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.     

一种求解矩形块装填问题的拟人算法

在货物装载、木材下料、超大规模集成电路（VLSI）设计等工作中提出了矩形块装填与切割问题，对这一问题，国内外学者提出了诸如模拟退火算法、遗传算法及其它一些启发式算法等求解算法。本文利用人类的智慧和他们上万年以来形成的经验，提出了一种求解矩形块装填问题的拟人算法。谊算法使用了两个主要的思想策略，即矩形块选择策略和矩形块放置策略。用本文提出的算法，对21个测试算例进行了实算测试，测试结果表明：算法所得装填结果的优度高，计算时间短。对这21个测试算例。用本文算法计算，得到了其中16个算例的最优解，而计算时间都在2秒以内。进一步的测试表明，本文提出的算法对求解矩形块装填问题十分有效。     

A graph-based immune-inspired constraint satisfaction search

We propose an artificial immune algorithm to solve constraint satisfaction problems (CSPs). Recently, bio-inspired algorithms have been proposed to solve CSPs. They have shown to be efficient in solving hard problem instances. Given that recent publications indicate that immune-inspired algorithms offer advantages to solve complex problems, our main goal is to propose an efficient immune algorithm which can solve CSPs. We have calibrated our algorithm using relevance estimation and value calibration (REVAC), which is a new technique recently introduced to find the parameter values for evolutionary algorithms. The tests were carried out using randomly generated binary constraint satisfaction problems and instances of the three-colouring problem with different constraint networks. The results suggest that the technique may be successfully applied to solve CSPs.     

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.     

An efficient heuristic algorithm for arbitrary shaped rectilinear block packing problem

Arbitrary shaped rectilinear block packing problem is a problem of packing a series of rectilinear blocks into a larger rectangular container, where arbitrary shaped rectilinear block is a polygonal block whose interior angle is either 90° or 270°. This problem involves many industrial applications, such as VLSI design, timber cutting, textile industry and layout of newspaper. Many algorithms based on different strategies have been presented to solve it. In this paper, we proposed an efficient heuristic algorithm which is based on principles of corner-occupying action and caving degree describing the quality of packing action. The proposed algorithm is tested on six instances from literatures and the results are rather satisfying. The computational results demonstrate that the proposed algorithm is rather efficient for solving the arbitrary shaped rectilinear block packing problem.     

Ensemble of multi-objective metaheuristic algorithms for multi-objective unconstrained binary quadratic programming problem

Metaheuristics have been widely utilized for solving NP-hard optimization problems. However, these algorithms usually perform differently from one problem to another, i.e., one may be effective on a problem but performs badly on another problem. Therefore, it is difficult to choose the best algorithm in advance for a given problem. In contrast to selecting the best algorithm for a problem, selection hyper-heuristics aim at performing well on a set of problems (instances). This paper proposes a selection hyper-heuristic based algorithm for multi-objective optimization problems. In the proposed algorithm, multiple metaheuristics exhibiting different search behaviors are managed and controlled as low-level metaheuristics in an algorithm pool, and the most appropriate metaheuristic is selected by means of a performance indicator at each search stage. To assess the performance of the proposed algorithm, an implementation of the algorithm containing four metaheuristics is proposed and tested for solving multi-objective unconstrained binary quadratic programming problem. Experimental results on 50 benchmark instances show that the proposed algorithm can provide better overall performance than single metaheuristics, which demonstrates the effectiveness of the proposed algorithm.     

A new heuristic algorithm for rectangle packing

The rectangle packing problem often appears in encasement and cutting as well as very large-scale integration design. To solve this problem, many algorithms such as genetic algorithm, simulated annealing and other heuristic algorithms have been proposed. In this paper, a new heuristic algorithm is recommended based on two important concepts, namely, the corner-occupying action and caving degree. Twenty-one rectangle-packing instances are tested by the algorithm developed, 16 of which having achieved optimum solutions within reasonable runtime. Experimental results demonstrate that the algorithm developed is fairly efficient for solving the rectangle packing problem.     

An efficient heuristic algorithm for rectangle-packing problem

Rectangle-packing problem involves many industrial applications, such as shipping, timber cutting, very large scale integration (VLSI) floor planning, and so on. This problem has shown to be NP hard, and many algorithms such as genetic algorithm, simulated annealing and other heuristic algorithms are presented to solve it. Based on the wisdom and experience of human being, an efficient heuristic algorithm is proposed in this paper. Two group benchmarks are used to test the performance of the produced algorithm, 19 instances of first group and 3 instances of second group having achieved optimal solutions. The experimental results demonstrate that the presented algorithm is rather efficient for solving the rectangle-packing problem.     

求解带约束优化问题的混合式多策略萤火虫算法

目前多目标优化算法主要针对如何处理多个目标之间的冲突,对于如何处理约束考虑较少,鉴于此,提出一种求解带约束优化问题的混合式多策略萤火虫算法(HMSFA-PC).首先,提出一种改进的动态罚函数策略对约束优化问题进行预处理,将其转换为非约束优化问题;其次,对萤火虫算法本身进行改进,采用Lévy flights搜索机制有效地增大搜索范围;接着,引入随机扩张因子改进算法吸引模型,使种群突破束缚,有效避免早熟收敛,提出自适应维度重组机制,根据不同迭代时期选择差异性较大的个体进行信息交互、相互学习.为检验算法处理无约束优化问题的性能,将其在基准测试函数上与部分典型算法进行比较;为检验算法处理约束优化问题的性能,将其在实际约束测试问题中与一些顶尖约束求解算法进行比较.结果表明,HMSFA-PC在处理无约束优化问题时具有收敛速度快、收敛精度高等优势,并且在动态罚函数的协作下求解实际约束优化问题时仍具有良好的优化性能.     

求解矩形packing问题的贪心算法

在货物装载、木材下料、超大规模集成电路设计等工作中提出了矩形packing问题。对这一问题,国内外学者提出了诸如模拟退火算法、遗传算法及其它一些启发式算法等求解算法。该文利用人类的智慧及历史上形成的经验,提出了一种求解矩形packing问题的贪心算法。并对21个公开测试实例进行了实算测试,所得结果的平均面积未利用率为0.28%,平均计算时间为17.86s,并且还得到了其中8个实例的最优解。测试结果表明,该算法对求解矩形packing问题相当有效。     

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。     

Optimization of heuristic search using recursive algorithm selection and reinforcement learning

The traditional approach to computational problem solving is to use one of the available algorithms to obtain solutions for all given instances of a problem. However, typically not all instances are the same, nor a single algorithm performs best on all instances. Our work investigates a more sophisticated approach to problem solving, called Recursive Algorithm Selection, whereby several algorithms for a problem (including some recursive ones) are available to an agent that makes an informed decision on which algorithm to select for handling each sub-instance of a problem at each recursive call made while solving an instance. Reinforcement learning methods are used for learning decision policies that optimize any given performance criterion (time, memory, or a combination thereof) from actual execution and profiling experience. This paper focuses on the well-known problem of state-space heuristic search and combines the A* and RBFS algorithms to yield a hybrid search algorithm, whose decision policy is learned using the Least-Squares Policy Iteration (LSPI) algorithm. Our benchmark problem domain involves shortest path finding problems in a real-world dataset encoding the entire street network of the District of Columbia (DC), USA. The derived hybrid algorithm exhibits better performance results than the individual algorithms in the majority of cases according to a variety of performance criteria balancing time and memory. It is noted that the proposed methodology is generic, can be applied to a variety of other problems, and requires no prior knowledge about the individual algorithms used or the properties of the underlying problem instances being solved.     

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.     

On an evolutionary approach for constrained optimization problem solving

Over the last few decades, many different evolutionary algorithms have been introduced for solving constrained optimization problems. However, due to the variability of problem characteristics, no single algorithm performs consistently over a range of problems. In this paper, instead of introducing another such algorithm, we propose an evolutionary framework that utilizes existing knowledge to make logical changes for better performance. The algorithmic aspects considered here are: the way of using search operators, dealing with feasibility, setting parameters, and refining solutions. The combined impact of such modifications is significant as has been shown by solving two sets of test problems: (i) a set of 24 test problems that were used for the CEC2006 constrained optimization competition and (ii) a second set of 36 test instances introduced for the CEC2010 constrained optimization competition. The results demonstrate that the proposed algorithm shows better performance in comparison to the state-of-the-art algorithms.     

Bid evaluation in combinatorial auctions: optimization and learning

In combinatorial auctions bidders can post bids on groups of items. The problem of selecting the winning bids, called Winner Determination Problem, is NP‐hard. In this paper, we consider an interesting variant of this problem, called Bid Evaluation Problem (BEP), where items are services and are subject to precedence constraints and temporal windows. The BEP has many practical applications, such as, for instance, in transportation routes auctions and in take off and landing time slot allocation problems. We have developed a number of optimization algorithms based on Constraint Programming, on Integer Programming and on the combination of the two techniques. We first show that all algorithms proposed outperform the only commercial system for solving BEP instances called Multi AGent Negotiation Testbed, a more general tool for agent negotiation. Then, we evaluate the developed algorithms and use the decision tree machine learning technique for finding a relation between the instance structure and the solving algorithm and providing an automatic algorithm selection procedure. We show that we can achieve an accuracy of 90% in predicting the best algorithm given the instance to be solved with a significant time saving w.r.t. a single solving technique or a costless, but less accurate, prediction technique. Copyright © 2009 John Wiley & Sons, Ltd.     

A majorization-minimization approach to the sparse generalized eigenvalue problem

Generalized eigenvalue (GEV) problems have applications in many areas of science and engineering. For example, principal component analysis (PCA), canonical correlation analysis (CCA) and Fisher discriminant analysis (FDA) are specific instances of GEV problems, that are widely used in statistical data analysis. The main contribution of this work is to formulate a general, efficient algorithm to obtain sparse solutions to a GEV problem. Specific instances of sparse GEV problems can then be solved by specific instances of this algorithm. We achieve this by solving the GEV problem while constraining the cardinality of the solution. Instead of relaxing the cardinality constraint using a ? ₁-norm approximation, we consider a tighter approximation that is related to the negative log-likelihood of a Student??s t-distribution. The problem is then framed as a d.c. (difference of convex functions) program and is solved as a sequence of convex programs by invoking the majorization-minimization method. The resulting algorithm is proved to exhibit global convergence behavior, i.e., for any random initialization, the sequence (subsequence) of iterates generated by the algorithm converges to a stationary point of the d.c. program. Finally, we illustrate the merits of this general sparse GEV algorithm with three specific examples of sparse GEV problems: sparse PCA, sparse CCA and sparse FDA. Empirical evidence for these examples suggests that the proposed sparse GEV algorithm, which offers a general framework to solve any sparse GEV problem, will give rise to competitive algorithms for a variety of applications where specific instances of GEV problems arise.     

Multi-operator based evolutionary algorithms for solving constrained optimization problems

Over the last two decades, many sophisticated evolutionary algorithms have been introduced for solving constrained optimization problems. Due to the variability of characteristics in different COPs, no single algorithm performs consistently over a range of problems. In this paper, for a better coverage of the problem characteristics, we introduce an algorithm framework that uses multiple search operators in each generation. The appropriate mix of the search operators, for any given problem, is determined adaptively. The framework is tested by implementing two different algorithms. The performance of the algorithms is judged by solving 60 test instances taken from two constrained optimization benchmark sets from specialized literature. The first algorithm, which is a multi-operator based genetic algorithm (GA), shows a significant improvement over different versions of GA (each with a single one of these operators). The second algorithm, using differential evolution (DE), also confirms the benefit of the multi-operator algorithm by providing better and consistent solutions. The overall results demonstrated that both GA and DE based algorithms show competitive, if not better, performance as compared to the state of the art algorithms.     

Distributed assembly permutation flow shop problem; Single seekers society algorithm

The distributed manufacturing and assembly systems have an important role at the point of overcoming the difficulties faced by today's mass-production industry. By using both of these systems together in the same production system, the advantages of this integration can make industries more flexible and stronger. Besides, optimizing these systems is more complicated since the multiple production systems can undoubtfully affect the production system’s performance. In this paper, two new mixed-integer linear programming (MILP) models are proposed for the distributed assembly permutation flow shop problem (DAPFSP), inspiring by the multiple-travelling salesman structure. Moreover, a single seekers society (SSS) algorithm is proposed for solving the DAPFSP to minimize the maximum completion time of all products. The performance of the proposed MILP models is evaluated using 900 small-sized benchmark instances. The proposed MILP models were effective and were able to find more optimal solutions or improve the best-found solutions for the small-sized DAPFSP benchmark instances. Similarly, the SSS algorithm is statistically compared with the best-known algorithms developed for solving the DAPFSP on 900 small and 810 large-sized benchmark instances. The proposed SSS algorithm shows superior performance compared to other algorithms in solving the small-sized DAPFSP instances in terms of finding better solutions. In addition, it is as effective as the best performing algorithms developed to solve the large-sized DAPFSP instances. Furthermore, the best-found solutions for 40 numbers of test problems reported to be improved in this paper.     

The ant system applied to the quadratic assignment problem

In recent years, there has been growing interest in algorithms inspired by the observation of natural phenomena to define computational procedures that can solve complex problems. We describe a distributed heuristic algorithm that was inspired by the observation of the behavior of ant colonies, and we propose its use for the quadratic assignment problem. The results obtained in solving several classical instances of the problem are compared with those obtained from other evolutionary heuristics to evaluate the quality of the proposed system     

Selection and scheduling in a virtual organisation environment with a service broker

This paper addresses the problems of selection and scheduling related to a virtual organisation in which a central service broker communicates with groups of service providers offering services of different types. Our objective is to show that these problems can be formulated in a precise manner and that well-established algorithms can be developed for solving them. Especially, Tabu Search and Simulated Annealing with Variable Neighbourhood search are proposed as options for finding near-optimal solutions to the selection and scheduling problems. In our computational experiments, the algorithms were first validated with the help of benchmark job-shop problems and then applied to a specific cutting stock problem with randomly generated instances.