供应链环境下需求不确定的批量生产计划

针对由单个制造商、单一产品和多个客户构成的供应链系统,建立了分散控制下系统利润最大化模型,提出了新的客户选择可变方案,分别设计了遗传算法和分枝定界法对问题进行了求解。通过实例仿真与前人提出的客户选择不可变方案进行了比较分析,结果证明,分枝定界法更适合求解规模较小的问题,而遗传算法可以通过调整种群规模和遗传算子来解决规模较大的问题;与客户选择不可变相比,当客户选择可变时,系统能获取较大的期望利润。     

Optimization of Dynamic Hardware Reconfigurations

Recent generations of Field Programmable Gate Arrays (FPGA) allow the dynamic reconfiguration of cells on the chip during run-time. For a given problem consisting of a set of tasks with computation requirements modeled by rectangles of cells, several optimization problems such as finding the array of minimal size to accomplish the tasks within a given time limit are considered. Existing approaches based on ILP formulations to solve these problems as multi-dimensional packing problems turn out not to be applicable for problem sizes of interest. Here, a breakthrough is achieved in solving these problems to optimality by using the new notion of packing classes. It allows a significant reduction of the search space such that problems of the above type may be solved exactly using a special branch-and-bound technique. We validate the usefulness of our method by providing computational results.     

A Best-first Branch-and-bound Algorithm for Orthogonal Rectangular Packing Problems

In this paper we discuss the problem of packing a set of small rectangles (pieces) in an enclosing final rectangle. We present first a best-first branch-and-bound exact algorithm and second a heuristic approach in order to solve exactly and approximately this problem. The performances of the proposed approaches are evaluated on several randomly generated problem instances. Computational results show that the proposed exact algorithm is able to solve small and medium problem instances within reasonable execution time. The derived heuristic performs very well in the sense that it produces high-quality solutions within small computational time.     

求解矩形装箱问题的一种近似算法

提出了利用近似算法求解二维矩形装箱问题的最小高度的一种方法。该方法基于启发式递归策略和遗传算法。利用启发式递归策略把所有大小各异的矩形都装入宽度固定的矩形容器中，并计算装完后所需容器的高度，用遗传算法的进化能力优化高度，使得所需容器的高度尽可能小。计算数据证明这种方法能够得到很好的结果，特别是对数据量大的测试问题，效果更好。     

A branch-and-bound algorithm for single machine scheduling with quadratic earliness and tardiness penalties

This paper considers the problem of scheduling a single machine, in which the objective function is to minimize the weighted quadratic earliness and tardiness penalties and no machine idle time is allowed. We develop a branch and bound algorithm involving the implementation of lower and upper bounding procedures as well as some dominance rules. The lower bound is designed based on a lagrangian relaxation method and the upper bound includes two phases, one for constructing initial schedules and the other for improving them. Computational experiments on a set of randomly generated instances show that one of the proposed heuristics, used as an upper bound, has an average gap less than 1.3% for instances optimally solved. The results indicate that both the lower and upper bounds are very tight and the branch-and-bound algorithm is the first algorithm that is able to optimally solve problems with up to 30 jobs in a reasonable amount of time.     

A recursive branch-and-bound algorithm for the rectangular guillotine strip packing problem

A heuristic recursive algorithm for the two-dimensional rectangular strip packing problem is presented. It is based on a recursive structure combined with branch-and-bound techniques. Several lengths are tried to determine the minimal plate length to hold all the items. Initially the plate is taken as a block. For the current block considered, the algorithm selects an item, puts it at the bottom-left corner of the block, and divides the unoccupied region into two smaller blocks with an orthogonal cut. The dividing cut is vertical if the block width is equal to the plate width; otherwise it is horizontal. Both lower and upper bounds are used to prune unpromising branches. The computational results on a class of benchmark problems indicate that the algorithm performs better than several recently published algorithms.     

Scheduling two-machine no-wait open shops to minimize makespan

This paper examines the problem of scheduling two-machine no-wait open shops to minimize makespan. The problem is known to be strongly NP-hard. An exact algorithm, based on a branch-and-bound scheme, is developed to optimally solve medium-size problems. A number of dominance rules are proposed to improve the search efficiency of the branch-and-bound algorithm. An efficient two-phase heuristic algorithm is presented for solving large-size problems. Computational results show that the branch-and-bound algorithm can solve problems with up to 100 jobs within a reasonable amount of time. For large-size problems, the solution obtained by the heuristic algorithm has an average percentage deviation of 0.24% from a lower bound value.     

用于二维不规则排样的离散临界多边形模型

提出了一个用于求解二维不规则排样问题的离散临界多边形模型.Burke等人的BLF算法是求解排样问题的一种有效算法,但其算法对一些特殊实例会产生非法的解.为了解决这个问题,提出了一种基于离散临界多边形模型,并对其正确性作了严格证明.新模型是只含有点和区间的简单模型,在大大降低原问题几何复杂性的同时,也使许多启发式策略可以更容易地求解该问题.计算结果表明,基于离散临界多边型模型的排样算法是很有效的.     

Branch-and-bound algorithm for the maximum triangle packing problem

This work addresses the problem of finding the maximum number of unweighted vertex-disjoint triangles in an undirected graph G. It is a challenging NP-hard combinatorial problem and it is well-known to be APX-hard. A branch-and-bound algorithm which uses a lower bound based on neighborhood degree is presented. A naive upper bound is proposed as well as another one based on a surrogate relaxation of the related integer linear program which is analogous to a multidimensional knapsack problem. Further, a Greedy Search algorithm and a genetic algorithm are described to improve the lower bound. A computational comparison of lower bounds, branch-and-bound algorithm and CPLEX solver is provided using randomly generated benchmarks and well-known DIMACS implementation challenges. The empirical study shows that the branch-and-bound finds the optimal triangle packing solution for small randomly generated MTP instances (up to 100 vertices and 200 triangles) and some DIMACS graphs. For some larger instances and DIMACS challenges graphs, we remark that our lower bound outperforms CPLEX solver regarding the triangle packing solution and the computation time.     

Dantzig-Wolfe decomposition and branch-and-price solving in G12

The G12 project is developing a software environment for stating and solving combinatorial problems by mapping a high-level model of the problem to an efficient combination of solving methods. Model annotations are used to control this process. In this paper we explain the mapping to branch-and-price solving. Dantzig-Wolfe decomposition is automatically performed using the additional information given by the model annotations. The models obtained can then be solved using column generation and branch-and-price. G12 supports the selection of specialised subproblem solvers, the aggregation of identical subproblems to reduce symmetries, automatic disaggregation when required by branch-and-bound, the use of specialised subproblem constraint-branching rules, and different master problem solvers including a hybrid solver based on the volume algorithm. We demonstrate the benefits of the G12 framework on three examples: a trucking problem, cutting stock, and two-dimensional bin packing.     

Branch and bound for the cutwidth minimization problem

The cutwidth minimization problem consists of finding a linear arrangement of the vertices of a graph where the maximum number of cuts between the edges of the graph and a line separating consecutive vertices is minimized. We first review previous approaches for special classes of graphs, followed by lower bounds and then a linear integer formulation for the general problem. We then propose a branch-and-bound algorithm based on different lower bounds on the cutwidth of partial solutions. Additionally, we introduce a Greedy Randomized Adaptive Search Procedure (GRASP) heuristic to obtain good initial solutions. The combination of the branch-and-bound and GRASP methods results in optimal solutions or a reduced relative gap (difference between upper and lower bounds) on the instances tested. Empirical results with a collection of previously reported instances indicate that the proposed algorithm is able to solve all the small instances (up to 32 vertices) as well as some of the large instances tested (up to 158 vertices) using less than 30 minutes of CPU time. We compare the results of our method with previous lower bounds, and with the best previous linear integer formulation solved using Cplex. Both comparisons favor the proposed procedure.     

Packing of one-dimensional bins with contiguous selection of identical items: An exact method of optimal solution

Consideration was given to the one-dimensional bin packing problem under the conditions for heterogeneity of the items put into bins and contiguity of choosing identical items for the next bin. The branch-and-bound method using the “next fit” principle and the “linear programming” method were proposed to solve it. The problem and its solution may be used to construct an improved lower bound in the problem of two-dimensional packing.     

Heuristic and exact algorithms for the max–min optimization of the multi-scenario knapsack problem

We are concerned with a variation of the standard 0–1 knapsack problem, where the values of items differ under possible S scenarios. By applying the ‘pegging test’ the ordinary knapsack problem can be reduced, often significantly, in size; but this is not directly applicable to our problem. We introduce a kind of surrogate relaxation to derive upper and lower bounds quickly, and show that, with this preprocessing, the similar pegging test can be applied to our problem. The reduced problem can be solved to optimality by the branch-and-bound algorithm. Here, we make use of the surrogate variables to evaluate the upper bound at each branch-and-bound node very quickly by solving a continuous knapsack problem. Through numerical experiments we show that the developed method finds upper and lower bounds of very high accuracy in a few seconds, and solves larger instances to optimality faster than the previously published algorithms.     

The total completion time open shop scheduling problem with a given sequence of jobs on one machine

This paper addresses the open shop scheduling problem to minimize the total completion time, provided that one of the machines has to process the jobs in a given sequence. The problem is NP-hard in the strong sense even for the two-machine case. A lower bound is derived based on the optimal solution of a relaxed problem in which the operations on every machine may overlap except for the machine with a given sequence of jobs. This relaxed problem is NP-hard in the ordinary sense, however it can be quickly solved via a decomposition into subset-sum problems. Both heuristic and branch-and-bound algorithm are proposed. Experimental results show that the heuristic is efficient for solving large-scaled problems, and the branch-and-bound algorithm performs well on small-scaled problems.Scope and purposeShop scheduling problems, widely used in the modeling of industrial production processes, are receiving an increasing amount of attention from researchers. To model practical production processes more closely, additional processing restrictions can be introduced, e.g., the resource constraints, the no-wait in process requirement, the precedence constraints, etc. This paper considers the total completion time open shop scheduling problem with a given sequence of jobs on one machine. This model belongs to a new class of shop scheduling problems under machine-dependent precedence constraints. This problem is NP-hard in the strong sense. A heuristic is proposed to efficiently solve large-scaled problems and a branch-and-bound algorithm is presented to optimally solve small-scaled problems. Computational experience is also reported.     

基于离散粒子群优化算法求解矩形件排样问题

改进了一种近似排样算法,并将改进的近似排样算法与离散粒子群优化算法结合求解矩形件排样问题.设计了应用离散粒子群优化算法求解矩形件排样问题的相关操作和定义,给出了离散粒子群优化算法求解矩形件排样问题的详细步骤,最后通过实验测试,验证了算法的有效性.     

An iterative merging algorithm for soft rectangle packing and its extension for application of fixed-outline floorplanning of soft modules

We address an important variant of the rectangle packing problem, the soft rectangle packing problem, and explore its problem extension for the fixed-outline floorplanning with soft modules. For the soft rectangle packing problem with zero deadspace, we present an iterative merging packing algorithm that merges all the rectangles into a final composite rectangle in a bottom-up order by iteratively merging two rectangles with the least areas into a composite rectangle, and then shapes and places each pair of sibling rectangles based on the dimensions and position of their composite rectangle in an up-bottom order. We prove that the proposed algorithm can guarantee feasible layout under some conditions, which are weaker as compared with a well-known zero-dead-space packing algorithm. We then provide a deadspace distribution strategy, which can systematically assign deadspace to modules, to extend the iterative merging packing algorithm to deal with soft packing problem with deadspace. For the fixed-outline floorplanning with soft modules problem, we propose an iterative merging packing based hierarchical partitioning algorithm, which adopts a general hierarchical partitioning framework as proposed in the popular PATOMA floorplanner. The framework uses a recursive bipartitioning method to partition the original problem into a set of subproblems, where each subproblem is a soft rectangle packing problem and how to solve the subproblem plays a key role in the final efficiency of the floorplanner. Different from the PATOMA that adopts the zero-dead-space packing algorithm, we adopt our proposed iterative merging packing algorithm for the subproblems. Experiments on the IBM-HB benchmarks show that the proposed packing algorithm is more effective than the zero-dead-space packing algorithm, and experiments on the GSRC benchmarks show that our floorplanning algorithm outperforms three state-of-the-art floorplanners PATOMA, DeFer and UFO, reducing wirelength by 0.2%, 4.0% and 2.3%, respectively.     

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.     

求解矩形布局问题的自适应算法

矩形布局问题属于NP-Hard 问题,其求解算法多为启发式算法。该文侧重 于构造布局求解算法中定位函数（规则）的优化,将模拟退火算法的思想融入到遗传算法中, 提出了求解矩形布局问题的自适应算法,其利用自适应交叉、变异及接收劣质解的概率等方 法对定位函数中各参数进行优化。算法通过两种方式确定初始种群的数目,具有较强的适应 性。在算法搜索的后期,利用差异性较大的个体进行交叉操作,从而保持种群的多样性。最 后通过实例证明了该算法能够很好的应用于矩形布局问题的求解。     

gBoost: a mathematical programming approach to graph classification and regression

Graph mining methods enumerate frequently appearing subgraph patterns, which can be used as features for subsequent classification or regression. However, frequent patterns are not necessarily informative for the given learning problem. We propose a mathematical programming boosting method (gBoost) that progressively collects informative patterns. Compared to AdaBoost, gBoost can build the prediction rule with fewer iterations. To apply the boosting method to graph data, a branch-and-bound pattern search algorithm is developed based on the DFS code tree. The constructed search space is reused in later iterations to minimize the computation time. Our method can learn more efficiently than the simpler method based on frequent substructure mining, because the output labels are used as an extra information source for pruning the search space. Furthermore, by engineering the mathematical program, a wide range of machine learning problems can be solved without modifying the pattern search algorithm.