An exact algorithm for the budget-constrained multiple knapsack problem

This paper is concerned with a variant of the multiple knapsack problem (MKP), where knapsacks are available by paying certain ‘costs’, and we have a fixed budget to buy these knapsacks. Then, the problem is to determine the set of knapsacks to be purchased, as well as to allocate items into the accepted knapsacks in such a way as to maximize the net total profit. We call this the budget-constrained MKP and present a branch-and-bound algorithm to solve this problem to optimality. We employ the Lagrangian relaxation approach to obtain an upper bound. Together with the lower bound obtained by a greedy heuristic, we apply the pegging test to reduce the problem size. Next, in the branch-and-bound framework, we make use of the Lagrangian multipliers obtained above for pruning subproblems, and at each terminal subproblem, we solve MKP exactly by calling the MULKNAP code [D. Pisinger, An exact algorithm for large multiple knapsack problem, European J. Oper. Res. 114 (1999), pp. 528–541]. Thus, we were able to solve test problems with up to 160,000 items and 150 knapsacks within a few minutes in our computing environment. However, solving instances with relatively large number of knapsacks, when compared with the number of items, still remains hard. This is due to the weakness of MULKNAP to this type of problems, and our algorithm inherits this weakness as well.     

离散杂草优化算法在0/1背包问题中的应用

为解决粒子群优化算法在求解0/1背包问题中的早熟收敛问题,将杂草优化算法应用到离散问题,提出了一种离散杂草优化算法(DIWO)。根据组合优化问题的特点,对原算法中正态分布于父代周围的子代进行离散化分析,引入遗传操作中的一种改进的变异机制,保证了新算法的有效性,使其具有局部的随机搜索能力。通过三个仿真实例验证,对比粒子群算法,新算法在种群数量较小、迭代次数较少的情况下能取得更好的结果。     

A beam search approach to the container loading problem

The single container loading problem is a three-dimensional packing problem in which a container has to be filled with a set of boxes. The objective is to maximize the space utilization of the container. This problem has wide applications in the logistics industry. In this work, a new constructive approach to this problem is introduced. The approach uses a beam search strategy. This strategy can be viewed as a variant of the branch-and-bound search that only expands the most promising nodes at each level of the search tree. The approach is compared with state-of-the-art algorithms using 16 well-known sets of benchmark instances. Results show that the new approach outperforms all the others for each set of instances.     

LP-based bounds for the container and multi-container loading problem

New relaxations are developed in this paper for problems of optimal packing of small (rectangular-shaped) pieces within one or several larger containers. Based on these relaxations tighter bounds for the Container Loading Problem (CLP) and the Multi-Container Loading Problem (MCLP) are obtained.
The new relaxations for the CLP and MCLP lead to linear programming problems. A corresponding solution approach is discussed which is based on a column generation technique. Results of computational tests are also given.     

A parallel hybrid local search algorithm for the container loading problem

In this contribution, a parallel hybrid local search algorithm for the three‐dimensional container loading problem (CLP) is proposed. First a simulated annealing method for the CLP is developed, which is then combined with an existing tabu search algorithm to form a hybrid metaheuristic. Finally, parallel versions are introduced for these algorithms. The emphasis is on CLP instances with a weakly heterogeneous load. Numerical tests based on the well‐known 700 test instances from Bischoff and Ratcliff are performed, and the outcome is compared with methods from other authors. The results show a high solution quality obtained with reasonable computing time.     

A hybrid differential evolution algorithm for multiple container loading problem with heterogeneous containers

We consider a multiple container loading problem, commonly known as the three-dimensional bin packing problem (3D-BPP), which deals with maximizing container space utilization while the containers available for packing are heterogeneous, i.e., varying in size. The problem has wide applications in cargo transportation, warehouse management, medical packaging, and so on. We develop a differential evolution (DE) algorithm hybridized with a novel packing heuristic strategy, best-match-first (BMF), which generates a compact packing solution based on a given box packing sequence and a container loading sequence. The effectiveness of the proposed algorithm is evaluated on a set of industrial instances and randomly generated instances. The results show that the proposed algorithm outperforms existing solution approaches in terms of solution quality.     

An approximate dynamic programming approach to convex quadratic knapsack problems

Quadratic knapsack problem (QKP) has a central role in integer and combinatorial optimization, while efficient algorithms to general QKPs are currently very limited. We present an approximate dynamic programming (ADP) approach for solving convex QKPs where variables may take any integer value and all coefficients are real numbers. We approximate the function value using (a) continuous quadratic programming relaxation (CQPR), and (b) the integral parts of the solutions to CQPR. We propose a new heuristic which adaptively fixes the variables according to the solution of CQPR. We report computational results for QKPs with up to 200 integer variables. Our numerical results illustrate that the new heuristic produces high-quality solutions to large-scale QKPs fast and robustly.     

Asymptotically optimal algorithms for approximate agreement

This paper introduces some algorithms to solve crash-failure, failure-by-omission and Byzantine failure versions of the Byzantine Generals or consensus problem, where non-faulty processors need only arrive at values that are close together rather than identical. For each failure model and each value ofS, we give at-resilient algorithm usingS rounds of communication. IfS=t+1, exact agreement is obtained. In the algorithms for the failure-by-omission and Byzantine failure models, each processor attempts to identify the faulty processors and corrects values transmited by them to reduce the amount of disagreement. We also prove lower bounds for each model, to show that each of our algorithms has a convergence rate that is asymptotic to the best possible in that model as the number of processors increases. Alan Fekete was born in Sydney Australia in 1959. He studied Pure Mathematics and Computer Science at the University of Sydney, obtaining a B.Sc.(Hons) in 1982. He then moved to Cambridge, Massachusetts, where he obtained a distributed Ph.D. degree, awarded by Harvard University's Mathematics department for work supervised by Nancy Lynch in M.I.T.'s Laboratory for Computer Science. He spend the year 1987–1988 at M.I.T. as a postdoctoral Research Associate, and is now Lecturer in Computer Science at the University of Sydney. His research concentrates on understanding the modularity in distributed algorithms, especially those used for concurrency control in distributed databases.A preliminary version of this paper has appeared in the Proceedings of the 5th ACM Symposium on Principles of Distributed Computing (August 1986). This work was begun in the Department of Mathematics, Harvard University, and completed at the Laboratory for Computer Science at Massachusetts Institute of Technology. The work was supported in part (through Professor N. Lynch) by the Office of Naval Research under Contract N00014-85-K-0168, by the Office of Army Research under contract DAAG29-84-K-0058, by the National Science Foundation under Grants MCS-8306854, DCR-83-02391, and CCR-8611442, and by the Defense Advanced Research Projects Agency (DARPA) under Contract N00014-83-K-0125     

Problem reduction heuristic for the 0-1 multidimensional knapsack problem

This paper introduces new problem-size reduction heuristics for the multidimensional knapsack problem. These heuristics are based on solving a relaxed version of the problem, using the dual variables to formulate a Lagrangian relaxation of the original problem, and then solving an estimated core problem to achieve a heuristic solution to the original problem. We demonstrate the performance of these heuristics as compared to legacy heuristics and two other problem reduction heuristics for the multi-dimensional knapsack problem. We discuss problems with existing test problems and discuss the use of an improved test problem generation approach. We use a competitive test to highlight the performance of our heuristics versus the legacy heuristic approaches. We also introduce the concept of computational versus competitive problem test data sets as a means to focus the empirical analysis of heuristic performance.     

浅谈求解组合优化问题的几种近似算法

现实生活中,为了最大限度地利用资源、节省开支,出现了许多最优化利用资源的问题,往往是要求求出最大值或最小值的。在优化问题中,比较常见的是组合优化问题。针对此类问题,也出现了不少求解的算法。该文对其中比较常用的几种近似算法进行了总结,并通过一种典型的组合优化问题——装箱问题的实例对各算法的优劣进行了比较。     

A heuristic for the skiving and cutting stock problem in paper and plastic film industries

This paper investigates the skiving and cutting stock problem (SCSP) encountered in the paper and plastic film industries, in which a set of nonstandard reels generated from previous cutting processes are used to produce finished rolls through the skiving and cutting process. First, reels are skived together lengthwise to form a reel‐pyramid (a polygon), and then the reel‐pyramid is cut into finished rolls of small widths. Depending on if a reel can be divided lengthwise into subreels to form the reel‐pyramid, the problem can be classified into divisible SCSP (DSCSP) and indivisible SCSP (ISCSP). In this paper, two integer programming (IP) models are proposed for DSCSP and ISCSP, respectively. A sequential value correction procedure combined with the two IP models (SVCTIP) is developed to solve the two SCSPs. The effectiveness of the SVCTIP is demonstrated through extensive computational tests.     

多选择背包问题的人工蜂群算法*

多选择背包问题是组合优化中的NP难题之一,采用一种新的智能优化算法——人工蜂群算法进行求解。该算法通过雇佣蜂、跟随蜂和侦察蜂的局部寻优来实现全局最优。基于算法实现的核心思想,用MATLAB编程实现,对参考文献的算例进行仿真测试。与其他算法进行了比较,获得了满意的结果。这说明了算法在解决该问题上的可行性与有效性,拓展了人工蜂群算法的应用领域。     

A review of stochastic algorithms with continuous value function approximation and some new approximate policy iteration algorithms for multidimensional continuous applications

We review the literature on approximate dynamic programming, with the goal of better understanding the theory behind practical algorithms for solving dynamic programs with continuous and vector-valued states and actions and complex information processes. We build on the literature that has addressed the well-known problem of multidimensional (and possibly continuous) states, and the extensive literature on model-free dynamic programming, which also assumes that the expectation in Bellman’s equation cannot be computed. However, we point out complications that arise when the actions/controls are vector-valued and possibly continuous. We then describe some recent research by the authors on approximate policy iteration algorithms that offer convergence guarantees (with technical assumptions) for both parametric and nonparametric architectures for the value function.     

Solution of a Euclidean Combinatorial Optimization Problem by the Dynamic-Programming Method

A class of Euclidean combinatorial optimization problems is selected that can be solved by the dynamic programming method. The problem of allocation of servicing enterprises is solved as an example.     

Performance evaluation of vector implementations of combinatorial algorithms

We study the performance and the use of vector computers for the solution of combinatorial optimization problems, particularly dynamic programming and shortest path problems. A general model for performance evaluation and vector implementations for the problems described above are studied. These implementations were done on a CRAY-1 vector computer and the computational results obtained show (i) the adequacy of the performance evaluation model and (ii) very important gains concerning computing times, showing that vector computers will be of great importance in the field of combinatorial optimization.     

Approximate Dynamic Programming for Stochastic Resource Allocation Problems

A stochastic resource allocation model, based on the principles of Markov decision processes (MDPs), is proposed in this paper. In particular, a general-purpose framework is developed, which takes into account resource requests for both instant and future needs. The considered framework can handle two types of reservations (i.e., specified and unspecified time interval reservation requests), and implement an overbooking business strategy to further increase business revenues. The resulting dynamic pricing problems can be regarded as sequential decision-making problems under uncertainty, which is solved by means of stochastic dynamic programming (DP) based algorithms. In this regard, Bellman’s backward principle of optimality is exploited in order to provide all the implementation mechanisms for the proposed reservation pricing algorithm. The curse of dimensionality, as the inevitable issue of the DP both for instant resource requests and future resource reservations, occurs. In particular, an approximate dynamic programming (ADP) technique based on linear function approximations is applied to solve such scalability issues. Several examples are provided to show the effectiveness of the proposed approach.      

A matheuristic for the firefighter problem on graphs

In this paper, we describe a computational study conducted on The Firefighter Problem (FFP ), which models fire spreading and containment in a graph. Once the fire breaks out on a set of vertices, the goal is to save as many vertices as possible with limited resources. Practical applications of the FFP occur in areas such as disease control and network security. The FFP is NP‐hard and heuristics have been proposed earlier. Our main contributions include improvements to an existing integer linear programming formulation that led to an average speedup of two to compute exact solutions. Moreover, we developed a novel matheuristic, a technique based on the interoperation between metaheuristics and mathematical programming. We performed extensive experiments on public benchmarks both for parameter tuning and for comparison of our results with those from the literature. A rigorous statistical analysis shows that our new matheuristic outperforms the existing approaches.     

求解背包问题的一种改进遗传算法

讨论了遗传算法在问题求解中的早熟现象,引进一个参数用以衡量种群中染色体的相似程度,用以增加种群的多样性;在杂交和变异运算过程中,混合了模拟退火思想作为新个体的接受准则;通常的变异算子需要扫描每一个染色体中每一个等位基因,提出一种新的变异方式,大大提高了算法搜索效率。通过实际计算比较表明,该改进遗传算法在背包问题求解中具有很好的收敛性、稳定性和计算效率。     

Shift-and-merge technique for the DP solution of the time-constrained backpacker problem

We formulate the time-constrained backpacker problem as an extension of the classical knapsack problem (KP), where a ‘backpacker’ travels from a origin to a destination on a directed acyclic graph, and collects items en route within the capacity of his knapsack and within a fixed time limit. We present a dynamic programming (DP) algorithm to solve this problem to optimality, and a ‘shift-and-merge’ DP algorithm to solve larger instances. The latter is an extension of the list-type DP, which has been successful for one-dimensional KPs, to the two-dimensional case. Computational experiments on a series of instances demonstrate advantage of the shift-and-merge technique over commercial MIP solvers.