Two-dimensional Strip Packing Problem Research Articles

Robust Optimization for the Two-Dimensional Strip-Packing Problem with Variable-Sized Bins

The two-dimensional strip-packing problem (2D-SPP) emerges as a notable variant of the cutting and packing (C&P) problem, aiming to optimize the arrangement of small rectangular items within unique strips with a fixed width and infinite height to minimize the usage of height. Despite extensive academic exploration, applying 2D-SPP solutions in industrial settings remains challenging. Two significant issues, often overlooked in academia yet frequently encountered in industrial contexts, are the uncertain demand for items, exacerbated by the bullwhip effect, and the need for diverse types of strips to cater to varying customer needs. Our paper addresses this academia–industry gap by proposing a robust optimization model for the uncertain 2D-SPP with variable-sized bins, aiming to manage the demand fluctuations within a box uncertainty set framework. Additionally, we employ the contiguous one-dimensional relaxation technique in conjunction with column generation to tighten the lower bound of the problem, thereby augmenting solution accuracy. Furthermore, we leverage the Karush–Kuhn–Tucker (KKT) condition to transform the model into a more tractable form, subsequently leading to an exact solution. Based on datasets from a real-life plastic-cutting company, comprehensive experiments validate the effectiveness and efficiency of our proposed relaxation method and algorithm, showcasing the potential for an improved industrial application of 2D-SPP solutions.

Impact of minimum distance constraints on sheet metal waste for plasma cutting.

We approached the two-dimensional rectangular strip packing problem (2D-SPP), where the main goal is to pack a given number of rectangles without any overlap to minimize the height of the strip. Real-life constraints must be considered when developing 2D-SPP algorithms to deliver solutions that will improve the cutting processes. In the 2D-SPP literature, a gap related to studies approaching constraints in real-life scenarios was identified. Therefore, the impact of real-life constraints found in the plasma cutting process in sheet metal waste was analyzed. A mathematical model from the literature was modified to obtain packing arrangements with plasma cutting constraints. The combination of size and number of rectangles, as well as strip width, was the main factor that affected the packing arrangement, limiting the allocation of rectangles and generating empty spaces. In summary, considering the sheet metal waste context, instances with smaller widths should be avoided in practical operations for high minimum distance constraint values, returning the worst packing arrangements. For low minimum distance constraint values, smaller width instances can be used in practical operations, as the packing arrangement is acceptable. Finally, this article can reduce material waste and enhance the cutting process in the sheet metal industry, by showing packing characteristics which lead to higher amounts of raw material waste.

Heuristics for the two-dimensional irregular bin packing problem with limited rotations

This paper studies a two-dimensional irregular bin packing problem with some specific degrees of rotations. Despite a large number of studies on the two-dimensional irregular strip packing problem, much less attention has been paid to the two-dimensional irregular bin packing problem (2DIBPP), and most of the current research on 2DIBPP is about free rotations. However, limited by the texture of the material, sometimes only some specific degrees of rotations are allowed. To pack a set of irregular pieces with some specific degrees into multi-stock sheets and maximize the utilization, we propose a heuristic algorithm that combines some pieces into one block for packing and uses residual spaces with reference lines to help select the next piece to be packed. In order to further increase the utilization of bins, we adopt fine-tuning, movement, filling, replacement and swap operations, as well as partial and full repack strategies. The computational results show that the proposed algorithm can obtain superior results on the instances with small number of pieces, relatively large number of types of pieces, large bins and more specific degrees of rotations.

Solving the Multi-Choice Two Dimensional Shelf Strip Packing Problem with Time Windows

In the tool coating field, scheduling of production lines requires solving an optimisation problem which we call the multi-choice two-dimensional shelf strip packing problem with time windows. A set of rectangular items needs to be packed in two stages: items are placed on shelves, which in turn are placed on one of several available strips. Crucially, the item's width depends on the chosen strip and each item is associated with a time window such that items can only be placed on the same shelf if their time windows overlap. In collaboration with an industrial partner, this real-world optimisation problem is tackled in this paper by both exact and heuristic methods. The exact method is an arc-flow-based integer linear programming formulation, solved with the commercial solver CPLEX. Experimental evaluation shows that this approach can solve instances to proven optimality with up to 20 different item sizes. Larger, more realistic instances are solved heuristically by an adaptive large neighbourhood search, using first fit and best fit decreasing approaches as repair heuristics. In this way, we obtain high-quality solutions with a remaining optimality gap below 3.3% for instances with up to 2000 different item sizes. The work reported is due to be incorporated into an end-to-end decision support system with the industrial partner.

A framework to select heuristics for the rectangular two-dimensional strip packing problem

Defining the algorithm capable of best fit the characteristics observed for a problem is a complex task in the context of combinatorial optimization problems. As a decision-making process, one of the most practical and useful ways to treat and solve algorithm selection problems is using supervised machine learning techniques to search for patterns between explanatory variables characteristics of the problem and the algorithms available to be selected. The present article deals with the development of a framework to fit classification models based on supervised machine learning techniques to select improvement heuristics for the rectangular 2D strip packing problem (2D-SPP) with 90-degrees rotation. Classification models were fitted to predict the best improvement heuristic for constructive heuristics bottom-left, bottom-left-fill, best-fit, best-fit with bottom-left-fill, fast-heuristic, and fast-heuristic with bottom-left-fill, using 19 features provided by problem characteristics. A total of 15,666 benchmark problem instances from the literature were used to represent the rectangular 2D-SPP characteristics variations found in real-world applications to train and test the fitted classification models. The framework proved to be consistent to predict improvement heuristics with acceptable accuracy, being able to be applied for the prediction of other cutting and packing problems algorithms.

Optimization of a Rural Portfolio Credit Granting System Using Improved Two-Dimensional Strip Packing Grouping Delay Problem

Rural preferential loans usually take the form of portfolio credits. From the perspective of public interest, the total delay time for obtaining loans is expected to be minimized. To use rural portfolio credits effectively, the two-dimensional strip packing grouping delay problem (2SPGDP) is improved to optimize the rural portfolio credit granting system. First, 2SPGDP is established by adding grouping constraints and the latest start time constraints to the two-dimensional strip packing problem, and the total delay is taken as the optimization objective. Second, based on the depth search reverse spanning tree (DSRST) and the insert spare space (ISS) method, the branch-and-bound reverse order insert algorithm (BB-RIA) is designed. Finally, the lag pruning operator (LPO) is designed to reduce lag. The improved model (2SPGDP) and BB-RIA-LPO algorithm are used to solve several classical two-dimensional strip packing problems and a specific rural portfolio credit case. Compared with the Bottom-Left and Branch and Bound Algorithm, our model and algorithm improve the success rate by 25% and reduce the total delay by 6%. The case of rural portfolio credit illustrates the operability and effectiveness of this method.

Cargo securing under multi-drop and axle weight constraints

This study emerges from a real-world cargo securing application to ensure safer road transportation and falls into the category of container loading problems with practical constraints. The literature has lacked efficient methods for the secure loading of items with non-identical dimensions, weights and rotations into containers while ensuring that the securing efforts necessary are minimal when also taking into account multi-drop and axle weight constraints. This paper puts forward a new way of ensuring cargo stability that gives rise to a novel combinatorial optimization problem, which is a generalization of the two-dimensional rectangular strip packing problem with orthogonal rotations. We formally demonstrate the intractability of the problem and provide a mixed integer programming formulation. The formulation is based on a discretization of the packing polyhedron, enabling us to model a range of complicated and practical constraints. A group of practical constraints tends to be large in number and makes it difficult to solve the formulation in a reasonable amount of time. In order to overcome this difficulty, we develop an exact algorithmic framework. This framework initially solves certain relaxations of the problem to obtain strong lower bounds before subsequently embedding those lower bounds into a branch-and-cut algorithm. The experimental study serves three purposes: (i) evaluating the performance of the algorithmic framework and the mathematical formulation to assess the merits of the two methods, (ii) identifying the characteristics of hard problem instances and (iii) extracting insights regarding challenges in cargo securing to help managers and practitioners in decision making.

An extended model formulation for the two-dimensional irregular strip packing problem considering general industry-relevant aspects

Two-dimensional cutting and packing problems including irregular items (nesting problems) are common, e.g., in the clothing, paper, glass and leather industries. In the irregular strip packing problem considered in this work a finite number of rotations of convex as well as non-convex polygons with and without holes are permitted. To deal with the geometry of irregular items, direct trigonometry is applied. The focus is on aspects and characteristics that are typical for many affected industries and have been neglected so far. In the mentioned sectors, it is conceivable that items can be created from smaller parts by assembling them using various techniques. There might be several possible combinations of parts to be joined together to result in the desired item, i.e., there might be several cutting patterns to choose from. Also, whether the large material, i.e., large object is single-colored or has a particular structure or design is of great importance. In the latter case, special attention must be paid to the rotation of certain items or parts in order to achieve the desired (uniform or non-uniform) appearance of the final product. The utilized data structure is introduced, to address the mentioned aspects in the presented mixed-integer linear model which is an extension of a formulation published by previous authors. Furthermore, the method of calculating “critical vertices” is introduced, which requires only a reduced number of comparisons between edges and vertices of two items to ensure overlap-free positioning. Industry-relevant examples are highlighted in the computational study to illuminate the versatility of the model.

GRASP Optimization for the Strip Packing Problem with Flags, Waste Functions, and an Improved Restricted Candidate List

This research addresses the two-dimensional strip packing problem to minimize the total strip height used, avoiding overlapping and placing objects outside the strip limits. This is an NP-hard optimization problem. We propose a greedy randomized adaptive search procedure (GRASP), incorporating flags as a new approach for this problem. These flags indicate available space after accommodating an object; they hold the available width and height for the following objects. We also propose three waste functions as surrogate objective functions for the GRASP candidate list and use and enhanced selection for the restricted candidate list, limiting the object options to better elements. Finally, we use overlapping functions to ensure that the object fits in the flag because there are some cases where a flag’s width can be wrong due to new object placement. The tests showed that our proposal outperforms the most recent state-of-the-art metaheuristic. Additionally, we make comparisons against two exact algorithms and another metaheuristic.

A hybrid metaheuristic for the two-dimensional strip packing problem

In this paper we present a hybrid metaheuristic approach called PVS (Progress and Verify Strategy) for the two-dimensional strip packing problem (2SPP). PVS relies on two procedures: a local search algorithm that delivers satisfying placements of the items on the horizontal axis, and an exact procedure that searches for the positions of the items on the vertical axis. This last one explores all the possibilities, starting with the most promising ones, and can be stopped at any moment. PVS follows a specific anytime strategy which continuously improves the current solution until it is provably optimal or a given time limit is reached. Experimental results show that the method is competitive on moderate-sized instances compared to the best known approaches.

The effect of benchmark data characteristics during empirical strip packing heuristic performance evaluation

In this paper, we consider the two-dimensional strip packing problem in which a fixed set of items has to be packed into a single object of unlimited height with the aim of minimising the packing height. A new, systematic way of comparing the relative performances of packing algorithms is proposed. A large set of strip packing benchmark instances from various repositories in the literature is clustered into different classes of test problems based on their underlying features. We compare a representative sample of existing strip packing heuristics in terms of solution quality in respect of the clustered data. The effectiveness of the designs of the different algorithms is contrasted for each of the data categories. It is found that the aspects of the test problems affect the solution qualities and relative rankings achieved by the various packing algorithms, but that no single heuristic is superior in respect of all the data categories.

Efficient Load Control Based Demand Side Management Schemes Towards a Smart Energy Grid System

In this paper, we propose efficient load scheduling based demand side management schemes for the objective of peak load reduction. We propose two heuristic algorithms, named G-MinPeak and LevelMatch, which are based on the generalized two-dimensional strip packing problem, where each of the appliances has their specific timing requirements to be fulfilled. Furthermore, we have proposed some improvement schemes that try to modify the resulted schedule from the proposed heuristic algorithms to reduce the peak. All the proposed algorithms and improvement schemes are experimented using benchmark data sets for performance evaluation. Extensive simulation studies have been conducted using practical data to evaluate the performance of the algorithms in real life. The results obtained show that all the proposed methodologies are thoroughly effective in reducing peak load, resulting in smoother load profiles. Specifically, for the benchmark datasets, the deviation from the optimal values has been about 6% and 7% for LevelMatch and G-MinPeak algorithms respectively and by using the improvement schemes the deviations are further reduced up to 3% in many cases. For the practical datasets, the proposed improvement schemes reduce the peak by 5.21− 7.35 % on top of the peaks obtained by the two proposed heuristic algorithms without much computation overhead.

Improved metaheuristics for the two-dimensional strip packing problem

Given a fixed set of rectangular items and a single rectangular object of fixed width and unlimited height, the two-dimensional strip packing problem consists of packing all the items into the object in a non-overlapping manner, such that the resulting packing height is a minimum. Two improved strip packing metaheuristics are proposed in this paper. The first algorithm is a hybrid approach in which the method of simulated annealing is combined with a heuristic construction algorithm, while the second algorithm involves application of the method of simulated annealing directly in the space of completely defined packing layouts, without an encoding of solutions. These two algorithms are compared with a representative sample of metaheuristics from the literature in terms of solution quality achieved in the context of a large set of 1718 benchmark instances, clustered into four sets of test problems, each with differing characteristics. It is found that the new algorithms indeed compare favourably with, and in some cases outperform, existing strip packing metaheuristics in the literature.

An Effective Discrete Grey Wolf Optimization Algorithm for Solving the Packing Problem

This article proposes a novel discrete grey wolf optimization for the packing problem, called the two-dimensional strip packing (2DSP) problem without guillotine constraint. The 2DSP involves cutting pieces from a stock sheet with the objective of minimizing waste. To solve the 2DSP problem by the discrete grey wolf algorithm, many strategies are originally proposed. The searching and attacking operators in the algorithm are redesigned to guarantee coding effectiveness. A novel approach to measure the distance between the wolves is presented. In addition, an improved best-fit strategy is developed to solve this packing problem. The best-fit strategy divides the situation into five cases based on the width and length of the rectangle. Computational results on widely used benchmark instances show that the novel discrete grey wolf algorithm can solve the 2DSP problem effectively, and surpasses most of the previously reported meta-heuristic algorithms.

Fragmentary Structures in a Two-Dimensional Strip Packing Problem

The general problem of two-dimensional packing in a semi-bounded strip is considered. It is shown that the problem can be considered as an optimization problem on a fragmentary structure and is reduced to the problem of combinatorial optimization on a set of permutations. A universal approach to representing two-dimensional figures and an algorithm for packing them in a strip are considered. An approach to modifying the original problem to attain an optimal solution is proposed.

Hybrid simulated annealing and reduced variable neighbourhood search for an aircraft scheduling and parking problem

Aircraft stands and runways at airports are critical airport resources for aircraft scheduling and parking. Making use of limited apron and runway resources to improve airport efficiency is becoming increasingly important. In this paper, we study a realistic Aircraft Scheduling and Parking Problem (ASPP) with the goal of simultaneously determining the takeoff and landing time of each aircraft with consideration for wake vortex effect constraints and parking positions in the limited parking apron at a target airport. The objective of the ASPP is to minimise the total service time for aircraft. We developed a mixed-integer linear programme formulation for the ASPP. A novel improved bottom-left/right strategy is applied to construct solutions and a Hybrid Simulated Annealing and Reduced Variable Neighborhood Search (HSARVNS) is proposed to identify near-optimal solutions. Numerical experiments on randomly generated ASPP instances and on a large set of benchmarks for a reduced version of the ASPP (i.e. the classical Two-Dimensional Strip-Packing Problem (2D-SPP)) demonstrate the effectiveness and efficiency of the proposed approach. For the ASPP, HSARVNS can find optimal solutions for small instances in a fraction of a second and can find high-quality solutions for instances with up to 250 aircraft within a reasonable timeframe. For the 2D-SPP, the HSARVNS can find optimal solutions for 32 of 38 tested benchmarks within 90 s on average.

Improved approximation for two dimensional Strip Packing with polynomial bounded width

We study the well-known two-dimensional Strip Packing problem. Given a set of rectangular axis-parallel items and a strip of width W with infinite height, the objective is to find a packing of all items into the strip, which minimizes the packing height. Lately, it has been shown that the lower bound of 3/2 of the absolute approximation ratio can be beaten when we allow a pseudo-polynomial running-time of type (nW)f(1/ε). If W is polynomially bounded by the number of items, this is a polynomial running-time. The currently best pseudo-polynomial approximation algorithm by Nadiradze and Wiese achieves an approximation ratio of 1.4+ε. We present a pseudo-polynomial algorithm with improved approximation ratio 4/3+ε. Furthermore, the presented algorithm has a significantly smaller running-time as the 1.4+ε approximation algorithm.

Models for the two-dimensional level strip packing problem – a review and a computational evaluation

The two-dimensional level strip packing problem has received little attention from the scientific community. To the best of our knowledge, the most competitive model is the one proposed in 2004 by Lodi et al., where the items are packed by levels. In 2015, an arc flow model addressing the two-dimensional level strip cutting problem was proposed by Mrad. The literature presents some mathematical models, despite not addressing specifically the two-dimensional level strip packing problem, they are efficient and can be adapted to the problem. In this paper, we adapt two mixed integer linear programming models from the literature, rewrite the Mrad’s model for the strip packing problem and add well-known valid inequalities to the model proposed by Lodi et al. Computational results were performed on instances from the literature and show that the model put forward by Lodi et al. with valid inequalities outperforms the remaining models with respect to the number of optimal solutions found.

An open space based heuristic for the 2D strip packing problem with unloading constraints

This paper studies the two-dimensional strip packing problem with unloading constraints. Given a set of items (represented by rectangles) belonging to different customers and a rectangular strip with fixed width and open length, the objective is to pack all the items into the strip such that the used length is minimized. In addition, the resulting packing must satisfy the unloading constraints. Based on the characteristics of this problem, this paper employs the open space to represent the candidate position. An open space based first-fit heuristic is proposed to generate a packing pattern for a given sequence of items. An implementation using the segment tree is provided. Finally, a randomized local search without any parameter is used to improve the solution by trying different sequences. The computational results on well-known instances show that the proposed approach outperforms existing approaches in the literature. In additional, we test our approach on the two-dimensional orthogonal packing problem with unloading constraints, the results show that our approach is capable of finding feasible solutions for 283 open instances.

Data mining based framework to assess solution quality for the rectangular 2D strip-packing problem

In this paper, we explore the use of reference values (predictors) for the optimal objective function value of hard combinatorial optimization problems, instead of bounds, obtained by data mining techniques, and that may be used to assess the quality of heuristic solutions for the problem. With this purpose, we resort to the rectangular two-dimensional strip-packing problem (2D-SPP), which can be found in many industrial contexts. Mostly this problem is solved by heuristic methods, which provide good solutions. However, heuristic approaches do not guarantee optimality, and lower bounds are generally used to give information on the solution quality, in particular, the area lower bound. But this bound has a severe accuracy problem. Therefore, we propose a data mining-based framework capable of assessing the quality of heuristic solutions for the 2D-SPP. A regression model was fitted by comparing the strip height solutions obtained with the bottom-left-fill heuristic and 19 predictors provided by problem characteristics. Random forest was selected as the data mining technique with the best level of generalisation for the problem, and 30,000 problem instances were generated to represent different 2D-SPP variations found in real-world applications. Height predictions for new problem instances can be found in the regression model fitted. In the computational experimentation, we demonstrate that the data mining-based framework proposed is consistent, opening the doors for its application to finding predictions for other combinatorial optimisation problems, in particular, other cutting and packing problems. However, how to use a reference value instead of a bound, has still a large room for discussion and innovative ideas. Some directions for the use of reference values as a stopping criterion in search algorithms are also provided.