Applying self-adaptive evolutionary algorithms to two-dimensional packing problems using a four corners’ heuristic

This paper proposes a four corners’ heuristic for application in evolutionary algorithms (EAs) applied to two-dimensional packing problems. The four corners’ (FC) heuristic is specifically designed to increase the search efficiency of EAs. Experiments with the FC heuristic are conducted on 31 problems from the literature both with rotations permitted and without rotations permitted, using two different EA algorithms: a self-adaptive parallel recombinative simulated annealing (PRSA) algorithm, and a self-adaptive genetic algorithm (GA). Results on bin packing problems yield the smallest trim losses we have seen in the published literature; with the FC heuristic, zero trim loss was achieved on problems of up to 97 rectangles. A comparison of the self-adaptive GA to fixed-parameter GAs is presented and the benefits of self-adaption are highlighted.     

A hybrid heuristic algorithm for the 2D variable-sized bin packing problem

In this paper, we consider the two-dimensional variable-sized bin packing problem (2DVSBPP) with guillotine constraint. 2DVSBPP is a well-known NP-hard optimization problem which has several real applications. A mixed bin packing algorithm (MixPacking) which combines a heuristic packing algorithm with the Best Fit algorithm is proposed to solve the single bin problem, and then a backtracking algorithm which embeds MixPacking is developed to solve the 2DVSBPP. A hybrid heuristic algorithm based on iterative simulated annealing and binary search (named HHA) is then developed to further improve the results of our Backtracking algorithm. Computational experiments on the benchmark instances for 2DVSBPP show that HHA has achieved good results and outperforms existing algorithms.     

New perspectives in VLSI design automation: deterministic packing by Sequence Pair

In the paper we consider a problem of packing rectangular blocks on a plane, which is known as Block Packing Problem (BPP). This problem is a central issue of the modern VLSI chips design methods. Basing on a new interpretation of the Sequence-Pair representation of the packing solution-space, which is based on Complementary Mirror Constraint Graphs (CMCG), we develop the efficient method of neighborhood exploration. This method might be able to improve the efficiency of other neighborhood-based search methods, such as simulated annealing or tabu search, as well as, to construct efficient heuristics. We illustrate application of the developed method by constructing a heuristic algorithm for solving BPP and comparing its efficiency and effectiveness to the algorithms commonly used so far.     

A tree search method for the container loading problem with shipment priority

This paper addresses a special kind of container loading problem with shipment priority. We present a tree search method, which is based on a greedy heuristic. In the greedy heuristic, blocks made up of identical items with the same orientation are selected for packing into a container. Five evaluation functions are proposed for block selection, and the different blocks selected by each evaluation function constitute the branches of the search tree. A method of space splitting and merging is also embedded in the algorithm to facilitate efficient use of the container space. In addition, the proposed algorithm covers an important constraint called shipment priority to solve practical problems. The validity of the proposed algorithm is examined by comparing the present results with those of published algorithms using the same data.     

A heuristic for the circle packing problem with a variety of containers

In this paper we present a heuristic algorithm based on the formulation space search method to solve the circle packing problem. The circle packing problem is the problem of finding the maximum radius of a specified number of identical circles that can be fitted, without overlaps, into a two-dimensional container of fixed size. In this paper we consider a variety of containers: the unit circle, unit square, rectangle, isosceles right-angled triangle and semicircle. The problem is formulated as a nonlinear optimization problem involving both Cartesian and polar coordinate systems.Formulation space search consists of switching between different formulations of the same problem, each formulation potentially having different properties in terms of nonlinear optimization. As a component of our heuristic we solve a nonlinear optimization problem using the solver SNOPT.Our heuristic improves on previous results based on formulation space search presented in the literature. For a number of the containers we improve on the best result previously known. Our heuristic is also a computationally effective approach (when balancing quality of result obtained against computation time required) when compared with other work presented in the literature.     

Port space allocation with a time dimension

In the Port of Singapore, as in many other ports, space has to be allocated in yards for inbound and transit cargo. Requests for container space occur at different times during the planning period, and are made for different quantities and sizes of containers. In this paper, we study space allocation under these conditions. We reduce the problem to a two-dimensional packing problem with a time dimension. Since the problem is NP-hard, we develop heuristic algorithms, using tabu search, simulated annealing, a genetic algorithm and ‘squeaky wheel’ optimization, as solution approaches. Extensive computational experiments compare the algorithms, which are shown to be effective for the problem.     

A skyline heuristic for the 2D rectangular packing and strip packing problems

In this paper, we propose a greedy heuristic for the 2D rectangular packing problem (2DRP) that represents packings using a skyline; the use of this heuristic in a simple tabu search approach outperforms the best existing approach for the 2DRP on benchmark test cases. We then make use of this 2DRP approach as a subroutine in an “iterative doubling” binary search on the height of the packing to solve the 2D rectangular strip packing problem (2DSP). This approach outperforms all existing approaches on standard benchmark test cases for the 2DSP.     

A simulated annealing hyper-heuristic methodology for flexible decision support

Most of the current search techniques represent approaches that are largely adapted for specific search problems. There are many real-world scenarios where the development of such bespoke systems is entirely appropriate. However, there are other situations where it would be beneficial to have methodologies which are generally applicable to more problems. One of our motivating goals for investigating hyper-heuristic methodologies is to provide a more general search framework that can be easily and automatically employed on a broader range of problems than is currently possible. In this paper, we investigate a simulated annealing hyper-heuristic methodology which operates on a search space of heuristics and which employs a stochastic heuristic selection strategy and a short-term memory. The generality and performance of the proposed algorithm is demonstrated over a large number of benchmark datasets drawn from two very different and difficult problems, namely; course timetabling and bin packing. The contribution of this paper is to present a method which can be readily (and automatically) applied to different problems whilst still being able to produce results on benchmark problems which are competitive with bespoke human designed tailor made algorithms for those problems.     

An Application of Tabu Search Heuristic for the Maximum Edge-Weighted Subgraph Problem

The purpose of this article is to describe an efficient search heuristic for the Maximum Edge-weighted Subgraph (MEwS) problem. This problem requires to find a subgraph such that the sum of the weights associated with the edges of the subgraph is maximized subject to a cardinality constraint. In this study a tabu search heuristic for the MEwS problem is proposed. Different algorithms to obtain an initial solution are presented. One neighborhood search strategy is also proposed. Preliminary computational results are reported for randomly generated test problems of MEwS problem with different densities and sizes. For most of test problems, the tabu search heuristic found good solutions. In addition, for large size test problems, the tabu search outperformed the local search heuristic appearing in the literature.     

Memetic cooperative models for the tool switching problem

This work deals with memetic-computing agent-models based on the cooperative integration of search agents endowed with (possibly different) optimization strategies, in particular memetic algorithms. As a proof-of-concept of the model, we deploy it on the tool switching problem (ToSP), a hard combinatorial optimization problem that arises in the area of flexible manufacturing. The ToSP has been tackled by different algorithmic methods ranging from exact to heuristic methods (including local search meta-heuristics, population-based techniques and hybrids thereof, i.e., memetic algorithms). Here we consider an ample number of instances of this cooperative memetic model, whose agents are adapted to cope with this problem. A detailed experimental analysis shows that the meta-models promoting the cooperation among memetic algorithms provide the best overall results compared with their constituent parts (i.e., memetic algorithms and local search approaches). In addition, a parameter sensitivity analysis of the meta-models is developed in order to understand the interplay among the elements of the proposed topologies.     

Local search algorithms for the rectangle packing problem with general spatial costs

We propose local search algorithms for the rectangle packing problem to minimize a general spatial cost associated with the locations of rectangles. The problem is to pack given rectangles without overlap in the plane so that the maximum cost of the rectangles is minimized. Each rectangle has a set of modes, where each mode specifies the width and height of the rectangle and its spatial cost function. The spatial costs are general piecewise linear functions which can be non-convex and discontinuous. Various types of packing problems and scheduling problems can be formulated in this form. To represent a solution of this problem, a pair of permutations of n rectangles is used to determine their horizontal and vertical partial orders, respectively. We show that, under the constraint specified by such a pair of permutations, optimal locations of the rectangles can be efficiently determined by using dynamic programming. The search for finding good pairs of permutations is conducted by local search and metaheuristic algorithms. We report computational results on various implementations using different neighborhoods, and compare their performance. We also compare our algorithms with other existing heuristic algorithms for the rectangle packing problem and scheduling problem. These computational results exhibit good prospects of the proposed algorithms. Key words.rectangle packing – sequence pair – general spatial cost – dynamic programming – metaheuristicsMathematics Subject Classification (1991):20E28, 20G40, 20C20     

Random search in the one-dimensional cutting stock problem

The paper reports on a randomized approach to the one-dimensional cutting stock problem. In a sequential heuristic the next pattern to be processed is chosen according to the best outcome of 200 trials of random bin packing. The quality of the solutions is discussed and the problem of multiple occurrence of equivalent solutions with different patterns is explained by the cross over technique known from genetic algorithms. For one data instance with 27 demand lengths, reported by Haessler (1975), four different optimal solutions are given.     

Efficient bounds for the stable set,vertex cover and set packing problems

In this paper we describe a collection of efficient algorithms that deliver approximate solution to the weighted stable set, vertex cover and set packing problems. All algorithms guarantee bounds on the ratio of the heuristic solution to the optimal solution.     

A two-stage intelligent search algorithm for the two-dimensional strip packing problem

This paper presents a two-stage intelligent search algorithm for a two-dimensional strip packing problem without guillotine constraint. In the first stage, a heuristic algorithm is proposed, which is based on a simple scoring rule that selects one rectangle from all rectangles to be packed, for a given space. In the second stage, a local search and a simulated annealing algorithm are combined to improve solutions of the problem. In particular, a multi-start strategy is designed to enhance the search capability of the simulated annealing algorithm. Extensive computational experiments on a wide range of benchmark problems from zero-waste to non-zero-waste instances are implemented. Computational results obtained in less than 60 seconds of computation time show that the proposed algorithm outperforms the supposedly excellent algorithms reported recently, on average. It performs particularly better for large instances.     

Algorithms for single machine total tardiness scheduling with sequence dependent setups

We consider the problem of scheduling a single machine to minimize total tardiness with sequence dependent setup times. We present two algorithms, a problem space-based local search heuristic and a Greedy Randomized Adaptive Search Procedure (GRASP) for this problem. With respect to GRASP, our main contributions are—a new cost function in the construction phase, a new variation of Variable Neighborhood Search in the improvement phase, and Path Relinking using three different search neighborhoods. The problem space-based local search heuristic incorporates local search with respect to both the problem space and the solution space. We compare our algorithms with Simulated Annealing, Genetic Search, Pairwise Interchange, Branch and Bound and Ant Colony Search on a set of test problems from literature, showing that the algorithms perform very competitively.     

Human-guided search

We present a survey of techniques and results from the Human-Guided Search (HuGS) project, an effort to investigate interactive optimization. HuGS provides simple and general visual metaphors relating to local search operations that allow users to guide the exploration of the search space. These metaphors apply to a wide variety of problems and combinatorial optimization algorithms, which we demonstrate by describing the HuGS toolkit and as well as eight diverse applications we developed using it. User experiments show that human guidance can improve the performance of powerful heuristic search algorithms. HuGS is also a valuable development environment for understanding and improving optimization algorithms. Although HuGS was designed for human-computer interaction, for two different problems we have used the HuGS code base to develop completely automatic heuristic algorithms that produced at the time new best automatic results on benchmark problem instances.     

New and improved level heuristics for the rectangular strip packing and variable-sized bin packing problems

We consider two types of orthogonal, oriented, rectangular, two-dimensional packing problems. The first is the strip packing problem, for which four new and improved level-packing algorithms are presented. Two of these algorithms guarantee a packing that may be disentangled by guillotine cuts. These are combined with a two-stage heuristic designed to find a solution to the variable-sized bin packing problem, where the aim is to pack all items into bins so as to minimise the packing area. This heuristic packs the levels of a solution to the strip packing problem into large bins and then attempts to repack the items in those bins into smaller bins in order to reduce wasted space. The results of the algorithms are compared to those of seven level-packing heuristics from the literature by means of a large number of strip-packing benchmark instances. It is found that the new algorithms are an improvement over known level-packing heuristics for the strip packing problem. The advancements made by the new and improved algorithms are limited in terms of utilised space when applied to the variable-sized bin packing problem. However, they do provide results faster than many existing algorithms.     

A Method for Solving Container Packing for a Single Size of Box

There is an extensive literature on heuristic algorithms for two-dimensional cutting problems and three-dimensional packing, but there seems to be very little on the three-dimensional single-box type packing problem. This paper gives a structure for dealing with that problem as a heuristic. It also presents a set of upper bounds on the optimal fit. Finally, the paper compares a particular application of the algorithmic structure with the George and Robinson multiple-box type heuristic.     

A heuristic algorithm for the strip packing problem

The two-dimensional strip packing problem is to pack a given set of rectangles into a strip with a given width and infinite height so as to minimize the required height of the packing. From the computational point of view, the strip packing problem is an NP-hard problem. With the B*-tree representation, this paper first presents a heuristic packing strategy which evaluates the positions used by the rectangles. Then an effective local search method is introduced to improve the results and a heuristic algorithm (HA) is further developed to find a desirable solution. Computational results on randomly generated instances and popular test instances show that the proposed method is efficient for the strip packing problem.     

Constrained order packing: comparison of heuristic approaches for a new bin packing problem

Bin packing problems consist of allocating a set of small parts to a set of large bins by minimizing the number of used bins. Although several boundary conditions have been stated in the past, for example conflicts or restrictions on cutting and rotations, we introduce a set of constraints, which lead to a new problem structure. These constraints are motivated by the precast-concrete-part industry and represent requirements on the ordering of parts and their positions, machinery restrictions and due dates. Furthermore, we solve the problem using several heuristic approaches that are based on algorithms for the standard bin packing problem. Therefore, existing concepts are classified and adapted to fit the new problem, including Simulated Annealing, Genetic Algorithms, Tabu Search methods and SubSetSum based search routines. Finally, all proposed algorithms are tested and obtained results are discussed.