem generalizes the classical Bin Packing problem and also makespan minimization on identical machines, and thus it is already strongly \(\mathsf{NP}\) -hard. Moreover, due to a reduction from the Partition problem, it is \(\mathsf{NP}\) -hard to obtain a polynomial-time \((3/2-\varepsilon)\) -approximation algorithm for GSP for any \(\varepsilon&gt;0\) (exactly as Strip Packing ). We provide a matching polynomial time \((3/2+\varepsilon)\) -approximation algorithm for GSP. Furthermore, we present a pseudo-polynomial time \((1+\varepsilon)\) -approximation algorithm for GSP. This is surprising as it is \(\mathsf{NP}\) -hard to obtain a \((5/4-\varepsilon)\) -approximation algorithm for (general) Strip Packing in pseudo-polynomial time. Thus, our results essentially settle the approximability of GSP for both the polynomial and the pseudo-polynomial settings.

This paper examines the value-based reinforcement learning method applied to the optimization of the rectangular strip packing problem with three different approaches to define the state and the action. The episode in the reinforcement learning is defined as a round of placement of all the given pieces. We analyze the drawbacks of two previously designed approaches and propose that the state is defined by the stage along with the selected piece. We also record the fitness value of the placement during the packing and design a fitness-based reward. The three methods are evaluated on a group of random packing problems in terms of time consumption by searching for the particular state, memory consumption by recording the past state, and the efficiency of packing optimization. The results show that the proposed reinforcement learning with fitness-based reward delivers a good comprehensive performance. The proposed method is also tested on a few well-known benchmark problems, and the results indicate that the proposed method could be an effective tool. We discuss the similarities and differences between the reinforcement learning method and the local search method.

A multi-objective evolutionary algorithm based on incremental support vector regression for the irregular strip packing problem

Using helical polyhedron for online irregular strip packing problem with free rotations

High Multiplicity Strip Packing with Three Rectangle Types

An exact approach for the two-dimensional strip packing problem with defects

We address the well-known NP-hard problem of packing rectangular items into a strip, a problem of significant importance in electronics (e.g., packing components on printed circuit boards and macro-cell placement in Very-Large- Scale Integration design) and telecommunications (e.g., allocating data packets over transmission channels). Traditional heuristics and metaheuristics struggle with generalization, efficiency, and adaptability, as they rely on predefined rules or require extensive computational effort for each new problem instance. In this paper, we propose a neural-driven constructive heuristic that leverages a lightware neural network trained via black-box optimization to dynamically evaluate item placement decisions. Instead of relying on static heuristic rules, our approach adapts to the characteristics of each problem instance, enabling more efficient and effective packing strategies. To train the neural network, we employ the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), a state-ofthe- art derivative-free optimization method. Our method learns decision policies by optimizing fill factor improvements over a large dataset of problem instances. Unlike conventional heuristics, our approach dynamically adapts placement decisions based on a broad set of features describing the current partial solution and remaining items. Through extensive computational experiments, we compare our method against well-known strip packing heuristics, including MaxRects and Skyline-based algorithms. The results demonstrate that our approach consistently outperforms the best traditional heuristics, achieving up to 6.74 percentage points of improvement in packing efficiency. Furthermore, our method improves 87.87% of tested instances. Our study highlights the potential of machine learning-driven heuristics in combinatorial optimization and opens avenues for further research into adaptive decision-making strategies in packing and scheduling problems

A backtracking heuristic algorithm (BHA) was proposed for a two-dimensional rectangular strip packing problem with rotations and without guillotine cutting, which has many applications. An improved fitness strategy was used to select the fittest rectangle to be packed on a strip with a certain height. Next, a backtracking constructive heuristic was repeatedly used at a higher height until all the rectangles were packed. A multi-start improvement procedure then found the best solution by taking a different rectangle as the first rectangle, whereas the sequence of the other rectangles remained unchanged. Finally, in order to further expand the scope of the solution, a simple randomized local search procedure based on random sequences of rectangles with the first rectangle unchanged was applied to search for the optimal solution. BHA has only two parameters; it is simple and effective. Computational results on benchmark problems (zero-waste instances and non-zero-waste instances) with different scales (from 10 to 75,032 rectangles) indicate the following: (1) though it is non-deterministic, the difference between the results after each running is tiny and (2) the proposed algorithm outperforms most of the other algorithms under comparison on the whole, especially for large-scale instances with more than 1000 rectangles, which is further verified by statistical analysis and greatly meaningful in mass industrial production like metal cutting.

We approach an open dimension problem, in specific, a two-dimensional strip packing problem variation found in sheet metal laser cutting, where rectangular items must be cut from a metal sheet, aiming to increase the packing layout added value. Therefore, this research objective is to analyze the packing layout added value with raw material reuse and practical constraints found in real-life laser cutting operations. The Best Fit Decreasing Height heuristic was modified to reuse raw material and calculate the packing layout added value, being compared with three construction heuristics using a set of literature and generated instances. We show the modified best fit decreasing height heuristic obtained better results when compared to the selected heuristics, with a high sheet metal utilization by the original instance rectangles and efficient raw material reuse. Thus, for sheet metal laser cutting practical operations, the modified best fit decreasing height heuristic is suitable for generating good packing layouts, resulting in industrial benefits including cost savings, increased productivity, greater competitiveness, and sustainability. Approaching raw material reuse increased the packing layout added value in most solutions found, and should be considered in real-life laser cutting operations. However, prioritizing only raw material reuse is not ideal, since a high number of additional rectangles can cause manufacturing wastes including overproduction, stock, and extra processing.

Transformer-Based Placement Heuristic for Online 2d Strip Packing Problem

In this study, we theoretically compare integer programming models for the two-dimensional two-staged knapsack problem. Including the well-known level packing model, we introduce two pattern-based models called the strip packing model and the staged pattern model derived from integer programming models for the two-dimensional two-staged cutting stock problem. We show that the level packing model provides weaker linear programming (LP) relaxation bounds than pattern-based models. Furthermore, we also present upper bounds on the LP-relaxation bound of the level packing model, which can be obtained from the LP-relaxation bounds of the pattern-based models.

In this study, the 1.5-dimensional cutting stock problem with technical constraints is considered. In the literature, this problem is also defined as a strip packing or open dimension problem. When given a strip of infinite length and bounded width, the problem is to define a packing of rectangular objects into a strip that minimizes its final length. Technical constraints, such as the order type and the number of strips, are indispensable in real life; however, they are often neglected in the literature because they make the problem difficult to solve. Only one study was reached in the literature that took into account technical constraints, but in that mentioned study, only a mathematical model was proposed for the problem. In this context, our aim is to solve the problem with a more effective approach. The research question in this study is the usability of the column generation technique to solve the 1.5-dimensional cutting stock problem. In this study, the column generation approach was proposed for the first time for the considered problem. To demonstrate the performance of the proposed solution method, randomly generated test problems were solved with GAMS/Cplex. As we report the results, proposed column generation approach (CG) reaches very close (such as %1 and %2 error) solutions to integrated mathematical model (IM) for small sized problems in a second. On the other hand, while CG solved all the problems in a reasonable time, IM could not produce a feasible solution to some problems. Numerical experiments showed that the column generation algorithm outperforms the integrated mathematical model for the problem.