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This document is a special issue of the journal Mathematics dedicated to recent advances in discrete optimization and scheduling. The issue includes 10 papers covering a range of topics such as batch loading and scheduling, optimization algorithms, scheduling surgeries, routing problems, and mathematical models for simplification. The papers present various approaches and algorithms for solving these problems, and they have been reviewed by experts in the field. The authors express their gratitude to the contributors, reviewers, and editorial staff, and hope that the readers will find stimulating ideas for further research in this area. [Extracted from the article]

To improve cargo loading efficiency and achieve diverse needs of companies for the loading process, this paper innovatively establishes a multiple objective mixed integer programming model for the three-dimensional multiple bin-size bin packing problem (3D-MBSBPP). The model is designed to maximize container space utilization rate and cargo load balance, minimize container usage costs, and incorporates some practical constraints. On this basis, we propose a novel dynamic feedback algorithm based on spatial corner fitness (SCF_DFA) to solve this model, which consists of three stages. Specifically, Stage 1 employs a heuristic algorithm based on spatial corner fitness to optimize the search of the remaining spaces. Stage 2 employs a container type selection algorithm to dynamically adjust and optimize container types. Stage 3 uses an improved genetic algorithm to improve the quality of the solutions of the first two stages. We demonstrate the effectiveness of the proposed algorithm through comparative experiments on benchmark instances, and apply it to solve the real-life instances for the 3D-MBSBPP. The results show that the proposed algorithm can make the average container space utilization rate reach 85.38%, which is 1.48% higher than that of baseline method, while the loading results obtained are more balanced, indicating the advantages of the SCF_DFA in solving 3D-MBSBPP. Furthermore, we conduct ablation experiments to confirm the effectiveness of each component within the algorithm. [ABSTRACT FROM AUTHOR]

Weather radar is an active remote sensing device used to monitor the full lifecycle changes in severe convective weather with high spatial and temporal resolution. Effective radar calibration is a crucial foundation for ensuring the high-quality application of observational data. This paper utilizes a UAV platform equipped with a high-precision RTK system and standard metal spheres to study the principles and methods of metal sphere calibration, constructing a complete calibration process and calibration accuracy evaluation metrics. Additionally, a collocated radar comparison observation experiment was conducted for cross-validation, and metal sphere calibration tests were performed on problematic radars. The experimental results indicate the following: (1) The combined application of a high-precision RTK system and a laser range camera can provide real-time position information on the metal sphere, improving the efficiency of radar target acquisition. (2) The calibration method based on UAV-suspended metal spheres can periodically conduct the quantitative calibration of Z and ZDR, achieving calibration accuracies within 0.5 dB and 0.2 dB, respectively, and supports the qualitative inspection of key parameters such as beamwidth and pulse width. (3) During field tests, a high success rate "coarse adjustment + fine adjustment + staring" sphere-finding technique was established, based on automatic switching between RHI, PPI, and FIX scanning modes. This method directs the UAV to adjust the metal sphere to the center of the radar distance bin, reducing the impact of uneven beam filling and bin crossing, ensuring the accuracy of scattering characteristic measurements. [ABSTRACT FROM AUTHOR]

This paper introduces the Color Histogram Contouring (CHC) method, a new training-less approach to object detection that emphasizes the distinctive features in chrominance components. By building a chrominance-rich feature vector with a bin size of 1, the proposed CHC method exploits the precise information in chrominance features without increasing bin sizes, which can lead to false detections. This feature vector demonstrates invariance to lighting changes and is designed to mimic the opponent color axes used by the human visual system. The proposed CHC algorithm iterates over non-zero histogram bins of unique color features in the model, creating a feature vector for each, and emphasizes those matching in both the scene and model histograms. When both model and scene histograms for these unique features align, it ensures the presence of the model in the scene image. Extensive experiments across various scenarios show that the proposed CHC technique outperforms the benchmark training-less Swain and Ballard method and the algorithm of Viola and Jones. Additionally, a comparative experiment with the state-of-the-art You Only Look Once (YOLO) technique reveals that the proposed CHC technique surpasses YOLO in scenarios with limited training data, highlighting a significant advancement in training-less object detection. This approach offers a valuable addition to computer vision, providing an effective training-less solution for real-time autonomous robot localization and mapping in unknown environments. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper introduces the three‐dimensional case picking problem (3D‐CPP) and proposes a multistart biased‐randomized algorithm (BRA) to solve it. The 3D‐CPP combines two important topics in modern warehouse logistics: the pallet loading problem and the routing of pickers in manual warehouses. The proposed optimization procedure aims at minimizing the overall distance traveled by the pickers, and is achieved by combining a routing problem (i.e., the order in which picking positions are visited) with a loading problem (i.e., the way in which cases are placed onto the pallet). We also consider additional constraints regarding the weight, vertical support, and strength of the cases. In order to solve this problem, we first propose a constructive heuristic which combines routing and packing procedures. This initial heuristic is then extended into a multistart BRA by employing a skewed probability distribution to introduce a certain degree of randomness during the solution‐construction process. A series of computational experiments allow us to assess the quality of the proposed approach, through a comparison with other algorithms as well as using real‐life data provided by an industrial partner. [ABSTRACT FROM AUTHOR]

Given a non-decreasing sequence S = (s 1 , s 2 , ... , s k) of positive integers, an S -packing coloring of a graph G is a partition of the vertex set of G into k subsets { V 1 , V 2 , ... , V k } such that for each 1 ≤ i ≤ k , the distance between any two distinct vertices u and v in V i is at least s i + 1. In this paper, we study the problem of S -packing coloring of cubic Halin graphs, and we prove that every cubic Halin graph is (1 , 1 , 2 , 3) -packing colorable. In addition, we prove that such graphs are (1 , 2 , 2 , 2 , 2 , 2) -packing colorable. [ABSTRACT FROM AUTHOR]

The online 3D bin packing problem(3D-BPP) is widely used in the logistics industry and is of great practical significance for promoting the intelligent transformation of the industry. The heuristic algorithm relies too much on manual experience to formulate more perfect packing rules. In recent years, many scholars solve 3D-BPP via deep reinforcement learning(DRL) algorithms. However, they ignore many skills used in manual packing, one of the most important skill is workers put the item aside if the item is packed improperly. Inspired by this skill, we propose a DRL algorithm with a buffer zone. Firstly, we define the wasted space and the buffer zone. And then, we integrate them into the DRL algorithm framework. Importantly, we compare the bin utilization with di erent thresholds of wasted space and di erent buffer zone sizes. Experimental results show that our algorithm outperforms existing heuristic algorithms and DRL algorithms. [ABSTRACT FROM AUTHOR]

This paper studies the assignment problem of multi product assembly jobs to days. The problem aims to minimize the amount of overtime while avoiding assembly delays for jobs that can be fragmented into smaller sub-tasks. When sequencedependent setup times are negligible, the problem considered transforms into the bin packing problem with restricted item fragmentation where jobs represent items and days stand for bins. We present a mixed integer programming model of the problem by extending earlier formulations in the literature. Computational experiments show that the mathematical model obtained optimal solutions for majority of instances tested within reasonable computation times. [ABSTRACT FROM AUTHOR]

Many packing, routing, and knapsack problems can be expressed in terms of integer linear programming models based on set covering. These models have been exploited in a range of successful heuristics and exact techniques for tackling such problems. In this paper, we show that integer linear programming models based on set covering can be very useful for their use within an algorithm called “Construct, Merge, Solve & Adapt”(CMSA), which is a recent hybrid metaheuristic for solving combinatorial optimization problems. This is because most existing applications of CMSA are characterized by the use of an integer programming solver for solving reduced problem instances at each iteration. We present applications of CMSA to the variable-sized bin packing problem and to the electric vehicle routing problem with time windows and simultaneous pickups and deliveries. In both applications, CMSA based on a set covering model strongly outperforms CMSA when using an assignment-type model. Moreover, state-of-the-art results are obtained for both considered optimization problems. [ABSTRACT FROM AUTHOR]

The 3D packing problem has a wide range of applications. However, the complex geometry of irregular objects leads to a sharp increase in the number of placement combinations, making it a challenging problem. In this paper, we propose a packing pipeline based on rigid body dynamics simulation to deal with two types of 3D packing problems. One is the variant bin packing problem, which involves placing more objects into a container of given dimensions to maximize space utilization. The other is the open dimension problem, where the goal is to minimize the container that can accommodate all objects. We first use heuristic placement strategies and a fast collision detection algorithm to efficiently obtain initial packing results. Then, we simulate the shaking of the container according to the dynamic principle. Combined with the vacant space filling operation, shaking the container drives the movement of objects in the container to make the arrangement of objects more compact. For the open dimension packing, the container height is optimized by adjusting the constraints of simulation in the basic pipeline. Experimental results show that our method has advantages over existing methods in both speed and packing density. [Display omitted] • Flexible framework for various packing problems, using physical simulation. • FFT-based collision metric and DBLF strategy for efficient packing initialization. • Effective optimization based on physical simulation, including shaking and filling. [ABSTRACT FROM AUTHOR]

The online bin packing problem is a well-known optimization challenge that finds application in a wide range of real-world scenarios. In the paper, we propose a novel algorithm called FuzzyPatternPack(FPP), which leverages fuzzy inference and pattern-based predictions of the distribution of item sizes in online bin packing. In comparison to traditional heuristics like BestFit(BF) and FirstFit(FF), as well as the more recent PatternPack(PaP) and ProfilePacking(PrP) algorithm based on online predictions, FPP demonstrates competitive and superior performance in solving various benchmark problems. Particularly, it excels in addressing problems with evolving distributions, making it a promising solution for real-world applications where the item sizes may change over time. This research unveils the promising potential of employing fuzzy logic to effectively address uncertainty in scheduling and planning problems. • Proposed algorithm utilizes pattern extraction and handles uncertainty effectively. • Employment of a fuzzy classification system adds flexibility in modeling ambiguity. • Superior performance and robustness were seen under evolving input distributions. [ABSTRACT FROM AUTHOR]

The bin packing problem is a classical problem widely existing in practical applications, such as shipping, warehousing and manufacturing industries. Whereas current research mainly focuses on reducing packing costs by optimizing the packing scheme, this study explores a novel approach by redesigning the bin sizes to fit the items ready to be packed. Specifically, this study considers a general three-dimensional open dimension problem (3D-ODP) where all dimensions, i.e. length, width and height, of a number of heterogeneous bin types are unknown and need to be decided. Based on the designed bin types, the corresponding packing scheme with minimal total costs is optimized, which is referred to as a three-dimensional multiple-bin-size bin packing problem (3D-MBSBPP). The combination of 3D-ODP and 3D-MBSBPP is redefined as a three-dimensional bin design and packing problem (3D-BDPP), for which a two-layer heuristic is developed. It consists of an outer heuristic framework (i.e. genetic algorithm or differential evolution algorithm) to design bin types, and an inner deterministic constructive heuristic (i.e. placement algorithm) to generate an effective solution for the 3D-MBSBPP. Extensive simulation tests on the classical benchmark and a case study derived from an e-commerce company are conducted to verify the algorithms and deduce managerial insights. [ABSTRACT FROM AUTHOR]

Due to the competitive nature of the construction industry, the efficiency of requirement analysis is important in enhancing client satisfaction and a company's reputation. For example, determining the optimal configuration of panels (generally called panelization) that form the structure of a building is one aspect of cost estimation. However, existing methods typically rely on rule-based approaches that may lead to suboptimal material usage, particularly in complex designs featuring angled walls and openings. Such inefficiency can increase costs and environmental impact due to unnecessary material waste. To address these challenges, this research proposes a Panelization Algorithm for Architectural Designs, referred to as PAAD, which utilizes a genetic evolutionary strategy built on the 2D bin packing problem. This method is designed to balance between strict adherence to manufacturing constraints and the objective of optimizing material usage. PAAD starts with multiple potential solutions within the predefined problem space, facilitating dynamic exploration of panel configurations. It approaches structural rules as flexible constraints, making necessary corrections in post-processing, and through iterative developments, the algorithm refines panel sets to minimize material use. The methodology is validated through an analysis against an industry implementation and expert-derived solutions, highlighting PAAD's ability to surpass existing results and reduce the need for manual corrections. Additionally, to motivate future research, a synthetic data generator, the architectural drawing encodings used, and a preliminary interface are also introduced. This not only highlights the algorithm's practical applicability but also encourages its use in real-world scenarios. [ABSTRACT FROM AUTHOR]

We present a new generalization of the bin covering problem that is known to be a strongly NP-hard problem. In our generalization there is a positive constant Δ , and we are given a set of items each of which has a positive size. We would like to find a partition of the items into bins. We say that a bin is near exact covered if the total size of items packed into the bin is between 1 and 1 + Δ . Our goal is to maximize the number of near exact covered bins. If Δ = 0 or Δ > 0 is given as part of the input, our problem is shown here to have no approximation algorithm with a bounded asymptotic approximation ratio (assuming that P ≠ N P ). However, for the case where Δ > 0 is seen as a constant, we present an asymptotic fully polynomial time approximation scheme (AFPTAS) that is our main contribution. [ABSTRACT FROM AUTHOR]

We consider proof complexity in light of the unusual binary encoding of certain combinatorial principles. We contrast this proof complexity with the normal unary encoding in several refutation systems, based on Resolution and Sherali--Adams. We first consider Res(s), which is an extension of Resolution working on s-DNFs (Disjunctive Normal Form formulas). We prove an exponential lower bound of n \Omega (k)/\mathrm{d}(s) for the size of refutations of the binary version of the k-Clique Principle in Res(s), where s=o((log log n) 1/3) and d(s) is a doubly exponential function. Our result improves that of Lauria et al., who proved a similar lower bound for Res(1), i.e., Resolution. For the k-Clique and other principles we study, we show how lower bounds in Resolution for the unary version follow from lower bounds in Res(log n) for the binary version, so we start a systematic study of the complexity of proofs in Resolution-based systems for families of contradictions given in the binary encoding. We go on to consider the binary version of the (weak) Pigeonhole Principle Bin PHPmn. We prove that for any \delta, \epsilon >0, Bin PHPmn requires refutations of size 2n1 \delta in Res(s) for s =O(log 1 2 \epsilon n). Our lower bound cannot be improved substantially with the same method since for m\geq 2 \surd n\mathrm{l}\mathrm{o}\mathrm{g} n we can prove there are 2O(\surd n\mathrm{l}\mathrm{o}\mathrm{g} n) size refutations of Bin PHPmn in Res(log n). This is a consequence of the same upper bound for the unary weak Pigeonhole Principle of Buss and Pitassi. We contrast unary versus binary encoding in the Sherali--Adams (SA) refutation system where we prove lower bounds for both rank and size. For the unary encoding of the Pigeonhole Principle and the Ordering Principle, it is known that linear rank is required for refutations in SA, although both admit refutations of polynomial size. We prove that the binary encoding of the (weak) Pigeonhole Principle Bin PHPmn requires exponentially sized (in n) SA refutations, whereas the binary encoding of the Ordering Principle admits logarithmic rank, polynomially sized SA refutations. We continue by considering a natural refutation system we call "SA+Squares," which is intermediate between SA and Lasserre (Sum-of-Squares). This has been studied under the name static-LS\infty + by Grigoriev et al. In this system, the unary encoding of the Linear Ordering Principle LOPn requires O(n) rank while the unary encoding of the Pigeonhole Principle becomes constant rank. Since Potechin has shown that the rank of LOPn in Lasserre is O(\surd nlog n), we uncover an almost quadratic separation between SA+Squares and Lasserre in terms of rank. Grigoriev et al. noted that the unary Pigeonhole Principle has rank 2 in SA+Squares and therefore polynomial size. Since we show the same applies to the binary Bin PHPn+1 n, we deduce an exponential separation for size between SA and SA+Squares. [ABSTRACT FROM AUTHOR]

A homothetic packing of squares is any set of various-size squares with the same orientation where no two squares have overlapping interiors. If all n squares have the same size then we can have up to roughly 4 n contacts by arranging the squares in a grid formation. The maximum possible number of contacts for a set of n squares will drop drastically, however, if the size of each square is chosen more-or-less randomly. In the following paper we describe a necessary and sufficient condition for determining if a set of n squares with fixed sizes can be arranged into a homothetic square packing with more than 2 n − 2 contacts. Using this, we then prove that any (possibly not homothetic) packing of n squares will have at most 2 n − 2 face-to-face contacts if the various widths of the squares do not satisfy a finite set of linear equations. [ABSTRACT FROM AUTHOR]

Packing irregular objects composed by generalized spheres is considered. A generalized sphere is defined by an arbitrary norm. For three classes of packing problems, balance, homothetic and sparse packing, the corresponding new (generalized) models are formulated. Non-overlapping and containment conditions for irregular objects composed by generalized spheres are presented. It is demonstrated that these formulations can be stated for any norm. Different geometrical shapes can be treated in the same way by simply selecting a suitable norm. The approach is applied to generalized spheres defined by Lp norms and their compositions. Numerical solutions of small problem instances obtained by the global solver BARON are provided for two-dimensional objects composed by spheres defined in Lp norms to demonstrate the potential of the approach for a wide range of engineering optimization problems. [ABSTRACT FROM AUTHOR]

The bin packing is a well-known NP-Hard problem in the domain of artificial intelligence, posing significant challenges in finding efficient solutions. Conversely, recent advancements in quantum technologies have shown promising potential for achieving substantial computational speedup, particularly in certain problem classes, such as combinatorial optimization. In this study, we introduce QAL-BP, a novel Quadratic Unconstrained Binary Optimization (QUBO) formulation designed specifically for bin packing and suitable for quantum computation. QAL-BP utilizes the Augmented Lagrangian method to incorporate the bin packing constraints into the objective function while also facilitating an analytical estimation of heuristic, but empirically robust, penalty multipliers. This approach leads to a more versatile and generalizable model that eliminates the need for empirically calculating instance-dependent Lagrangian coefficients, a requirement commonly encountered in alternative QUBO formulations for similar problems. To assess the effectiveness of our proposed approach, we conduct experiments on a set of bin packing instances using a real Quantum Annealing device. Additionally, we compare the results with those obtained from two different classical solvers, namely simulated annealing and Gurobi. The experimental findings not only confirm the correctness of the proposed formulation, but also demonstrate the potential of quantum computation in effectively solving the bin packing problem, particularly as more reliable quantum technology becomes available. [ABSTRACT FROM AUTHOR]

Over the last decades, Structure from Motion (or image orientation) has been widely studied in the fields of photogrammetry and computer vision, mainly thanks to its feasibility for dealing with various image datasets (such as crowdsourced or UAV images). However, due to the fact that images are becoming easy to obtain, nowadays, it is challenging to deal with large scale of datasets, wherein the feature matching and pose-graph generation are the key limitations in terms of time efficiency. In this work, we proposed an efficient method to accelerate the generation of correspondences and two-view geometries. Specifically, based on some already estimated two-view geometries, unknown two-view geometries can be computed via A* algorithm. Then, the corresponding feature matching can be perform in a guided way using an epiploar-hash bins that is derived from the estimated two-view epipolar geometries. The experimental results demonstrated that, our method can improve the speed of generating pose-graph by 3–4 times comparing to two popular packages (colmap and OpenMVG) and is also faster than one SOTA method of Barath et al., (2021), yet the results of SfM are typically on par with them and reprojection error of our works are even better. [ABSTRACT FROM AUTHOR]

In online three-dimensional packing problems (3D-PPs), unlike offline problems, items arrive sequentially and require immediate packing decisions without any information about the quantities and sizes of the items to come. Heuristic methods are of great importance in solving online problems to find good solutions in a reasonable amount of time. However, the literature on heuristics for online problems is sparse. As our first contribution, we developed a pool of heuristics applicable to online 3D-PPs with complementary performance on different sets of instances. Computational results showed that in terms of the number of used bins, in all problem instances, at least one of our heuristics had a better or equal performance compared to existing heuristics in the literature. The developed heuristics are also fully applicable to an intermediate class between offline and online problems, referred to in this paper as a specific type of "semi-online with full look-ahead", which has several practical applications. In this class, as in offline problems, complete information about all items is known in advance (i.e., full look-ahead); however, due to time or space constraints, as in online problems, items should be packed immediately in the order of their arrival. As our second contribution, we presented an algorithm selection framework, building on developed heuristics and utilizing prior information about items in this specific class of problems. We used supervised machine learning techniques to find the relationship between the features of problem instances and the performance of heuristics and to build a prediction model. The results indicate an 88% accuracy in predicting (identifying) the most promising heuristic(s) for solving any new instance from this class of problems. • Several heuristic methods to solve online 3D-packing problems (3D-PPs) are proposed. • Proposed heuristics are applicable to semi-online 3D-PPs with full look-ahead. • An algorithm selection framework to predict the most promising heuristic is proposed. • The prediction performance of three learning algorithms is compared in the framework. • The multi-label neural network demonstrates a notably high prediction accuracy. [ABSTRACT FROM AUTHOR]

In this paper we study the problem of selfish bin packing with punishment. Different from the selfish bin packing problem proposed by Bilò in 2006, where the items are astute to minimize their shared proportional cost in a bin, we consider the case that items are into the utility defined as the load of bin it is packed in and, a unilateral moving may incur punishment. We define and consider several kinds of punishment, which can be roughly classified into two types: the punishment according to the behavior and the punishment based on the result. For both types we present the tight bound for Price of Anarchy (PoA) under the objective of minimizing the number of bins used. The results proved show that punishment does not always work, a punishment based on the result can be viewed as a delicate designed threat in advance which could perform much better, with an approximate 1.48 upper bound comparing to the optimal solution. • Initiate the consideration of punishment into the bin packing game. • Define three versions of punishment for the selfish bin packing problem. • Prove that punishment may not help to get better performance ratio. • Propose a dissuasive mechanism which could encourage agents to form a better equilibrium profile. [ABSTRACT FROM AUTHOR]