Bin packing and related problems: General arc-flow formulation with graph compression

We present an exact method, based on an arc-flow formulation with side constraints, for solving bin packing and cutting stock problems—including multi-constraint variants—by simply representing all the patterns in a very compact graph. Our method includes a graph compression algorithm that usually reduces the size of the underlying graph substantially without weakening the model.Our formulation is equivalent to Gilmore and Gomory׳s, thus providing a very strong linear relaxation. However, instead of using column-generation in an iterative process, the method constructs a graph, where paths from the source to the target node represent every valid packing pattern.The same method, without any problem-specific parameterization, was used to solve a large variety of instances from several different cutting and packing problems. In this paper, we deal with vector packing, bin packing, cutting stock, cardinality constrained bin packing, cutting stock with cutting knife limitation, bin packing with conflicts, and other problems. We report computational results obtained with many benchmark test datasets, some of them showing a large advantage of this formulation with respect to the traditional ones.     

Approximate algorithms for the container loading problem

In this paper we develop several algorithms for solving three–dimensional cutting/packing problems. We begin by proposing an adaptation of the approach proposed in Hifi and Ouafi (1997) for solving two–staged unconstrained two–dimensional cutting problems. We show how the algorithm can be polynomially solved for producing a constant approximation ratio. We then extend this algorithm for developing better approximate algorithms. By using hill–climbing strategies, we construct some heuristics which produce a good trade–off between the computational time and the solution quality. The performance of the proposed algorithms is evaluated on different problem instances of the literature, with different sizes and densities (a total of 144 problem instances).     

Arc-flow model for the two-dimensional guillotine cutting stock problem

We describe an exact model for the two-dimensional cutting stock problem with two stages and the guillotine constraint. It is an integer linear programming (ILP) arc-flow model, formulated as a minimum flow problem, which is an extension of a model proposed by Valério de Carvalho for the one dimensional case. In this paper, we explore the behavior of this model when it is solved with a commercial software, explicitly considering all its variables and constraints. We also derive a new family of cutting planes and a new lower bound, and consider some variants of the original problem. The model was tested on a set of real instances from the wood industry, with very good results. Furthermore the lower bounds provided by the linear programming relaxation of the model compare favorably with the lower bounds provided by models based on assignment variables.     

Tight bounds for the identical parallel machine-scheduling problem: Part II

A companion paper introduces new lower bounds and heuristics for the problem of minimizing makespan on identical parallel machines. The objective of this paper is threefold. First, we describe further enhancements of previously described lower bounds. Second, we propose a new heuristic that requires solving a sequence of 0–1 knapsack problems. Finally, we show that embedding these newly derived bounds in a branch‐and‐bound procedure yields a very effective exact algorithm. Moreover, this algorithm features a new symmetry‐breaking branching strategy. We present the results of computational experiments that were carried out on a large set of instances and that attest to the efficacy of the proposed algorithm. In particular, we report proven optimal solutions for some benchmark problems that have been open for some time.     

An efficient artificial bee colony algorithm with application to nonlinear predictive control

In this paper a constrained nonlinear predictive control algorithm, that uses the artificial bee colony (ABC) algorithm to solve the optimization problem, is proposed. The main objective is to derive a simple and efficient control algorithm that can solve the nonlinear constrained optimization problem with minimal computational time. Indeed, a modified version, enhancing the exploring and the exploitation capabilities, of the ABC algorithm is proposed and used to design a nonlinear constrained predictive controller. This version allows addressing the premature and the slow convergence drawbacks of the standard ABC algorithm, using a modified search equation, a well-known organized distribution mechanism for the initial population and a new equation for the limit parameter. A convergence statistical analysis of the proposed algorithm, using some well-known benchmark functions is presented and compared with several other variants of the ABC algorithm. To demonstrate the efficiency of the proposed algorithm in solving engineering problems, the constrained nonlinear predictive control of the model of a Multi-Input Multi-Output industrial boiler is considered. The control performances of the proposed ABC algorithm-based controller are also compared to those obtained using some variants of the ABC algorithms.     

Heuristics for two-dimensional knapsack and cutting stock problems with items of irregular shape

In this paper, the two-dimensional cutting/packing problem with items that correspond to simple polygons that may contain holes are studied in which we propose algorithms based on no-fit polygon computation. We present a GRASP based heuristic for the 0/1 version of the knapsack problem, and another heuristic for the unconstrained version of the knapsack problem. This last heuristic is divided in two steps: first it packs items in rectangles and then use the rectangles as items to be packed into the bin. We also solve the cutting stock problem with items of irregular shape, by combining this last heuristic with a column generation algorithm. The algorithms proposed found optimal solutions for several of the tested instances within a reasonable runtime. For some instances, the algorithms obtained solutions with occupancy rates above 90% with relatively fast execution time.     

The usable leftover one‐dimensional cutting stock problem—a priority‐in‐use heuristic

We consider a one‐dimensional cutting stock problem in which the material not used in the cutting patterns, if large enough, is kept for use in the future. Moreover, it is assumed that leftovers should not remain in stock for a long time, hence, such leftovers have priority‐in‐use compared to standard objects (objects bought by the industry) in stock. A heuristic procedure is proposed for this problem, and its performance is analyzed by solving randomly generated dynamic instances where successive problems are solved in a time horizon. For each period, new demands arise and a new problem is solved on the basis of the information about the stock of the previous periods (remaining standard objects in the stock) and usable leftovers generated during those previous periods. The computational experiments show that the solutions presented by the proposed heuristic are better than the solutions obtained by other heuristics from the literature.     

一维下料的改进遗传算法优化

针对一维下料优化问题,在对一维下料方案数学模型分析的基础上,提出了基于改进遗传算法的优化求解方案。主要思想是把零件的一个顺序作为一种下料方案,定义了遗传算法中的关键问题:编码、解码方法、遗传算子和适应度函数的定义。该算法设计了一种新颖的遗传算子,包括顺序交叉算子、线性变异算子、扩展选择算子。根据这一算法开发出了一维下料方案的优化系统。实际应用表明,该算法逼近理论最优值,而且收敛速度快,较好地解决了一维下料问题。     

An efficient deterministic heuristic for two-dimensional rectangular packing

This paper proposes a deterministic heuristic, a best fit algorithm (BFA), for solving the NP-hard two-dimensional rectangular packing problem to maximize the filling rate of a rectangular sheet. There are two stages in this new approach: the constructive stage and the tree search stage. The former aims to rapidly generate an initial solution by employing the concepts of action space and fit degree in evaluating different placements. The latter seeks to further improve the solution and searches for promising placements by a partial tree search procedure. We then compare BFA with other approaches in terms of solution quality and computing time. We carry out computational experiments on two sets of well-known benchmark instances, C21 proposed by Hopper and Turton, and N13 proposed by Burke et al. BFA gained an average filling rate of 100% for the C21 instances within short times, indicating that all the layouts obtained are optimal. To the best of our knowledge, this is the first time that optimal layouts on all the 21 instances were obtained by a deterministic algorithm. As for the N13 instances, to date, researchers have found optimal solutions to the first three instances, whereas BFA solved seven, including the first three, within a reasonable period. An additional work is to adapt BFA to solve a relevant problem, the constrained two-dimensional cutting (or packing) problem (CTDC). Though BFA is not for the CTDC in the original design such that some specific characteristics of CTDC are not considered, the adapted algorithm still performed well on 21 public CTDC instances.     

On the concept of density control and its application to a hybrid optimization framework: Investigation into cutting problems

The Generate-and-Solve (GS) methodology is a hybrid approach that combines a metaheuristic component with an exact solver. GS has been recently introduced in the literature in order to solve cutting and packing problems, showing promising results. The GS framework includes a metaheuristic engine (e.g., a genetic algorithm) that works as a generator of reduced instances of the original optimization problem, which are, in turn, formulated as mathematical programming problems and solved by an integer programming solver. In this paper, we present an extended version of GS, focusing primarily on the concept of a new Density Control Operator (DCO). The role of this operator is to adaptively control the dimension of the reduced instances in such a way as to allow a much steadier progress towards a better solution, thereby avoiding premature convergence. In order to assess the potentials of this novel version of the GS methodology, we conducted computational experiments on a set of difficult benchmark instances of the constrained non-guillotine cutting problem. The results achieved are quantitatively and qualitatively discussed in terms of effectiveness and efficiency, showing that the proposed variant of the GS hybridization framework is highly suitable when effectiveness is a major requirement.     

Heuristics for the one-dimensional cutting stock problem with limited multiple stock lengths

This paper deals with the classical one-dimensional integer cutting stock problem, which consists of cutting a set of available stock lengths in order to produce smaller ordered items. This process is carried out in order to optimize a given objective function (e.g., minimizing waste). Our study deals with a case in which there are several stock lengths available in limited quantities. Moreover, we have focused on problems of low demand. Some heuristic methods are proposed in order to obtain an integer solution and compared with others. The heuristic methods are empirically analyzed by solving a set of randomly generated instances and a set of instances from the literature. Concerning the latter, most of the optimal solutions of these instances are known, therefore it was possible to compare the solutions. The proposed methods presented very small objective function value gaps.     

A Branch and Bound Algorithm for An Uncapacitated Facility Location Problem with a Side Constraint

In this paper, a branch and bound algorithm for solving an uncapacitated facility location problem (UFLP) with an aggregate capacity constraint is presented. The problem arises as a subproblem when Lagrangean relaxation of the capacity constraints is used to solve capacitated facility location problems. The algorithm is an extension of a procedure used by Christofides and Beasley (A tree search algorithm for the p-median problem. European Journal of Operational Research , Vol. 10, 1982, pp. 196–204) to solve p -median problems and is based on Lagrangean relaxation in combination with subgradient optimization for lower bounding, simple Lagrangean heuristics to produce feasible solutions, and penalties to reduce the problem size. For node selection, a jump-backtracking rule is proposed, and alternative rules for choosing the branching variable are discussed. Computational experience is reported.     

Algorithms for 3D guillotine cutting problems: Unbounded knapsack, cutting stock and strip packing

We present algorithms for the following three-dimensional (3D) guillotine cutting problems: unbounded knapsack, cutting stock and strip packing. We consider the case where the items have fixed orientation and the case where orthogonal rotations around all axes are allowed. For the unbounded 3D knapsack problem, we extend the recurrence formula proposed by [1] for the rectangular knapsack problem and present a dynamic programming algorithm that uses reduced raster points. We also consider a variant of the unbounded knapsack problem in which the cuts must be staged. For the 3D cutting stock problem and its variants in which the bins have different sizes (and the cuts must be staged), we present column generation-based algorithms. Modified versions of the algorithms for the 3D cutting stock problems with stages are then used to build algorithms for the 3D strip packing problem and its variants. The computational tests performed with the algorithms described in this paper indicate that they are useful to solve instances of moderate size.     

Designing radio-mobile access networks based on synchronous digital hierarchy rings

In this paper, we address the SDH network design problem (SDHNDP) which arises while designing the fixed part of global system for mobile communications access networks using synchronous digital hierarchy (SDH) rings.An SDH ring is a simple cycle that physically links a subset of antennae to a single concentrator. Inside a ring, a concentrator handles the total traffic induced by antennae. Technological considerations limit the number of antennae and the total length of a ring.The SDHNDP is a new problem. It belongs to a class of location-routing problems that introduce location into the multi-depot vehicle routing problem. In this paper, we precisely describe the SDHNDP and propose a mixed integer programming-based model for it. Furthermore, we devise a heuristic algorithm that computes a feasible solution.We report the results of our computational experiments using the CPLEX software, on instances comprising up to 70 antennae or six concentrator sites. An analysis provides insight into the behavior of the lower bound obtained by the LP relaxation of the model, in response to the network density. This lower bound can be improved by adding some valid inequalities. We show that an interesting cut can be obtained by approximating the minimum number of rings in any feasible solution. This can be achieved by solving a “minimum capacitated partition problem”. Finally, we compare the lower bound to the heuristic solution value for a set of instances.     

Exact Solution of Cutting Stock Problems Using Column Generation and Branch-and-Bound

This paper describes an attempt to solve the one-dimensional cutting stock problem exactly, using column generation and branch-and-bound. A new formulation is introduced for the one-dimensional cutting stock problem that uses general integer variables, not restricted to be binary. It is an arc flow formulation with side constraints, whose linear programming relaxation provides a strong lower bound. In this model, a cutting pattern, which corresponds to a path, is decomposed into single arc variables. The decomposition serves the purpose of showing that it is possible to combine the branch-and-bound method with variable generation. Computational times are reported for one-dimensional cutting stock instances with a number of orders up to 30.     

求解#SMT问题的局部搜索算法

#SMT问题是SMT问题的扩展,它需要计算一阶逻辑公式F所有可满足解的个数.目前,该问题已被广泛应用于编译器优化、硬件设计、软件验证和自动化推理等领域.随着#SMT问题的广泛应用,设计可以求解较大规模#SMT实例的求解器亟待解决.基于以上原因,设计了一种求解较大规模#SMT实例的近似求解器——VolComputeWithLocalSearch.它在现有的#SMT精确求解算法的基础上加入差分进化算法,通过调用体积计算工具qhull,进而给出#SMT问题的近似解.算法采用群体规则减少体积计算的次数,差分进化方法快速地枚举各个有解的区域.另外,从理论上证明了VolComputeWithLocalSearch求解器可以得到精确解的下界,使其可以应用在软件测试等只需要知道问题下界的领域.实验结果表明：VolComputeWithLocalSearch求解器是稳定的、具有快速的求解能力,并在高维问题上具有很好的表现.     

Lower tolerance-based Branch and Bound algorithms for the ATSP

In this paper, we develop a new tolerance-based Branch and Bound algorithm for solving NP-hard problems. In particular, we consider the asymmetric traveling salesman problem (ATSP), an NP-hard problem with large practical relevance. The main algorithmic contribution is our lower bounding strategy that uses the expected costs of including arcs in the solution to the assignment problem relaxation of the ATSP, the so-called lower tolerance values. The computation of the lower bound requires the calculation of a large set of lower tolerances. We apply and adapt a finding from [23] that makes it possible to compute all lower tolerance values efficiently. Computational results show that our Branch and Bound algorithm exhibits very good performance in comparison with state-of-the-art algorithms, in particular for difficult clustered ATSP instances.     

Two-machine group scheduling problems in discrete parts manufacturing with sequence-dependent setups

This paper focuses on minimizing the total completion time in two-machine group scheduling problems with sequence-dependent setups that are typically found in discrete parts manufacturing. As the problem is characterized as strongly NP-hard, three search algorithms based on tabu search are developed for solving industry-size scheduling problems. Four different lower bounding mechanisms are developed to identify a lower bound for all problems attempted, and the largest of the four is aptly used in the evaluation of the percentage deviation of the search algorithms to assess their efficacy. The problem sizes are classified as small, medium and large, and to accommodate the variability that might exist in the sequence-dependent setup times on both machines, three different scenarios are considered. Such finer levels of classification have resulted in the generation of nine different categories of problem instances, thus facilitating the performance of a very detailed statistical experimental design to assess the efficacy and efficiency of the three search algorithms. The search algorithm based on long-term memory with maximal frequencies either recorded a statistically better makespan or one that is indifferent when compared with the other two with all three scenarios and problem sizes. Hence, it is recommended for solving the research problem. Under the three scenarios, the average percentage deviation for all sizes of problem instances solved has been remarkably low. In particular, a mathematical programming based lower bounding mechanism, which focuses on converting (reducing) the original sequence-dependent group scheduling problem with several jobs in each group to a sequence-dependent job scheduling problem, has served well in identifying a high quality lower bound for the original problem, making it possible to evaluate a lower average percentage deviation for the search algorithm. Also, a 16–17-fold reduction in average computation time for solving a large problem instance with the recommended search algorithm compared with identifying just the lower bound of (not solving) the same instance by the mathematical programming based mechanism speaks strongly in favor of the search algorithm for solving industry-size group scheduling problems.     

矩形件简单块占角排样方式的动态规划

目的 针对矩形件无约束2维剪切排样问题,提出一种可简化板材切割工艺的简单块占角排样方式,并构造这种排样方式的动态规划生成算法。方法 该排样方式在板材左下角按照简单块方式排样若干行若干列同种矩形件,将板材剩余部分划分为两个子板;将子板按照上述方法继续递归排样和划分,直至子板排满矩形件为止。采用动态规划确定所有可能尺寸的板材左下角排样的最优矩形件、矩形件的最优行列数和板材剩余部分的最优子板划分。运用规范尺寸排除不必要的计算。结果 将本文算法与目前常见的算法进行比较,实验结果表明本文算法计算时间合理,排样价值较高。在第1组41道基准例题中,本文算法所有例题均求出了精确解,同质块T型算法、同质块两段算法和复合条带两段算法分别有7道、5道和4道例题未求出精确解。在第2组20道基准例题中,本文算法只有1道例题未求出精确解,普通三阶段算法、同质块T型算法、同质块两段算法和匀质条带三块算法分别有18道、15道、15道和20道例题未求出精确解。在第3组50道随机例题中,本文算法、普通两段算法和同质块两段算法板材利用率分别为99.913 7%、99.862 3%和99.796 1%。在第4组31道基准例题中,本文算法所有例题均求出了精确解,普通占角排样算法有2道例题未求出精确解。结论 本文算法计算时间远小于精确算法,优化效果接近精确算法;本文算法计算时间与多种启发式算法接近,但优化效果好于多种启发式算法。     

Genotype–phenotype heuristic approaches for a cutting stock problem with circular patterns

The cutting stock problem has been studied in the context of different industrial applications inducing NP-hard problems in most instances. However, the application in sawmill has not received the same attention. In this paper, we deal with the problem of determining the number of logs to cut over a period of several days and the geometry of sawmill patterns in order to satisfy the demand while minimizing the loss of material. First, the problem is formulated as an integer programming problem of the form of a constrained set covering problem where the knowledge of a priori cutting patterns is necessary to generate its columns. In our implementation, these patterns are obtained using a genetic algorithm (GA) or a simulated annealing method (SA). Then, two different approaches are introduced to solve the problem. The first approach includes two methods that combine a metaheuristic to generate the number of logs and a constructive heuristic to generate the cutting patterns for each of the logs. In the second approach, we use an exact procedure CPLEX to solve the integer programming model where the cutting patterns are generated with the GA method (GA+CPLEX) or the SA method (SA+CPLEX). These four methods are compared numerically on 11 semi-randomly generated problems similar to those found in real life. The best results for the loss are obtained with the two-stage GA+CPLEX approach that finds the best values for 7 problems.