Heuristics for the variable sized bin-packing problem

We investigate the one-dimensional variable-sized bin-packing problem. This problem requires packing a set of items into a minimum-cost set of bins of unequal sizes and costs. Six optimization-based heuristics for this problem are presented and compared. We analyze their empirical performance on a large set of randomly generated test instances with up to 2000 items and seven bin types. The first contribution of this paper is to provide evidence that a set covering heuristic proves to be highly effective and capable of delivering very-high quality solutions within short CPU times. In addition, we found that a simple subset-sum problem-based heuristic consistently outperforms heuristics from the literature while requiring extremely short CPU times.     

Automating the packing heuristic design process with genetic programming

The literature shows that one-, two-, and three-dimensional bin packing and knapsack packing are difficult problems in operational research. Many techniques, including exact, heuristic, and metaheuristic approaches, have been investigated to solve these problems and it is often not clear which method to use when presented with a new instance. This paper presents an approach which is motivated by the goal of building computer systems which can design heuristic methods. The overall aim is to explore the possibilities for automating the heuristic design process. We present a genetic programming system to automatically generate a good quality heuristic for each instance. It is not necessary to change the methodology depending on the problem type (one-, two-, or three-dimensional knapsack and bin packing problems), and it therefore has a level of generality unmatched by other systems in the literature. We carry out an extensive suite of experiments and compare with the best human designed heuristics in the literature. Note that our heuristic design methodology uses the same parameters for all the experiments. The contribution of this paper is to present a more general packing methodology than those currently available, and to show that, by using this methodology, it is possible for a computer system to design heuristics which are competitive with the human designed heuristics from the literature. This represents the first packing algorithm in the literature able to claim human competitive results in such a wide variety of packing domains.     

The load-balanced multi-dimensional bin-packing problem

The bin-packing problem is one of the most investigated and applicable combinatorial optimization problems. In this paper we consider its multi-dimensional version with the practical extension of load balancing, i.e. to find the packing requiring the minimum number of bins while ensuring that the average center of mass of the loaded bins falls as close as possible to an ideal point, for instance, the center of the bin. We formally describe the problem using mixed-integer linear programming models, from the simple case where we want to optimally balance a set of items already assigned to a single bin, to the general balanced bin-packing problem. Given the difficulty for standard solvers to deal even with small size instances, a multi-level local search heuristic is presented. The algorithm takes advantage of the Fekete–Schepers representation of feasible packings in terms of particular classes of interval graphs, and iteratively improves the load balancing of a bin-packing solution using different search levels. The first level explores the space of transitive orientations of the complement graphs associated with the packing, the second modifies the structure itself of the interval graphs, the third exchanges items between bins repacking proper n-tuples of weakly balanced bins. Computational experiments show very promising results on a set of 3D bin-packing instances from the literature.     

A constructive bin-oriented heuristic for the two-dimensional bin packing problem with guillotine cuts

A new heuristic algorithm for solving the two-dimensional bin-packing problem with guillotine cuts (2DBP|?|G)$(2\mathrm{DBP}|?|\mathrm{G})$ is presented. The heuristic constructs a solution by packing a bin at a time. Central to the adopted solution scheme is the principle of average-area sufficiency proposed by the authors for guiding selection of items to fill a bin. The algorithm is tested on a set of standard benchmark problem instances and compared with existing heuristics producing the best-known results. The results presented attest to the efficacy of the proposed scheme.     

Hybrid greedy heuristics based on linear programming for the three‐dimensional single bin‐size bin packing problem

In this paper, we propose to solve the three‐dimensional single bin‐size bin packing problem (3D‐SBSBPP) using a simple strategy based on integer linear programming (ILP) heuristics, without using any improvement based on metaheuristics. We first propose an ILP that is converted into a series of three‐dimensional single knapsack problems (3D‐SKP). Then, the first tailored heuristic can be viewed as a hybrid approach in which both “selection” and “positioning” phases are combined. The first phase serves to select a subset of items where each of these items is susceptible to belonging to an active container. The positioning phase serves to pack a subset of items already preselected by the selection phase. Then, both phases cooperate till packing all items into their corresponding containers. The second heuristic can be viewed as an extended version of the first one. Indeed, before deciding whether the current container is closed or a new container is activated, “a local reoptimization phase” is considered. Finally, both proposed heuristics are evaluated on a set of random instances obtained by using the standard generator scheme of the literature. The provided results show that both proposed heuristics remain competitive when compared to the results obtained by one of the best methods of the literature.     

A greedy gradient-simulated annealing selection hyper-heuristic

Educational timetabling problem is a challenging real world problem which has been of interest to many researchers and practitioners. There are many variants of this problem which mainly require scheduling of events and resources under various constraints. In this study, a curriculum based course timetabling problem at Yeditepe University is described and an iterative selection hyper-heuristic is presented as a solution method. A selection hyper-heuristic as a high level methodology operates on the space formed by a fixed set of low level heuristics which operate directly on the space of solutions. The move acceptance and heuristic selection methods are the main components of a selection hyper-heuristic. The proposed hyper-heuristic in this study combines a simulated annealing move acceptance method with a learning heuristic selection method and manages a set of low level constraint oriented heuristics. A key goal in hyper-heuristic research is to build low cost methods which are general and can be reused on unseen problem instances as well as other problem domains desirably with no additional human expert intervention. Hence, the proposed method is additionally applied to a high school timetabling problem, as well as six other problem domains from a hyper-heuristic benchmark to test its level of generality. The empirical results show that our easy-to-implement hyper-heuristic is effective in solving the Yeditepe course timetabling problem. Moreover, being sufficiently general, it delivers a reasonable performance across different problem domains.     

Case-based heuristic selection for timetabling problems

This paper presents a case-based heuristic selection approach for automated university course and exam timetabling. The method described in this paper is motivated by the goal of developing timetabling systems that are fundamentally more general than the current state of the art. Heuristics that worked well in previous similar situations are memorized in a case base and are retrieved for solving the problem in hand. Knowledge discovery techniques are employed in two distinct scenarios. Firstly, we model the problem and the problem solving situations along with specific heuristics for those problems. Secondly, we refine the case base and discard cases which prove to be non-useful in solving new problems. Experimental results are presented and analyzed. It is shown that case based reasoning can act effectively as an intelligent approach to learn which heuristics work well for particular timetabling situations. We conclude by outlining and discussing potential research issues in this critical area of knowledge discovery for different difficult timetabling problems.     

Extended guided tabu search and a new packing algorithm for the two-dimensional loading vehicle routing problem

In this paper, we develop an extended guided tabu search (EGTS) and a new heuristic packing algorithm for the two-dimensional loading vehicle routing problem (2L-CVRP). The 2L-CVRP is a combination of two well-known NP-hard problems, the capacitated vehicle routing problem, and the two-dimensional bin packing problem. It is very difficult to get a good performance solution in practice for these problems. We propose a meta-heuristic methodology EGTS which incorporates theories of tabu search and extended guided local search (EGLS). It has been proved that tabu search is a very good approach for the CVRP, and the guiding mechanism of the EGLS can help tabu search to escape effectively from local optimum. Furthermore, we have modified a collection of packing heuristics by adding a new packing heuristic to solve the loading constraints in 2L-CVRP, in order to improve the cost function significantly. The effectiveness of the proposed algorithm is tested, and proven by extensive computational experiments on benchmark instances.     

一种带脆度的尺寸可变装箱问题

作为经典装箱问题的扩展,尺寸可变装箱问题在现实生活中有着极高的应用背景。分析了尺寸可变装箱问题在解决货物装载运输问题上的不足,由此提出了一种带脆度的尺寸可变装箱问题。除了经典装箱问题中物品体积和箱子容量这两个参数,还引入了物品类型和箱子脆度等参数,给出了相关的数学模型。在经典的FFD（First Fit Decreasing）算法的基础上进行了推广,提出了新的启发式算法NFFD,它对箱子的特性进行了预处理,再进行装箱。分析了该算法的复杂性。对NFD、FFD和NFFD算法进行了数值模拟实验,实验结果表明,在相关参数符合均匀分布的条件下,NFFD算法的效果是最好的。     

CHAMP: Creating heuristics via many parameters for online bin packing

The online bin packing problem is a well-known bin packing variant and which requires immediate decisions to be made for the placement of a lengthy sequence of arriving items of various sizes one at a time into fixed capacity bins without any overflow. The overall goal is maximising the average bin fullness. We investigate a ‘policy matrix’ representation, which assigns a score for each decision option independently and the option with the highest value is chosen, for one-dimensional online bin packing. A policy matrix might also be considered as a heuristic with many parameters, where each parameter value is a score. We hence effectively investigate a framework which can be used for creating heuristics via many parameters. The proposed framework combines a Genetic Algorithm optimiser, which searches the space of heuristics in policy matrix form, and an online bin packing simulator, which acts as the evaluation function. The empirical results indicate the success of the proposed approach, providing the best solutions for almost all item sequence generators used during the experiments. We also present a novel fitness landscape analysis on the search space of policies. This study hence gives evidence of the potential for automated discovery by intelligent systems of powerful heuristics for online problems; reducing the need for expensive use of human expertise.     

Iterated local search using an add and delete hyper-heuristic for university course timetabling

Hyper-heuristics are (meta-)heuristics that operate at a higher level to choose or generate a set of low-level (meta-)heuristics in an attempt of solve difficult optimization problems. Iterated local search (ILS) is a well-known approach for discrete optimization, combining perturbation and hill-climbing within an iterative framework. In this study, we introduce an ILS approach, strengthened by a hyper-heuristic which generates heuristics based on a fixed number of add and delete operations. The performance of the proposed hyper-heuristic is tested across two different problem domains using real world benchmark of course timetabling instances from the second International Timetabling Competition Tracks 2 and 3. The results show that mixing add and delete operations within an ILS framework yields an effective hyper-heuristic approach.     

New heuristics for one-dimensional bin-packing

Several new heuristics for solving the one-dimensional bin packing problem are presented. Some of these are based on the minimal bin slack (MBS) heuristic of Gupta and Ho. A different algorithm is one based on the variable neighbourhood search metaheuristic. The most effective algorithm turned out to be one based on running one of the former to provide an initial solution for the latter. When tested on 1370 benchmark test problem instances from two sources, this last hybrid algorithm proved capable of achieving the optimal solution for 1329, and could find for 4 instances solutions better than the best known. This is remarkable performance when set against other methods, both heuristic and optimum seeking.Scope and purposePacking items into boxes or bins is a task that occurs frequently in distribution and production. A large variety of different packing problems can be distinguished, depending on the size and shape of the items, as well as on the form and capacity of the bins (H. Dyckhoff and U. Finke, Cutting and Packing in Production and Distribution: a Typology and Bibliography, Springer, Berlin, 1992). Similar problems occur in minimising material wastage while cutting pieces into particular smaller ones and in the scheduling of identical processors in order to minimise total completion time. This work addresses the basic packing problem, known as the one-dimensional bin packing problem, where it is required to pack a number of items into the smallest possible number of bins of pre-specified equal capacity. Even though this problem is simple to state, it is NP hard, i.e., it is unlikely that there exists an algorithm that could solve every instance of it in polynomial time. Solution of more general realistic packing problems is probably contingent upon the availability of effective and computationally efficient solution procedures for the basic problem. In this work we present several heuristics capable of doing that. Extensive computational testing attests to the power of these heuristics, as well as to their computational efficiency.     

A hyperheuristic approach to examination timetabling problems: benchmarks and a new problem from practice

Many researchers studying examination timetabling problems focus on either benchmark problems or problems from practice encountered in their institutions. Hyperheuristics are proposed as generic optimisation methods which explore the search space of heuristics rather than direct solutions. In the present study, the performance of tournament-based hyperheuristics for the exam timetabling problem are investigated. The target instances include both the Toronto and ITC 2007 benchmarks and the examination timetabling problem at KAHO Sint-Lieven (Ghent, Belgium). The Toronto and ITC 2007 benchmarks are post-enrolment-based examination timetabling problems, whereas the KAHO Sint-Lieven case is a curriculum-based examination timetabling problem. We drastically improve the previous (manually created) solution for the KAHO Sint-Lieven problem by generating a timetable that satisfies all the hard and soft constraints. We also make improvements on the best known results in the examination timetabling literature for seven out of thirteen instances for the To ronto benchmarks. The results are competitive with those of the finalists of the examination timetabling track of the International Timetabling Competition.     

Augmented neural networks and problem structure-based heuristics for the bin-packing problem

In this article, we report on a research project where we applied augmented-neural-networks (AugNNs) approach for solving the classical bin-packing problem (BPP). AugNN is a metaheuristic that combines a priority rule heuristic with the iterative search approach of neural networks to generate good solutions fast. This is the first time this approach has been applied to the BPP. We also propose a decomposition approach for solving harder BPP, in which subproblems are solved using a combination of AugNN approach and heuristics that exploit the problem structure. We discuss the characteristics of problems on which such problem structure-based heuristics could be applied. We empirically show the effectiveness of the AugNN and the decomposition approach on many benchmark problems in the literature. For the 1210 benchmark problems tested, 917 problems were solved to optimality and the average gap between the obtained solution and the upper bound for all the problems was reduced to under 0.66% and computation time averaged below 33?s per problem. We also discuss the computational complexity of our approach.     

Mixed integer programming based heuristics for the Patient Admission Scheduling problem

The Patient Admission Scheduling (PAS) problem is a combinatorial optimization problem where elective patients are automatically assigned to beds for the duration of their stay considering not only the medical necessity but also the patient preferences. Due to its combinatorial nature, solving the previously published problem instances to optimality is a difficult task. In this paper, we present two Mixed Integer Programming (MIP) based heuristics namely Fix-and-Relax (F&R) and Fix-and-Optimize (F&O) where PAS problem instances are decomposed into sub-problems and then the sub-problems are optimized. Our approach uses patient decomposition considering patient length-of-Stay (LoS), room preference, admission date, specialism choice, and age, as well as time decomposition considering different optimization window sizes. Quick solutions generated by F&R heuristic are used as an input to the F&O heuristic and are improved in an iterative nature. Main goal of the study is to provide high quality solutions in shorter run times. Computational findings show that the proposed heuristics provide promising results towards the solution of the problem in faster CPU times than the previously reported values where less than 15 percent gap from the best known solutions is achieved for most of the test instances, and as low as 5 percent gap for some of the sample data.     

Grammar-based generation of stochastic local search heuristics through automatic algorithm configuration tools

Several grammar-based genetic programming algorithms have been proposed in the literature to automatically generate heuristics for hard optimization problems. These approaches specify the algorithmic building blocks and the way in which they can be combined in a grammar; the best heuristic for the problem being tackled is found by an evolutionary algorithm that searches in the algorithm design space defined by the grammar.In this work, we propose a novel representation of the grammar by a sequence of categorical, integer, and real-valued parameters. We then use a tool for automatic algorithm configuration to search for the best algorithm for the problem at hand. Our experimental evaluation on the one-dimensional bin packing problem and the permutation flowshop problem with weighted tardiness objective shows that the proposed approach produces better algorithms than grammatical evolution, a well-established variant of grammar-based genetic programming. The reasons behind such improvement lie both in the representation proposed and in the method used to search the algorithm design space.     

Efficient algorithms for real-life instances of the variable size bin packing problem

In this paper we present a local search heuristic for real-life instances of the variable size bin packing problem, and an exact algorithm for small instances. One important issue our heuristic is able to satisfy is that solutions must be delivered within (milli)seconds and that the solution methods should be robust to last minute changes in the data. Furthermore we show that we are able to incorporate the concept of usable leftovers on a single bin, and the implementation of many additional constraints should be supported by the straightforward solution representation. The heuristic is compared to others from the literature, and comes out ahead on a large subset of the instances.     

Scatter search technique for exam timetabling

At universities where students enjoy flexibility in selecting courses, the Registrar’s office aims to generate an appropriate exam timetable for numerous courses and large number of students. An appropriate, real-world exam timetable should show fairness towards all students, respecting the following constraints: (a) eliminating or minimizing the number of simultaneous exams; (b) minimizing the number of consecutive exams; (c) minimizing the number of students with two or three exams per day (d) eliminating the possibility of more than three exams per day (e) exams should fit in rooms with predefined capacity; and (f) the number of exam periods is limited. These constraints are conflicting, which makes exam timetabling intractable. Hence, solving this problem in realistic time requires the use of heuristic approaches. In this work, we develop an evolutionary heuristic technique based on the scatter search approach for finding good suboptimal solutions for exam timetabling. This approach is based on maintaining and evolving a population of solutions. We evaluate our suggested technique on real-world university data and compare our results with the registrar’s manual timetable in addition to the timetables of other heuristic optimization algorithms. The experimental results show that our adapted scatter search technique generates better timetables than those produced by the registrar, manually, and by other meta-heuristics.     

A graph coloring constructive hyper-heuristic for examination timetabling problems

In this work we investigate a new graph coloring constructive hyper-heuristic for solving examination timetabling problems. We utilize the hierarchical hybridizations of four low level graph coloring heuristics, these being largest degree, saturation degree, largest colored degree and largest enrollment. These are hybridized to produce four ordered lists. For each list, the difficulty index of scheduling the first exam is calculated by considering its order in all lists to obtain a combined evaluation of its difficulty. The most difficult exam to be scheduled is scheduled first (i.e. the one with the minimum difficulty index). To improve the effectiveness of timeslot selection, a?roulette wheel selection mechanism is included in the algorithm to probabilistically select an appropriate timeslot for the chosen exam. We test our proposed approach on the most widely used un-capacitated Carter benchmarks and also on the recently introduced examination timetable dataset from the 2007 International Timetabling Competition. Compared against other methodologies, our results demonstrate that the graph coloring constructive hyper-heuristic produces good results and outperforms other approaches on some of the benchmark instances.     

求解二维装箱问题的强化学习启发式算法

二维带形装箱问题是一个经典的NP-hard的组合优化问题,该问题在实际的生活和工业生产中有着广泛的应用.研究该问题,对企业节约成本、节约资源以及提高生产效率有着重要的意义.提出了一个强化学习求解算法.新颖地使用强化学习为启发式算法提供一个初始的装箱序列,有效地改善启发式冷启动的问题.该强化学习模型能进行自我驱动学习,仅使用启发式计算的解决方案的目标值作为奖励信号来优化网络,使网络能学习到更好的装箱序列.使用简化版的指针网络来解码输出装箱序列,该模型由嵌入层、解码器和注意力机制组成.使用Actor-Critic算法对模型进行训练,提高了模型的效率.在714个标准问题实例和随机生成的400个问题实例上测试提出的算法,实验结果显示：提出的算法能有效地改善启发式冷启动的问题,性能超过当前最优秀的启发式求解算法.