For planning rearrangements of objects in a clutter, it is required to know the goal configuration of the objects. However, in real life scenarios, this information is not available most of the time. We introduce a novel method that computes a collision-free placement of objects on a cluttered surface, while minimizing the total number and amount of displacements of the existing moveable objects. Our method applies nested local searches that perform multi-objective optimizations guided by heuristics. Experimental evaluations demonstrate high computational efficiency and success rate of our method, as well as good quality of solutions.
Abstract Cutting and packing problems appear under various names in literature, e.g. cutting stock or trim loss problem, bin or strip packing problem, vehicle, pallet or container loading problem, nesting problem, knapsack problem etc. The paper develops a consistent and systematic approach for a comprehensive typology integrating the various kinds of problems. The typology is founded on the basic logical structure of cutting and packing problems. The purpose is to unify the different use of notions in the literature and to concentrate further research on special types of problems. We propose a new constructive algorithm, called HAPE3D, which is a heuristic algorithm based on the principle of minimum total potential energy for the 3D irregular packing problem, involving packing a set of irregularly shaped polyhedrons into a box-shaped container with fixed width and length but unconstrained height. The objective is to allocate all the polyhedrons in the container, and thus minimize the waste or maximize profit. HAPE3D can deal with arbitrarily shaped polyhedrons, which can be rotated around each coordinate axis at different angles. The most outstanding merit is that HAPE3D does not need to calculate no-fit polyhedron (NFP), which is a huge obstacle for the 3D packing problem. HAPE3D can also be hybridized with a meta-heuristic algorithm such as simulated annealing. Two groups of computational experiments demonstrate the good performance of HAPE3D and prove that it can be hybridized quite well with a meta-heuristic algorithm to further improve the packing quality. Abstract We study the problem of packing a given collection of arbitrary, in general concave, polyhedra into a cuboid of minimal volume. Continuous rotations and translations of polyhedra are allowed. In addition, minimal allowable distances between polyhedra are taken into account. We derive an exact mathematical model using adjusted radical free quasi phi-functions for concave polyhedra to describe non-overlapping and distance constraints. The model is a nonlinear programming formulation. We develop an efficient solution algorithm, which employs a fast starting point algorithm and a new compaction procedure. The procedure reduces our problem to a sequence of nonlinear programming subproblems of considerably smaller dimension and a smaller number of nonlinear inequalities. The benefit of this approach is borne out by the computational results, which include a comparison with previously published instances and new instances. This paper investigates the combinatorial and computational aspects of certain extremal geometric problems in two and three dimensions. Specifically, we examine the problem of intersecting a convex subdivision with a line in order to maximize the number of intersections. A similar problem is to maximize the number of intersected facets in a cross-section of a three-dimensional convex polytope. Related problems concern maximum chains in certain families of posets defined over the regions of a convex subdivision. In most cases we are able to prove sharp bounds on the asymptotic behavior of the corresponding extremal functions. We also describe polynomial algorithms for all the problems discussed. We present an efficient solution method for packing d-dimensional polytopes within the bounds of a polytope container. The central geometric operation of the method is an exact one-dimensional translation of a given polytope to a position which minimizes its volume of overlap with all other polytopes. We give a detailed description and a proof of a simple algorithm for this operation in which one only needs to know the set of (d-1)-dimensional facets in each polytope. Handling non-convex polytopes or even interior holes is a natural part of this algorithm. The translation algorithm is used as part of a local search heuristic and a meta-heuristic technique, guided local search, is used to escape local minima. Additional details are given for the three-dimensional case and results are reported for the problem of packing polyhedra in a rectangular parallelepiped. Utilization of container space is improved by an average of more than 14 percentage points compared to previous methods. The translation algorithm can also be used to solve the problem of maximizing the volume of intersection of two polytopes given a fixed translation direction. For two polytopes with complexity O(n) and O(m) and a fixed dimension, the running time is O(nmlog(nm)) for both the minimization and maximization variants of the translation algorithm.   The problem addressed in this paper is that of orthogonally packing a given set of rectangular-shaped items into the minimum number of three-dimensional rectangular bins. The problem is strongly NP-hard and extremely difficult to solve in practice. Lower bounds are discussed, and it is proved that the asymptotic worst-case performance ratio of the continuous lower bound is ?. An exact algorithm for filling a single bin is developed, leading to the definition of an exact branch-and-bound algorithm for the three-dimensional bin packing problem, which also incorporates original approximation algorithms. Extensive computational results, involving instances with up to 90 items, are presented: It is shown that many instances can be solved to optimality within a reasonable time limit.