Rectangular Bins Research Articles

Ground collision avoidance system with multi-trajectory risk assessment and decision function

AbstractThe traditional ground collision avoidance system (GCAS) makes avoidance decisions based on predicted collision time, without considering the impact of terrain environment and dynamic changes in load factor on avoidance decisions. This increases the risk of ground collisions for the aircraft. To solve the problem, a GCAS with multi-trajectory risk assessment and decision function is proposed. Firstly, a variety of predicted flight avoidance trajectories are established within the final manoeuvering capability of the aircraft. Secondly, for each predicted trajectory, the uncertain length between adjacent prediction points is used to construct a rectangular distance bin, and the terrain data below the avoided trajectory is extracted. Finally, the regret theory is used to establish a multi-attribute avoidance decision model to evaluate and prioritise the risk of collision avoidance trajectories, to provide effective collision avoidance decision for pilots. The algorithm is tested and verified with real digital elevation model and simulated flight data, and compared with traditional GCAS. Simulation results show that the proposed algorithm can comprehensively consider manoeuvering performance and threatening terrain, and provide safe and effective avoidance decisions for pilots.

A hybrid estimation of distribution algorithm for the offline 2D variable-sized bin packing problem

In this paper we present an evolutionary heuristic for the offline two-dimensional variable-sized bin packing problem. In this problem we have to pack a set of rectangles into two-dimensional variable-sized rectangular bins. The bins are divided into types, and the bins in different types have different sizes and possibly different weights (costs). There are (sufficiently) many bins from each type, and any rectangle fits into at least one bin-type. The goal is to pack the rectangles into the bins without overlap, parallel to the sides, so that the total area of the used bins (or total cost) is minimized. Our algorithm is a hybrid heuristic. It uses two different techniques to generate the descendants: either estimation of distribution algorithm and sampling the resulting probability model, or applying the usual operators of evolutionary algorithms (selection, mutation). To pack the rectangles into the bins the algorithm uses the strategy of randomly choosing one of two placement heuristics, that pack always only one group (one to three) of rectangles. It improves the quality of the solutions with three local search procedures. The algorithm has been tested on benchmark instances from the literature and has been compared with other heuristics and metaheuristics. Our algorithm outperformed the previously published results.

The evolution of rectangular bin packing problem — A review of research topics, applications, and cited papers

<p style='text-indent:20px;'>Bin packing problem (BPP) is one of the fastest-growing research issues within the field of combinatorial optimization. Over the last years, several studies carried out various BPP variants, mathematical models, and proposed methods to the BPPs. The classical BPP consists of packing a set of rectangular items in a minimum number of rectangular bins while respecting some constraints.</p><p style='text-indent:20px;'>Throughout the years, an improved typology was introduced by Wäscher et al. (2007), providing an excellent instrument for the organization and categorization criteria that defined the problem categories different from those of Dyckhoff (1990). Several early literature reviews have been conducted on various aspects of related packing problem variants.</p><p style='text-indent:20px;'>The contribution of this paper is to provide a comprehensive and refined taxonomy intended for BPPs. In addition to that, it is an up-to-date review based on a chronological taxonomy of the literature and depicts further research horizons.</p><p style='text-indent:20px;'>This systematic review allowed us to identify other characteristics and constraints, based on Wäscher's original ideas, mainly distinguished according to real cases studies. The detailed analysis provides a valuable framework for understanding the research gaps for future studies that should be acknowledged while proposing and solving new extensions.</p>

Covering, corner-searching and occupying: A three-stage intelligent algorithm for the 2d multishape part packing problem.

The bin packing problem has a wide range of applications in industry. With the upgrade of the task difficulty, the traditional 2d rectangular layout algorithm can no longer meet the needs of modern industry, such as express packing task and exoplanet ore collection task. The express or ore samples come in heterogeneous shapes so they cannot all be treated as rectangular pieces. In this paper, we propose a three-stage method called covering, corner-searching and occupying (C,S&O) to solve the two-dimensional multishape part packing problem. The objective of the packing problem variant is to ensure maximum use of the raw material and minimize the trim loss. The algorithm cannot make use of information about the sequence of future objects that are going to arrive, only knowing the shape and size of the coming one, and the coming part must be packed into the bin immediately after its arrival without buffering or readjusting. The method of C,S&O hybridizes the idea of “gold corners, silver edges and grass belly” in the Chinese game Go and the method of finding picture corners in machine vision. In the first stage, the rectangular bin and the coming part are transformed into matrix representation, and generating the position matrix that indicates possible ways of packing the part into the bin. In the second stage, the suitable layout position of the coming part is obtained using machine vision image processing technology for reference. The third stage is calculating the environment matching degree to determine the current optimal placement orientation. In order to facilitate the display of the simulation results, only three shapes of parts are considered in the simulation, rectangle, circle and triangle. The experimental results show the effectiveness of this method. Consulting the literature, it is found that this paper is the first to propose a layout method for multishape manufacturing parts.

Developing a Maximum Inscribed Rectangle Heuristic to Satisfy Rush Orders for Heavy Plate Steel

Steel service centers receive rush orders that must be fulfilled on very short notice. Each order only consumes a portion of one steel plate, so plate selection and job placement are the critical factors that affect the service center’s primary performance metric: plate yield. In conjunction with a steel service center, Artco Steel, we model this problem as a two-dimensional online bin-packing algorithm. Unique in the online bin-packing literature, we calculate the maximum inscribed rectangle (MIR) before and after job placement as the basis for heuristics that assign each job to a plate and position the job on the plate. Our work is the first paper to extend the online two-dimensional bin-packing problem to incorporate scrap, rectangular bin sizes, and a finite number of bins. The MIR procedure significantly outperformed Artco’s existing practice of giving priority to the most recently used plate, and the heuristic’s straightforward nature allowed easy adoption in 2010.

The Rectangular Bin Challenge: Using 3D Simulation to Explore New Bin Filler Methods for Apple Harvest Platforms

HighlightsBin filler design and bin dimensions determine the apple distribution in the bin.The performance of new apple distribution systems was estimated with 3D simulations.A simulated lemniscate path showed improved apple distribution in rectangular bins.3D simulation provides an efficient tool for design decisions before prototype production.Abstract. Mechanical harvest platforms equipped with conveyors are meant to replace the traditional apple-picking ladders and buckets; however, their feasibility has not yet been exhaustively established. The conveyors transport picked apples to the bin filler, which distributes the fruit into the bins. Commercial harvest platforms are designed to perform well with square bins; however, Israeli apple industry bins are rectangular. This results in non-uniform apple distribution, which requires human intervention and contributes to apple bruising and economic loss. In this study, 3D simulation was used to quantitatively evaluate the performance of three novel bin filler designs (back-and-forth, double disk, and lemniscate) and compare them to commercial bin fillers. The bin filler models were tested for the uniformity of the apple distribution in square and rectangular bins. The angle of repose and the standard deviation of the height of the top layer of apples were calculated. The commercial bin filler model performed better when filling a square bin (angle = 12°, SD height = 31 mm) than a rectangular bin (angle = 18°, SD height = 71 mm). The lemniscate model (angle = 10°, SD height = 41 mm) outperformed the two other novel models as well as the commercial bin filler model (angle = 18°, SD height = 67 mm). A machine design for the lemniscate model is proposed. The quantitative method described in this study illustrates the potential benefit of 3D simulation as an inexpensive means to explore the feasibility of new harvest devices. Keywords: 3D simulation, Apple, Bin filler, Fruit distribution, Harvest, Mechanical platform.

Packing and stacking rings into rectangular bins

In this paper, we propose a new variant of the circular packing problem whose objective is to pack rings with unequal outer and inner radii into larger rings and finally into rectangular bins of unequal sizes. This problem arises in industrial contexts where rings are cut from rectangular raw material. We divide this problem into three partial problems, formulate a model for these partial problems and provide heuristic algorithms. Our algorithms are based on some simplifications that make it possible to identify eco-efficient packing layouts also for large numbers of ring sizes and moderate numbers of bin sizes. In a computational evaluation of our algorithms, we demonstrate that the packing problem can be solved in approximately 70 seconds on average when 7,500 rings with 50 different outer circle sizes are packed into bins of seven different sizes.

An iteratively doubling local search for the two-dimensional irregular bin packing problem with limited rotations

This paper proposes an iteratively doubling local search approach for the two-dimensional irregular bin packing problem (2DIRBPP) with limited rotations, whose objective is to pack a given set of irregular pieces into the minimum number of rectangular bins. The allowable angles of each piece are limited. To solve this problem, a waste least first decreasing strategy is introduced to assign the piece to the bins. A simple greedy local search approach by moving the pieces from one bin to another is utilized to improve the solution. We adapt an overlap minimization approach to solving the one bin placement problem. The classical bottom-left method is utilized to generate the initial position for each piece, and the random local search by exchanging the positions of two pieces is used to minimize the overlap. Moreover, a novel warm start and an iteratively doubling search strategy are proposed to speed up the search process. The standard benchmark results show that our approach improves the results for most of the instances in the literature.

A lookahead matheuristic for the unweighed variable-sized two-dimensional bin packing problem

The unweighed oriented variable-sized two-dimensional guillotine bin packing problem consists in packing without overlap small rectangular items into large non-identical rectangular bins, with the items obtained via guillotine cuts. It minimizes the waste of the used bins. It is herein approximately solved using a hybrid matheuristic, which applies a sequence of low-level mixed-integer programs that reserve space for unpacked items and that are guided by feasibility constraints and by upper bounds on the objective function. The embedded constraints constitute a lookahead mechanism that prohibits the investigation of infeasible directions and constrains the search to improving ones. The matheuristic further employs high-level diversification and intensification mechanisms. The diversification incorporates a sequential value correction algorithm that tags a pseudo-price to each item to govern the fitness functions of mixed integer programs and subsequently their solution construction process. The intensification is a local search that investigates the neighbourhood of promising solutions. The extensive computational experiments provide evidence of the good performance of the proposed matheuristic. For the variable-sized bin packing benchmark instances, the matheuristic matches and improves 90.8% of the upper bounds. For the single bin-size bin packing benchmark instances, the matheuristic further proves the optimality of 82.6% upper bounds while it matches 14.4% and improves 2.6% existing bounds.

Multi-objective 3D bin packing problem with load balance and product family concerns

The classical three-dimensional bin packing problem (3D-BPP) orthogonally packs a set of rectangular items with varying dimensions into the minimum number of three-dimensional rectangular bins. While ensuring the minimum number of bins used, the safety of the logistic operations is addressed with the complementary load-balancing objective for which concepts such as orientation and stability are used in the literature though not at the same time. In this study, we extend the load-balanced 3D-BPP by combining both orientation and stability, and introducing a new concept called family unity which encourages packing a family of products (e.g., from the same order and with the same destination) together. Although item related concerns are very common in practice, there are no multi-objective studies in the bin packing literature that includes family unity concept. Therefore, this is the first study that proposes a multi-objective mixed integer programming model for the extended problem to determine the optimal packing plan that minimizes the number of bins used and the deviation of balance from the ideal barycenter while maximizing family unity ratio via a weighted objective function. A numerical example is provided to analyze the performance of the proposed model. Furthermore, a real-life container loading problem is solved and outputs of the study implies the practical advantages of including family unity and load-balance considerations in solving 3D-BPP problems.

Combinatorial Benders Decomposition for the Two-Dimensional Bin Packing Problem

The two-dimensional bin packing problem calls for packing a set of rectangular items into a minimal set of larger rectangular bins. Items must be packed with their edges parallel to the borders of the bins, cannot be rotated, and cannot overlap among them. The problem is of interest because it models many real-world applications, including production, warehouse management, and transportation. It is, unfortunately, very difficult, and instances with just 40 items are unsolved to proven optimality, despite many attempts, since the 1990s. In this paper, we solve the problem with a combinatorial Benders decomposition that is based on a simple model in which the two-dimensional items and bins are just represented by their areas, and infeasible packings are imposed by means of exponentially many no-good cuts. The basic decomposition scheme is quite naive, but we enrich it with a number of preprocessing techniques, valid inequalities, lower bounding methods, and enhanced algorithms to produce the strongest possible cuts. The resulting algorithm behaved very well on the benchmark sets of instances, improving on average on previous algorithms from the literature and solving for the first time a number of open instances. Summary of Contribution: We address the two-dimensional bin packing problem (2D-BPP), which calls for packing a set of rectangular items into a minimal set of larger rectangular bins. The 2D-BPP is a very difficult generalization of the standard one-dimensional bin packing problem, and it has been widely studied in the past because it models many real-world applications, including production, warehouse management, and transportation. We solve the 2D-BPP with a combinatorial Benders decomposition that is based on a model in which the two-dimensional items and bins are represented by their areas, and infeasible packings are imposed by means of exponentially many no-good cuts. The basic decomposition scheme is quite naive, but it is enriched with a number of preprocessing techniques, valid inequalities, lower bounding methods, and enhanced algorithms to produce the strongest possible cuts. The algorithm we developed has been extensively tested on the most well-known benchmark set from the literature, which contains 500 instances. It behaved very well, improving on average upon previous algorithms, and solving for the first time a number of open instances. We analyzed in detail several configurations before obtaining the best one and discussed several insights from this analysis in the manuscript.

The rectangular bin challenge: Using 3D simulation to explore new apple harvest platform bin filler methods

Highlights Bin filler design and bin dimensions determine apple distribution Performance of new apple distribution systems was estimated with 3D simulations Simulated lemniscate scattering path demonstrated improved apple distribution in rectangular bins 3D simulation provides an efficient, informed design decision tool before going into production Abstract. Mechanical harvest platform systems equipped with conveyors were meant to replace the traditional apple-picking ladder and bucket; however, their feasibility has not yet been exhausted. The conveyor belts transport the apples to the bin filler, whose role is to distribute the fruit into the bins. Commercial mechanical harvest platforms are designed to perform well with square bins; however, Israeli apple industry bins are rectangular. This results in a non-uniform apple distribution, which requires human intervention and contributes to apple bruising and economic loss. In the present study, a 3D simulation was used to quantitatively evaluate the performance of three novel bin filler designs (back-and-forth, double disk, and lemniscate) and compare them to commercial bin fillers. The bin filler models were tested for the uniformity of the apples distributed into square and rectangular bins. The angle of repose and the standard deviation of the height of the top layer of apples were calculated. The commercial bin filler model performed better when filling square bins (angle = 12°, SD height = 31 mm) than rectangular bins (angle = 18°, SD height = 71 mm). The lemniscate model (angle = 10°, SD height = 41 mm) outperformed the two others as well as the commercial bin filler model (collapsed: angle = 18°, height = 67 mm). A machine design for the lemniscate model is proposed. The quantitative methods described in this study point to the potential benefit of 3D simulation as an inexpensive means to explore the feasibility of new harvest devices.

Cleaning of water bodies using coastal sea bin (CSB)

The rivers in Mumbai are highly polluted with plastic waste and these rivers end up in the Arabian Sea, which brings all this waste back to the land during high tides. This garbage keeps piling up on land and it is difficult to manage this waste. Plastic waste causes harm to wildlife, environment, and human beings. Therefore. it is decided to design a rectangular suspended sea bin with the non-return flaps, facing the waves. The floating garbage that is flowing along the waves will enter the cage through the flaps and thus get trapped. The waves will thus progress without any hinderance due to the cage openings on the rear side. A comprehensive study is done with the weight, center of gravity and metacentric height of a rectangular coastal sea bin (CSB) for different materials. Aluminum is considered as an ideal material for the case study as it is durable, light- weight and non-corrosive. A simulated 3D CSB model with the scale ratio of 1:12.5 is also prepared.•Plastic waste causes harm to wildlife, environment, and human beings•A rectangular cage type rectangular coastal see bin (CSB) is designed with the non-return flaps, facing the waves for plastic collection•A simulated 3D CSB model with the scale ratio of 1:12.5 is also prepared.

Experimental investigation of segregation of granular mixtures during heap formation

The present study is on segregation of granular mixtures during heap formation in a quasi two-dimensional rectangular bin where binary mixture of a specified composition is poured intermittently into the auxiliary hopper and then allowed to pass through the gap (k = 10 mm) between the divider and the plate and finally settle on to the heap. The profiles of number fraction of big particles are plotted along the flow directions to study the segregation phenomena for surface profile. It shows that larger particles travel more distance and smaller particles settle near the pouring point for all cases studied.

Just-in-time two-dimensional bin packing

This paper considers the on-time guillotine cutting of small rectangular items from large rectangular bins. Items assigned to a bin define the bins’ processing time. Consequently, an item inherits the completion time of its assigned bin. Any deviation of an item’s completion time from its due date causes either earliness or tardiness penalties. This just-in-time two-dimensional bin packing problem (JITBP) combines two difficult discrete optimization problems: Bin packing and total weighted earliness tardiness single machine scheduling. It is herein modeled as an integrated constraint program, augmented with two sets of logically redundant constraints that speed the search. The first set uses the concept of dual feasible functions. It focuses on bin packing feasibility. The second is the result of a linear program that schedules filled bins on a single machine. As an alternative to this integrated model, this paper proposes two decomposition cut-and-check approaches that define the master problem (MP) as a relaxation of JITBP where the items are reduced to dimensionless entities. They then reestablish the geometric feasibility of the MPs’ solutions by iteratively augmenting MP with Benders cuts generated from the subproblems. The two approaches are similar in concept except that one implements MP as a constraint program (CP) while the second implements it as a mixed-integer program (MIP). Because JITBP is computationally challenging, we test all approaches under a number of heuristic assumptions, which include a maximum runtime for the MIP and CP solvers. The results provide computational evidence that the integrated constraint programming approach performs relatively well, and outperforms the decomposition approach whose MP is a CP. However, both approaches are outperformed by the decomposition approach whose MP is a warm-started MIP.

Analysis of granular rheology in a quasi-two-dimensional slow flow by means of discrete element method based simulations

The steady flow of spherical particles in a rectangular bin is studied using the discrete element method for different flow rates of the particles from the bin in the slow flow regime. The flow has two nonzero velocity components and is more complex than the widely studied unidirectional shear flows. The objective of the study is to characterize, in detail, the local rheology of the flowing material. The flow is shown to be of nearly constant density, with a symmetric stress tensor and the principal directions of the stress and rate of strain tensors being nearly colinear. The local rheology is analyzed using a coordinate transformation which enables direct computation of the viscosity and components of the pressure assuming the granular material to be a generalized Newtonian fluid. The scaled viscosity, fluctuation velocity, and volume fraction are shown to follow power law relations with the inertial number, a scaled shear rate, and data for different flow rates collapse to a single curve in each case. Results for flow of the particles on an inclined surface, presented for comparison, are similar to those for the bin flow but with a lower viscosity and a higher solid fraction due to layering of the particles. The in plane normal stresses are nearly equal and slightly larger than the third component. All three normal stresses correlate well with the corresponding fluctuation velocity components. Based on the empirical correlations obtained, a continuum model is presented for computation of granular flows.

Application of Nature Inspired Algorithms to Optimize Multi-objective Two-Dimensional Rectangle Packing Problem

This paper considers two-dimensional non-guillotine rectangular bin packing problem with multiple objectives in which small rectangular parts are to be arranged optimally on a large rectangular sheet. The optimization of rectangular parts is attained with respect to three objectives involving maximization of (1) utilization factor, minimization of (2) due dates of rectangles and (3) number of cuts. Three nature based metaheuristic algorithms — Cuckoo Search, Bat Algorithm and Flower Pollination Algorithm — have been used to solve the multi-objective packing problem. The purpose of this work is to consider multiple industrial objectives for improving the overall production process and to explore the potential of the recent metaheuristic techniques. Benchmark test data compare the performance of recent approaches with the popular approaches and also of the different objectives used. Different performance metrics analyze the behavior/performance of the proposed technique. Experimental results obtained in this work prove the effectiveness of the recent metaheuristic techniques used. Also, it was observed that considering multiple and independent factors as objectives for the production process does not degrade the overall performance and they do not necessarily conflict with each other.

Tilapia processing waste silage (TPWS): An alternative ingredient for Litopenaeus vannamei (Boone, 1931) diets in biofloc and clear-water systems

The aim of this study was to evaluate the growth performance of Litopenaeus vannamei juveniles reared under biofloc and clear-water conditions fed with different inclusion levels of tilapia processing waste silage (TPWS) based-diets. The experiment was performed in two individual systems: biofloc (BS) and clear-water systems (CWS). The trial used forty 40 L rectangular plastic bins (twenty per system) in a density of 63 shrimp/m2. The juveniles were distributed in a factorial completely randomized experimental design. The treatments were based on the percentage of silage inclusion (control or 0, 1.5%, 3.0%, 4.5% and 6.0% of inclusion) in BS or CWS, totalizing ten treatments and four replicates. Survival was above 80% in all treatments and was not affected by both systems and diet. Shrimp final weight and SGR were statistically influenced by system (P < 0.05) but not by the diet; and presented high values in BS. The inclusion of TPWS in L. vannamei diets did not affect shrimp performance. In addition, shrimp raised in BS demonstrated better growth performance as compared to CWS.

Autonomous Positioning Advisor for Trippers in Mineral Processing Facilities

Mineral processing facilities commonly have storage bins for temporary housing of the ore supplied to their processing lines. Facilities with large rectangular bins usually have a tripper machine to load and spread the ore along the entire extension of the bin, by means of forth-and-back traveling movements. The positioning control of a tripper is a relevant yet less concerned control problems, with significant impact on the productivity of mineral processing facilities, for which a poor controlled tripper is a common source of bottleneck. This work concerns the development and implementation of a strategy for autonomous position determination for a tripper in a large iron ore processing facility, with the ultimate goal of eliminating the need for a local operator to decide the tripper positioning under manual operation. Such “autonomous positioning advisor” acts as an expert system to determine the “best” position (bin cell) where the tripper should be placed to feed the bin in order to meet some operational requirements as: (1) to make the ore level more uniform across the bin; (2) to prevent excessive and unnecessary travels of the tripper; and (3) to suppress the ore discharge from the tripper to the bin when the tripper needs to travel over a fully filled cells to reach the destination cell. The positioning control strategy was based on operational knowledge and procedures used by the tripper operators when operating the tripper in manual mode. The results obtained with the autonomous positioning advisor allowed the tripper to operate consistently in automatic mode and showed improved operational performance.

Production performance of Litopenaeus vannamei (Boone, 1931) fed with different dietary levels of tilapia processing waste silage reared in biofloc system using two carbon sources

The evaluation of alternative protein sources in aquafeeds is a priority in the aquaculture industry. Moreover, carbon sources in biofloc systems can directly affects the nutritional composition of flocs and consequently improve the shrimp growth. In this context, the aim of the present study was to evaluate different levels of tilapia processing waste silage (TPWS) in L. vannamei diets reared in biofloc system using two carbon sources: sugar cane molasses (SCM) and biofloc cassava flour (CF). The trial used two experimental systems (one for each carbon source) with thirty 40 L rectangular plastic bins in a density of 250 shrimp m−3. The juveniles (2.86 g ± 0.33 g) were distributed in a factorial completely randomized experimental design. The treatments were based on the percentage of TPWS inclusion (control or 0, 1.5, 3.0, 4.5 and 6.0% of inclusion) and on the carbon sources used (SCM and CF) totalizing ten treatments and three replicates. After 45 days, water quality parameters remained within the recommended range for L. vannamei culture. Regarding to the production performance survival was above 70% in all treatments. Final biomass and survival were not affected by diet, rather than carbon source (P < .05) with higher values in SCM treatments. Shrimp final weight was affect by the diet; whether treatments presented no differences except by 4.5% of TPWS inclusion presented the lowest value (P < .05). Feed conversion ratio showed no significant differences. In this sense, the sugar cane molasses (SCM) and the inclusion of tilapia processing waste silage (TPWS) emerge as a good option to increase sustainability in shrimp culture.