Two-dimensional Orthogonal Packing Research Articles

An effective genetic algorithm for solving the capacitated vehicle routing problem with two-dimensional loading constraint

In this article, we focus on the symmetric capacitated vehicle routing problem where customer demand is composed of two-dimensional weighted items. The objective consists in designing a set of trips, starting and terminating at a central depot, that minimise the total transportation cost with a homogenous fleet of vehicles based on a depot node. Items in each vehicle trip must satisfy the two-dimensional orthogonal packing constraint. The capacitated vehicle routing problem with two-dimensional loading constraint is an NP-hard problem of high complexity. Given the importance of this problem, many solution approaches have been developed. However, it still a challenging problem. Then, we propose to use a new heuristic based on an adaptive genetic algorithm in order to find better solution. Our algorithm is tested with 150 benchmark instances and compared with state-of-the-art approaches. Results shown that our proposed approach is competitive in terms of the quality of the solutions found.

An effective genetic algorithm for solving the capacitated vehicle routing problem with two-dimensional loading constraint

In this article, we focus on the symmetric capacitated vehicle routing problem where customer demand is composed of two-dimensional weighted items. The objective consists in designing a set of trips, starting and terminating at a central depot, that minimise the total transportation cost with a homogenous fleet of vehicles based on a depot node. Items in each vehicle trip must satisfy the two-dimensional orthogonal packing constraint. The capacitated vehicle routing problem with two-dimensional loading constraint is an NP-hard problem of high complexity. Given the importance of this problem, many solution approaches have been developed. However, it still a challenging problem. Then, we propose to use a new heuristic based on an adaptive genetic algorithm in order to find better solution. Our algorithm is tested with 150 benchmark instances and compared with state-of-the-art approaches. Results shown that our proposed approach is competitive in terms of the quality of the solutions found.

A multi-objective genetic algorithm for a special type of 2D orthogonal packing problems

In this study, a multi-objective model is proposed for a specific two-dimensional orthogonal packing problem. The model has three objectives, namely: (i) to minimise the distances between the centres of rectangular items, (ii) to use a minimisation function to place the related rectangular items close to each other, and (iii) to minimise the area of each rectangular item that falls outside of the radius. In addition to the weighted-sum method, the model uses the conic scalarisation method for scalarisation of the nonlinear multi-objective optimisation problem. The proposed model is a hybridisation of a placement procedure and a multi-objective genetic algorithm, and introduces the concept of the defensive radius to the literature. The model is tested on a randomly generated set of instances and some test problems. The computational results validate the quality of the solutions and the effectiveness of the proposed multi-objective model compared to the cluster boundary algorithm. Besides, the results show that the conic scalarisation outperforms the weighted-sum method in terms of scalar results in most cases.

An open space based heuristic for the 2D strip packing problem with unloading constraints

This paper studies the two-dimensional strip packing problem with unloading constraints. Given a set of items (represented by rectangles) belonging to different customers and a rectangular strip with fixed width and open length, the objective is to pack all the items into the strip such that the used length is minimized. In addition, the resulting packing must satisfy the unloading constraints. Based on the characteristics of this problem, this paper employs the open space to represent the candidate position. An open space based first-fit heuristic is proposed to generate a packing pattern for a given sequence of items. An implementation using the segment tree is provided. Finally, a randomized local search without any parameter is used to improve the solution by trying different sequences. The computational results on well-known instances show that the proposed approach outperforms existing approaches in the literature. In additional, we test our approach on the two-dimensional orthogonal packing problem with unloading constraints, the results show that our approach is capable of finding feasible solutions for 283 open instances.

A New Search Procedure for the Two-dimensional Orthogonal Packing Problem

In this paper we propose a new exact procedure for the two-dimensional orthog- onal packing problem, based on F. Clautiaux et al. approach (Clautiaux et al. Eur. J. Oper. Res. 183(3), 1196-1211, 2007). The principle consists in searching first the positions of the items on the horizontal axis, so as that, at each position, the sum of the heights of the items does not exceed the height of the bin. Each time a valid placement of all the items is encountered, another procedure determines if it can be extended to a solution of the pack- ing problem, searching the positions of the items on the vertical axis. Novel aspects of our approach include a simple and efficient search procedure, which only generates restricted placements, at least in a first stage, in order to reduce the search space, and the memoriza- tion of unsuccessful configurations, which are then used to detect dead-ends. We tested our implementation on a selection of orthogonal packing problems and strip packing problems, and we compared our results with those of recent successful approaches.

A variable neighborhood search for the capacitated vehicle routing problem with two-dimensional loading constraints

Abstract This paper addresses the capacitated vehicle routing problem with two-dimensional loading constraints (2L-CVRP), which is a generalized capacitated vehicle routing problem in which customer demand is a set of two-dimensional, rectangular, weighted items. The objective is to design the route set of minimum cost for a homogenous fleet of vehicles, starting and terminating at a central depot, to serve all the customers. All the items packed in one vehicle must satisfy the two-dimensional orthogonal packing constraints. A variable neighborhood search is proposed to address the routing aspect, and a skyline heuristic is adapted to examine the loading constraints. To speed up the search process, an efficient data structure (Trie) is utilized to record the loading feasibility information of routes, but also to control the computational effort of the skyline spending on the same route. The effectiveness of our approach is verified through experiments on widely used benchmark instances involving two distinct versions of loading constraints ( unrestricted and sequential versions). Numerical experiments show that the proposed method outperforms all existing methods and improves or matches the majority of best known solutions for both problem versions.

A Hybrid Demon Algorithm for the Two-Dimensional Orthogonal Strip Packing Problem

This paper develops a hybrid demon algorithm for a two-dimensional orthogonal strip packing problem. This algorithm combines a placement procedure based on an improved heuristic, local search, and demon algorithm involved in setting one parameter. The hybrid algorithm is tested on a wide set of benchmark instances taken from the literature and compared with other well-known algorithms. The computation results validate the quality of the solutions and the effectiveness of the proposed algorithm.

An Exact Algorithm for the Two-Dimensional Orthogonal Packing Problem with Unloading Constraints

This paper describes an exact algorithm for solving a two-dimensional orthogonal packing problem with unloading constraints, which occurs as a subproblem of mixed vehicle routing and loading problems. The packing considered in this work is basically a feasibility problem involving a single bin. The problem is addressed through a decomposition approach wherein a branch-and-cut algorithm is designed for solving a one-dimensional relaxation of the original problem. When an integer solution is found in the branching tree, a subsidiary problem is solved to identify a two-dimensional packing that does not lead to any overlap and satisfies the unloading constraints. Cuts are added when the subsidiary problem proves to be infeasible. Several preprocessing techniques aimed at reducing the size of the solution space and uncovering infeasibility are also described. A numerical comparison with the best known exact method is reported at the end based on benchmark instances.

Scheduling inspired models for two-dimensional packing problems

We propose two exact algorithms for two-dimensional orthogonal packing problems whose main components are simple mixed-integer linear programming models. Based on the different forms of time representation in scheduling formulations, we extend the concept of multiple time grids into a second dimension and propose a hybrid discrete/continuous-space formulation. By relying on events to continuously locate the rectangles along the strip height, we aim to reduce the size of the resulting mathematical problem when compared to a pure discrete-space model, with hopes of achieving a better computational performance. Through the solution of a set of 29 test instances from the literature, we show that this was mostly accomplished, primarily because the associated search strategy can quickly find good feasible solutions prior to the optimum, which may be very important in real industrial environments. We also provide a comprehensive comparison to seven other conceptually different approaches that have solved the same strip packing problems.

A multi-start evolutionary local search for the two-dimensional loading capacitated vehicle routing problem

This paper addresses an extension of the capacitated vehicle routing problem where customer demand is composed of two-dimensional weighted items (2L-CVRP). The objective consists in designing a set of trips minimizing the total transportation cost with a homogenous fleet of vehicles based on a depot node. Items in each vehicle trip must satisfy the two-dimensional orthogonal packing constraints. A GRASP×ELS algorithm is proposed to compute solutions of a simpler problem in which the loading constraints are transformed into resource constrained project scheduling problem (RCPSP) constraints. We denote this relaxed problem RCPSP-CVRP. The optimization framework deals with RCPSP-CVRP and lastly RCPSP-CVRP solutions are transformed into 2L-CVRP solutions by solving a dedicated packing problem. The effectiveness of our approach is demonstrated through computational experiments including both classical CVRP and 2L-CVRP instances. Numerical experiments show that the GRASP×ELS approach outperforms all previously published methods.

A hybrid genetic algorithm-heuristic for a two-dimensional orthogonal packing problem

In this paper we address a two-dimensional (2D) orthogonal packing problem, where a fixed set of small rectangles has to be placed on a larger stock rectangle in such a way that the amount of trim loss is minimized. The algorithm we propose hybridizes a placement procedure with a genetic algorithm based on random keys. The approach is tested on a set of instances taken from the literature and compared with other approaches. The computation results validate the quality of the solutions and the effectiveness of the proposed algorithm.

A new exact method for the two-dimensional orthogonal packing problem

The two-dimensional orthogonal packing problem (2 OPP) consists in determining if a set of rectangles (items) can be packed into one rectangle of fixed size (bin). In this paper we propose two exact algorithms for solving this problem. The first algorithm is an improvement on a classical branch&bound method, whereas the second algorithm is based on a new relaxation of the problem. We also describe reduction procedures and lower bounds which can be used within enumerative methods. We report computational experiments for randomly generated benchmarks which demonstrate the efficiency of both methods: the second method is competitive compared to the best previous methods. It can be seen that our new relaxation allows an efficient detection of non-feasible instances.

A new constraint programming approach for the orthogonal packing problem

The two-dimensional orthogonal packing problem (2OPP) consists in determining if a set of rectangles can be packed in a larger rectangle of fixed size. We propose an exact method for 2OPP, based on a new constraint-based scheduling model. We provide a generalization of energetic reasoning techniques for the problem under investigation. Feasibility tests requiring the solution of subset-sum problems are described. Computational results confirm the efficiency of our method compared to others in the literature.

Application of a mixed simulated annealing-genetic algorithm heuristic for the two-dimensional orthogonal packing problem

In this paper a pure meta-heuristic (genetic algorithm) and a mixed meta-heuristic (simulated annealing-genetic algorithm) were applied to two-dimensional orthogonal packing problems and the results were compared. The major motivation for applying a modified genetic algorithm is as an attempt to alleviate the problem of pre-mature convergence. We found that in the long run, the mixed heuristic produces better results; while the pure heuristic produces only “good” results, but produces them faster.

PERFORMANCE BOUNDS FOR STATIC MULTIPROCESSOR SCHEDULING OF MULTI-TASK JOBS

Job scheduling on multiprocessor systems is studied here as a special case of oriented two-dimensional orthogonal bin packing. Each job has subtasks which can be processed in parallel, requiring multiple processors to be allocated to each job. Then each job corresponds to a rectangle with sides equal to the processor requirement and the processing time. We study two classes of algorithms: (i) Longest processing time first (LPT) algorithms, and (ii) Largest processor requirement first (LPR) algorithms. We obtain improved asymptotic upper bounds for these algorithms compared to the bounds of the corresponding algorithms for the general two-dimensional packing problem. This is due to the discrete nature of the processor requirement (dimension) of jobs. We find that the LPR algorithms have better asymptotic upper bound on the makespan compared to the LPT algorithms. Specifically, the bound is 7/4 for the LPR algorithms whereas it is 2 for the LPT algorithms. Moreover, LPR algorithms are found to be more suited for dynamic job scheduling.

The Three-Dimensional Pallet Chart: An Analysis of the Factors Affecting the Set of Feasible Layouts for a Class of Two-Dimensional Packing Problems

The two-dimensional orthogonal packing problem of packing identical rectangles into a large containing rectangle is important in pallet loading and has recently received much attention in O.R. publications. In this paper, we examine the conditions under which the set of feasible layouts remains unchanged and show that these conditions can be represented by a series of planes in three-dimensional space. We call this representation the three-dimensional pallet chart because it is an extension of the two-dimensional pallet charts presently used in many practical situations. The strength of this result is demonstrated by three examples of its use. Accurate two-dimensional charts are produced from the three-dimensional version with a minimum of calculation, and a complete sensitivity analysis to changes in box and pallet dimensions can be carried out visually by viewing the chart at different angles. Finally, the result is used to generate a new procedure for determining the maximum number of rectangles which can be fitted. This method is shown to be accurate for over 90% of observations in a random sample of 5000—an improvement of 20% over previous methods.