Andreas Bortfeldt

A hybrid genetic algorithm for the container loading problem

Abstract This paper presents a hybrid genetic algorithm (GA) for the container loading problem with boxes of different sizes and a single container for loading. Generated stowage plans include several vertical layers each containing several boxes. Within the procedure, stowage plans are represented by complex data structures closely related to the problem. To generate offspring, specific genetic operators are used that are based on an integrated greedy heuristic. The process takes several practical constraints into account. Extensive test calculations including procedures from other authors vouch for the good performance of the GA, above all for problems with strongly heterogeneous boxes.

A genetic algorithm for the two-dimensional strip packing problem with rectangular pieces

Given a set of rectangular pieces and a container of fixed width and variable length, the two-dimensional strip packing problem (2D-SPP) consists of orthogonally placing all the pieces within the container, without overlapping, such that the overall length of the layout is minimised. Until now mainly heuristics, for example genetic algorithms (GA), were proposed for the 2D-SPP which use encoded solutions that are manipulated by standard operators. In this paper a GA for the 2D-SPP is suggested that works without any encoding of solutions. Rather fully defined layouts are manipulated as such by means of specific genetic operators. Two additional constraints, namely the orientation constraint and the guillotine constraint, can be taken into account. The GA is subjected to a comprehensive test using benchmark instances with up to 5000 pieces. Compared to eleven competing methods from the literature the GA performs best.

A parallel tabu search algorithm for solving the container loading problem

This paper presents a parallel tabu search algorithm for the container loading problem with a single container to be loaded. The emphasis is on the case of a weakly heterogeneous load. The distributed-parallel approach is based on the concept of multi-search threads according to Toulouse et al. [Issues in designing parallel and distributed search algorithms for discrete optimization problems, Publication CRT-96-36, Centre de recherche sur les transports, Universitede Montreal, Canada, 1996] i.e., several search paths are investigated concurrently. The parallel searches are carried out by differently configured instances of a tabu search algorithm, which cooperate by the exchange of (best) solutions at the end of defined search phases. The parallel search processes are executed on a corresponding number of LAN workstations. The efficiency of the parallel tabu search algorithm is demonstrated by an extensive comparative test including well-known reference problems and loading procedures from other authors.

Constraints in container loading – A state-of-the-art review

Container loading is a pivotal function for operating supply chains efficiently. Underperformance results in unnecessary costs (e.g. cost of additional containers to be shipped) and in an unsatisfactory customer service (e.g. violation of deadlines agreed to or set by clients). Thus, it is not surprising that container loading problems have been dealt with frequently in the operations research literature. It has been claimed though that the proposed approaches are of limited practical value since they do not pay enough attention to constraints encountered in practice.

A parallel genetic algorithm for solving the container loading problem

This paper presents a parallel genetic algorithm (PGA) for the container loading problem with a single container to be loaded. The emphasis is on the case of a strongly heterogeneous load. The PGA follows a migration model. Several separate sub-populations are subjected to an evolutionary process independently of each other. At the same time the best individuals are exchanged between the sub-populations. The evolution of the different sub-populations is carried out on a corresponding number of LAN workstations. The quality of the PGA is demonstrated by an extensive comparative test including well-known reference problems and loading procedures from other authors.

A hybrid algorithm for the capacitated vehicle routing problem with three-dimensional loading constraints

The capacitated vehicle routing problem with three-dimensional loading constraints combines capacitated vehicle routing and three-dimensional loading with additional packing constraints concerning, for example, unloading operations. An efficient hybrid algorithm including a tabu search algorithm for routing and a tree search algorithm for loading is introduced. Computational results are presented for all publicly available test instances. Most of the best solutions previously reported in literature have been improved while the computational effort is drastically reduced compared to other methods.

A Tree Search Algorithm for Solving the Container Loading Problem

This paper presents a tree search algorithm for the three-dimensional container loading problem (3D-CLP). The 3D-CLP is the problem of loading a subset of a given set of rectangular boxes into a rectangular container so that the packing volume is maximized. The method includes two variants: the full-support variant guarantees full support from below for all packed boxes, although this constraint is not taken into account by the nonsupport variant. The guillotine cut constraint is considered by both variants. The method is mainly based on two concepts. On the one hand, the block building approach is generalized. Not only are blocks of identical boxes in the same spatial orientation applied but also blocks of different boxes with small inner gaps. On the other hand, the tree search is carried out in a special fashion called a partition-controlled tree search. This makes the search both efficient and diverse, enabling a sufficient search width as well as a suitable degree of foresight. The approach achieves excellent results for the well-known 3D-CLP instances suggested by Bischoff and Ratcliff [Bischoff, E. E., M. S. W. Ratcliff. 1995. Issues in the development of approaches to container loading. Omega23(4) 377--390] with reasonable computing time.

A tree search procedure for the container relocation problem

In the container relocation problem (CRP) n items are given that belong to G different item groups (g=1,?,G). The items are piled up in up to S stacks with a maximum stack height H. A move can either shift one item from the top of a stack to the top of another one (relocation) or pick an item from the top of a stack and entirely remove it (remove). A move of the latter type is only feasible if the group index of the item is minimum compared to all remaining items in all stacks. A move sequence of minimum length has to be determined that removes all items from the stacks. The CRP occurs frequently in container terminals of seaports. It has to be solved when containers, piled up in stacks, need to be transported to a ship or to trucks in a predefined sequence. This article presents a heuristic tree search procedure for the CRP. The procedure is compared to all known solution approaches for the CRP and turns out to be very competitive.

A heuristic for the three-dimensional strip packing problem

Abstract The contribution presents a heuristic for the three-dimensional strip packing problem (3D-SPP) with rectangular pieces (boxes). The considered 3D-SPP can be formulated as follows: for a given set of boxes and a given longitudinal open container, determine an arrangement of all boxes within the container so that the required container length is minimized. The presented heuristic was derived from a branch-and-bound approach for the container loading problem (CLP) that was recently proposed by Pisinger. Two approaches are investigated to adapt the CLP method to the 3D-SPP. The first approach looks directly at a container of practically infinite length. The second one solves an SPP instance by computing a series of CLP instances with descending container lengths. The best parts of previously best solutions are reused systematically within the second approach that proves to be more successful. The method is tested by means of 800 benchmark instances with up to 1000 boxes. A comparison to other methods from the literature shows the high performance of the 3D-SPP heuristic.

Ein Tabu Search-Verfahren für Containerbeladeprobleme mit schwach heterogenem Kistenvorrat

ZusammenfassungGegenstand des Beitrags ist die Entwicklung eines Tabu Search-Verfahrens (TSV) für Containerbeladeprobleme mit schwach heterogenem Kistenvorrat. Das TSV basiert auf einem zweistufigen Verfahrensansatz. Ein integrierter Greedy-Algorithmus dient unmittelbar der Erzeugung von Stauplänen. Vermittelt über eine geeignete Repräsentation zulässiger Problemlösungen wird in einer zweiten Verfahrensstufe eine Tabu-Suche durchgeführt. Das TSV berücksichtigt einige praxisrelevante Restriktionen. Die Performance des TSV wird anhand von Benchmarkproblemen aus der Literatur demonstriert, wobei Containerbeladeverfahren verschiedener Autoren zu Vergleichszwecken herangezogen werden.AbstractThe paper presents a tabu search algorithm (TSA) for weakly heterogeneous container loading problems. The approach includes two integrated components: The generation of different stowage plans by means of a greedy algorithm and a tabu search using an appropriate representation of these plans. The TSA considers some restrictions of practical relevance. The overall algorithm has been applied to numerous benchmark problems and its performance is demonstrated by a numerical comparison with several loading procedures suggested by other authors.