An improved tabu search approach with mixed objective function for one-dimensional cutting stock problems

In this paper, a methodology with both efficiency and effectiveness of using improved tabu search approach is applied successfully to solve one-dimensional cutting stock problem (1D-CSP). By introducing a varying mixed objective function, the chance of obtaining optimum solution in solving 1D-CSP, which is treated as a sequence problem, has increased. The total incentive trim loss, which functions as an incentive term in this paper, makes it possible to obtain good 1D-CSP results through the use of the new proposed TS approach. After testing against actual sample cases from shipyards and literatures, the solutions obtained from the proposed algorithm are superior. Computational results indicate that the methodology proposed outperforms the ones from other meta-heuristic methods.     

一种有效的求解一维下料问题的启发式算法

使用了一种改进的顺序启发式算法,在排样方式的生成过程中不断修正当前排入毛坯的价值,使之趋于合理,依次选取求解背包函数获得的最大单位价值的排样方式组成当前排样方案,迭代调用该过程多次,最终选取最优的排样方案。在保证较高材料利用率的同时考虑减少排样方式,增加最后一根材料余料长度等多个优化目标。通过多组实验结果比较,证实了算法的有效性。     

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.     

一维下料问题的AB分类法

为了解决大规模的一维下料问题的计算困难, 根据一维下料问题的特点,把贪心算法和随机搜索技术有机地结合起来,利用随机搜索技术对贪心算法进行了有效的改进,提出了一种简单实用的AB分类法。 实验表明,该算法对规模较大的问题也能较快地获得问题最优解或精度较高的近似最优解。     

A combined approach to the solution to the general one-dimensional cutting stock problem

In this article a combined method for the solution to the general one-dimensional cutting stock problem (G1D-CSP) is proposed. The main characteristic of G1D-CSP lies in that all stock lengths can be different. The new approach combines two existing methods: sequential heuristic procedure (SHP) and branch-and-bound. The algorithm based on the proposed method leads to almost optimal solutions, which are substantially better than the solutions of known methods at the expense of slightly increased time complexity, which is still low. So, the new method is suitable for all sizes of the problem. A comparison with the SHP is presented.     

Using Wang's two-dimensional cutting stock algorithm to optimally solve difficult problems

P. Y. Wang's classic bottom-up two-dimensional cutting stock algorithm generates cutting patterns by building rectangles both horizontally and vertically. This algorithm uses a parameter β ₁ to tradeoff the number of rectangles generated by the algorithm and hence the quality of the cutting pattern solution obtained versus the amount of computer resources required. Several researchers have made relatively straightforward modifications to Wang's basic algorithm resulting in improved computational times. However, even with these modifications, Wang's approach tends to require large amounts of computer resources in order to optimally or near-optimally solve difficult two-dimensional guillotine cutting stock problems. In this paper, we present an iterative approach that judiciously uses Wang's basic algorithm (with some previously defined modifications) to obtain optimal cutting patterns to difficult two-dimensional cutting stock problems in reasonable computer run times.     

On the one-dimensional stock cutting problem in the paper tube industry

The one-dimensional cutting stock problem (1D-CSP) is one of the representative combinatorial optimization problems which arises in many industrial applications. Although the primary objective of 1D-CSP is to minimize the total length of used stock rolls, the efficiency of cutting processes has become more important in recent years. The crucial bottleneck of the cutting process often occurs at handling operations in semiautomated manufacturers such as those in the paper tube industry. To reduce interruptions and errors at handling operations in the paper tube industry, we consider a variant of 1D-CSP that minimizes the total length of used stock rolls while constraining (C1) the number of setups of each stock roll type, (C2) the combination of piece lengths occurring in open stacks simultaneously, and (C3) the number of open stacks. For this problem, we propose a generalization of the cutting pattern called the “cutting group,” which is a sequence of cutting patterns that satisfies the given upper bounds of setups of each stock roll type and open stacks. To generate good cutting groups, we decompose the 1D-CSP into a number of auxiliary bin packing problems. We develop a tabu search algorithm based on a shift neighborhood that solves the auxiliary bin packing problems by the first-fit decreasing heuristic algorithm. Experimental results show that our algorithm improves the quality of solutions compared to the existing algorithm used in a paper tube factory.     

A new evolutionary approach to cutting stock problems with and without contiguity

Evolutionary algorithms (EAs) have been applied to many optimization problems successfully in recent years. The genetic algorithm (GAs) and evolutionary programming (EP) are two different types of EAs. GAs use crossover as the primary search operator and mutation as a background operator, while EP uses mutation as the primary search operator and does not employ any crossover. This paper proposes a novel EP algorithm for cutting stock problems with and without contiguity. Two new mutation operators are proposed. Experimental studies have been carried out to examine the effectiveness of the EP algorithm. They show that EP can provide a simple yet more effective alternative to GAs in solving cutting stock problems with and without contiguity. The solutions found by EP are significantly better (in most cases) than or comparable to those found by GAs.Scope and purposeThe one-dimensional cutting stock problem (CSP) is one of the classical combinatorial optimization problems. While most previous work only considered minimizing trim loss, this paper considers CSPs with two objectives. One is the minimization of trim loss (i.e., wastage). The other is the minimization of the number of stocks with wastage, or the number of partially finished items (pattern sequencing or contiguity problem). Although some traditional OR techniques (e.g., programming based approaches) can find the global optimum for small CSPs, they are impractical to find the exact global optimum for large problems due to combinatorial explosion. Heuristic techniques (such as various hill-climbing algorithms) need to be used for large CSPs. One of the heuristic algorithms which have been applied to CSPs recently with success is the genetic algorithm (GA). This paper proposes a much simpler evolutionary algorithm than the GA, based on evolutionary programming (EP). The EP algorithm has been shown to perform significantly better than the GA for most benchmark problems we used and to be comparable to the GA for other problems.     

基于交货期的多目标一维优化下料模型研究

根据目前对优化下料问题新特性的研究,指出优化下料问题呈现多目标和复杂约束状态,建立了以原料消耗最少和不同交货期的惩罚最小为目标函数的优化下料问题的数学模型.利用多目标智能优化算法求解模型,并结合算例验证了模型的有效性和实用性.     

Logic based Benders' decomposition for orthogonal stock cutting problems

We consider the problem of packing a set of rectangular items into a strip of fixed width, without overlapping, using minimum height. Items must be packed with their edges parallel to those of the strip, but rotation by 90° is allowed. The problem is usually solved through branch-and-bound algorithms. We propose an alternative method, based on Benders' decomposition. The master problem is solved through a new ILP model based on the arc flow formulation, while constraint programming is used to solve the slave problem. The resulting method is hybridized with a state-of-the-art branch-and-bound algorithm. Computational experiments on classical benchmarks from the literature show the effectiveness of the proposed approach. We additionally show that the algorithm can be successfully used to solve relevant related problems, like rectangle packing and pallet loading.     

Constrained two-dimensional cutting stock problems a best-first branch-and-bound algorithm

In this paper, we develop a new version of the algorithm proposed in Hifi (Computers and Operations Research 24/8 (1997) 727–736) for solving exactly some variants of (un)weighted constrained two-dimensional cutting stock problems. Performance of branch-and-bound procedure depends highly on particular implementation of that algorithm. Programs of this kind are often accelerated drastically by employing sophisticated techniques. In the new version of the algorithm, we start by enhancing the initial lower bound to limit initially the space search. This initial lower bound has already been used in Fayard et al. 1998 (Journal of the Operational Research Society, 49, 1270–1277), as a heuristic for solving the constrained and unconstrained cutting stock problems. Also, we try to improve the upper bound at each internal node of the developed tree, by applying some simple and effcient combinations. Finally, we introduce some new symmetric-strategies used for neglecting some unnecessary duplicate patterns . The performance of our algorithm is evaluated on some problem instances of the literature and other hard-randomly generated problem instances.     

The generalized assortment and best cutting stock length problems

This paper introduces two new one-dimensional cutting stock models: the generalized assortment problem (GAP) and the best cutting stock length (BSL) problem. These new models provide the potential to reduce waste to values lower than the optimum of current models, under the right management circumstances. In the GAP, management has a standard length and can select one or more of any additional custom stock lengths, and management wishes to minimize cutting stock waste. This model is different from existing models that assume that the selection is from a small fixed set of stock lengths. In the BSL problem, management chooses any number of custom stock lengths, but wishes to find the fewest custom stock lengths in order to have zero waste. Results show waste reductions of 80% with just one custom stock length compared with solutions from standard cutting stock formulations, when item lengths are long relative to the stock length. The models are most effective when the item lengths are nearly as long as the stock length. Solutions from the model have been implemented for a manufacturer. The model is easily generalized to allow multiple existing stock lengths and different costs.     

Near-optimal control for stochastic recursive problems

It is well documented (e.g. Zhou (1998) [8]) that the near-optimal controls, as the alternative to the “exact” optimal controls, are of great importance for both the theoretical analysis and practical application purposes due to its nice structure and broad-range availability, feasibility as well as flexibility. However, the study of near-optimality on the stochastic recursive problems, to the best of our knowledge, is a totally unexplored area. Thus we aim to fill this gap in this paper. As the theoretical result, a necessary condition as well as a sufficient condition of near-optimality for stochastic recursive problems is derived by using Ekeland’s principle. Moreover, we work out an ε-optimal control example to shed light on the application of the theoretical result. Our work develops that of [8] but in a rather different backward stochastic differential equation (BSDE) context.     

A hybrid heuristic to reduce the number of different patterns in cutting stock problems

We propose a hybrid procedure to obtain a reduced number of different patterns in cutting stock problems. Initially, we generate patterns with limited waste that fulfill the demands of at least two items when the patterns are repeatedly cut as much as possible but without overproducing any of the items. The problem is reduced and the residual problem is solved. Then, pattern reduction techniques (local search) are applied starting with the generated solution. The scheme is straightforward and can be used in cutting stock problems of any dimension. Variations of the procedure are also indicated. Computational tests performed indicated that the proposed scheme provides alternative solutions to the pattern reduction problem which are not dominated by other solutions obtained using procedures previously suggested in the literature.     

An optimization algorithm for cutting stock problems in the TFT-LCD industry

Due to lack of efficient approaches of mixed production, the present production approach of the TFT-LCD industry is batch production that each glass substrate is cut into LCD plates of one size only. This study proposes an optimization algorithm for cutting stock problems of the TFT-LCD industry. The proposed algorithm minimizes the number of glass substrates required to satisfy the orders, therefore reducing the production costs. Additionally, the solution of the proposed algorithm is a global optimum which is different from a local optimum or a feasible solution that is found by the heuristic algorithm. Numerical examples are also presented to illustrate the usefulness of the proposed algorithm.     

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.     

Near-optimal control problems for linear forward-backward stochastic systems

Herein, we study the near-optimality of linear forward-backward stochastic control systems. As the theoretical results, some sufficient and necessary conditions of the near-optimality are established in the form of Pontryagin stochastic maximum principle. As an illustration and practical application, one ε-optimal control example is figured out and solved using our theoretical results.     

改进非支配排序进化算法在下料问题中的应用

针对一维下料问题,提出了减少废料、减少下料设置时间和减少可回收余料的三目标优化模型,用改进的非支配排序进化算法求出问题的Pareto最优解集,运用逼近理想解方法从解集中选出一个满意解作为下料方案,各优化目标的权重用CRITIC法算出。仿真实验证明了所提出的方法可以有效解决该类多目标下料问题。