The container-loading problem aims to determine the arrangement of items in a container. Researchers approach this 3D, NP-hard problem 1 using heuristic methods. Usually, the CLP aims to maximize loading efficiency-that is, the container space use. Here, the problem we address involves only one container with known dimensions, and the cargo varies from weakly to strongly heterogeneous, independent of the total number of boxes. We consider three requirements related to the load's physical arrangement and to the transportation requirements: box orientation (for example, "this side up"), cargo stability, and container volume. Although considering both volume use and cargo stability could lead to a bi-objective CLP, we tackle cargo stability as a constraint in the constructive phase of the algorithm, and we explicitly consider volume use as the only objective in the constructive and local-search phases.In this article, we present GRMODGRASP, a new algorithm for the CLP based on the GRASP (greedy randomized adaptive search procedure) paradigm. 2 We evaluate GRMODGRASP's performance in terms of volume use and load stability and by comparing it with nine well-known algorithms. Our approach produces solutions that surpass other approaches' solutions in terms of volume use and cargo stability. The Modified George and Robinson heuristicWe based GRMODGRASP on GRMOD, an improved version of the George and Robinson heuristic. 3 This wall-building heuristic packs boxes in a container, with an opening in the front, from the back to the front along its length.One modification to the George and Robinson heuristic relates to the container length. 4 The original heuristic considers an infinite-length container (the 3D strip-packing problem), but it doesn't guarantee that the resulting packing will have a length equal to or less than the container's length; GRMOD deals with a finite-length container. With this modification, we can eliminate the George and Robinson algorithm's "unsuccessful packing" and "automatic repacking" procedures, which basically compare the final packing length with the container's length and reapply the heuristic with different parameters. Successively executing the George and Robinson heuristic might obtain a feasible solution if the cargo's total volume is equal to or less than the container's volume.Another modification addresses packing the container's final layers. 4 The George and Robinson heuristic uses a minimal-length parameter that inhibits constructing new layers at the end of the packing process. This causes layers with low volume use. In GRMOD (see figure 1), the layer depth dimension depends on the unpacked boxes' volume. So, the container's final layers have a smaller depth but better volume use. GRMOD incorporates the two modifications just described plus two improvements that we introduced to improve cargo stability. The first deals with new-space generation, and the other relates to the flexible-width value. Constructive heuristicLike the George and Robinson heuristic, the GRMOD constructive...

Real-world distribution problems raise some practical considerations usually not considered in a realistic way in more theoretical studies. One of these considerations is related to the vehicle capacity, not only in terms of cubic meters or weight capacity but also in terms of the cargo physical arrangements. In a distribution scene, two combinatorial optimization problems, the Vehicle Routing Problem with Time Windows and the Container Loading Problem, are inherently related to each other. This work presents a framework to integrate these two problems using two different resolutions methods. The first one treats the problem in a sequential approach while the second uses a hierarchical approach. In order to test the quality and efficiency of the proposed approaches some test problems were created based on the well-known

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Abstract. This work presents the Vehicle Routing with Time Windows and Loading Problem (VRTWLP) as a multi-objective optimization problem, implemented within a Genetic Algorithm. Specifically, the three dimensions of the problem to be optimized -the number of vehicles, the total travel distance and volume utilization -are considered to be separated dimensions of a multiobjective space. The quality of the solution obtained using this approach is evaluated and compared with results of other heuristic approaches previously developed by the author. The most significant contribution of this work is our interpretation of VRTWLP as a Multi-objective Optimization Problem.

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This paper deals with a two-stage packing problem that has to be solved in the daily distribution process of a Portuguese trading company. In the first stage, boxes including goods are to be packed on pallets, while in the second stage these pallets are loaded into one or more trucks. The boxes have to be transported to different customers, and the main goal is to guarantee a sufficient utilization of the truck loading space. A two-stage packing procedure is proposed to cover both problem stages. First, boxes are loaded onto pallets by means of a well-known container loading algorithm. Then, trucks are filled with loaded pallets by a tree search algorithm. The performance of the two-stage approach was evaluated using a set of instances that are based on actual company data.

In this paper, a capacitated vehicle routing problem with mixed linehauls and backhauls is presented. The addressed problem belongs to the subclass of pickup and delivery problems. Loading and sequencing constraints are also considered. These constraints arise in many realworld situations, and can improve efficiency since backhaul customers do not need to be delayed in a route when it is possible to load their items without rearrengements. The suggested approach is based on an insertion heuristic presented in the literature but extended to tackle the explicit consideration of loading constraints. Different strategies are presented and tested. An exhaustive computational study is performed on instances adapted from benchmark instances for integrated routing and loading problems.