The of the spatial arrangement of cargo inside

The Container LoadingProblem (CLP) is real-world driven, combinatorial optimization problem thatconfronts the optimization of the spatial arrangement of cargo inside thecontainers or transportation vehicles, by the maximum use of space.An assignment problemcan have two main basic objectives: the value of the cargo loaded is more thanthe number of containers to accommodate the cargo or the value of the cargoloaded is less than the number of containers to accommodate the cargo.The problem correlateswith the wider combinatorial optimization class of Cutting and Packing problems.As per the typology of Wäscher et al. (2007) for Cutting and Packing problems,the problem can be divided based on dimension, collection of large items,combination of small items, kind of assignment and shape of the small items.

Inthis paper, we deal with two types of problem with the maximum output on theobjective. Based on the cargo heterogeneity these problems can be classifiedeither as three-dimensional, rectangular single large object placement problems(3D-SLOPP) or as three-dimensional, rectangular single knapsack problems (3D-SKP).The CLP is eminentlyrelated to the field of transport management.

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The globalization have resultedin exchange of products and services by increased numbers and to distancesmaximizing the number of origins and destinations and in containerization whichis the definitive method of transporting goods and cargo worldwide. This willresult in increasing the number of challenges required to attain an efficienttransport system and also for upholding the prosperity and economicdevelopment. The problems relevant to the urban freight transportation(Sánchez-Díaz et al., 2015) or the designs of intermodal networks (Meng andWang, 2011) are the new problems that benefits with the decrease of thecongestion of cargo transport units with a resourceful use of the containerspace.

The conditions to befulfilled while preparing for loading the cargo into the containers: damage tocargo should not happen during transportation, efficient usage oftransportation space and worker’s safety should not be violated while loadingand unloading of cargo.However, the solutionto the real-world scenarios will be limited if the real-world constraints suchas cargo stability, container weight-limit and cargo orientation constraintsare not being fulfilled. One of the most important CLP constraint is cargostability because it leverages the safety of the cargo, both the workersassociated with loading operations and transportation and the also the vehicleduring transportation.With respect to the CLPliterature, the splitting of static and dynamic stability is dealt by cargostability. The cargo static stability has been assured by the establishment offull support constraint on the base of the boxes. Although assuring the cargostatic stability, it results in extreme restriction on the use of cargo spaceand on meeting the real-world needs when e.g.

, overhanging cargo is allowed. Therather oversimplified way static stability way has been treated by the majorityof the authors and also present in existing approaches to dynamic stability,where the mean number of boxes supporting the items excluding those placeddirectly on the floor and the percentage of boxes with insufficient lateralsupport defines the stability (Ramos et al., 2015).The CLP dealt in thiswork can be declared as follows: A given set of small items of parallelepipedshape of type k (k = 1 , . . .

, K) (known as boxes), B= b 1 , b 2 , . . . , b K , where each box type, inquantity n k , is defined by its depth, width and height ( d k , w k, h k ) are to be loaded into a large object of parallelepiped shape (knownas a container), C , characterized by its depth, width and height, ( D,W, H ), with the aim of attaining a maximum usage of the volume of thecontainer, while confronting the following geometric loading constraints: 1.     Each face of a box must be parallel toone of the faces of the container; 2.

     No overlap between the boxes must bepresent; 3.     All boxes must be placed entirely withinthe container;4.     According to one of the possibleorientations of the box, each box must be sited and each box type can have upto six possible orientations  The mechanicalproperties of the container and the boxes also require the following additionalpractical constraints:1.     Through the container entrance only thebox can be loaded; 2.     Static stability while loading cargoeach box must maintain its loading position undisturbed; 3.     All boxes are rigid; 4.     The geometric centre pretends the centreof gravity of each box.   The x, y and zaxes is placed parallel to the dimensions ( D, W, H ) of container C,respectively, of the first octant of a Cartesian coordinates system, with theback-bottom-left corner lying at the origin of the coordinates system.

Theplacement of a box b i in the container is set by its minimum andmaximum coordinates, ( x 1 i , y 1 i , z 1 i) and ( x 2 i , y 2 i , z 2 i ),respectively.The main objective of thework is to introduce an algorithm for the CLP that deals with the cargo stabilityunder the realistic framework. The proposed algorithm combines amulti-population biased random-key genetic algorithm with a constructiveheuristic that imposes a static stability constraint based on the static mechanicalequilibrium conditions applied to rigid bodies derived from Newton’s laws ofmotion. Empty spaces are managed by the constructive heuristic using maximal-spacesrepresentation and maximal-spaces are filled by the layer approach.