The of the spatial arrangement of cargo inside

The Container Loading
Problem (CLP) is real-world driven, combinatorial optimization problem that
confronts the optimization of the spatial arrangement of cargo inside the
containers or transportation vehicles, by the maximum use of space.

An assignment problem
can have two main basic objectives: the value of the cargo loaded is more than
the number of containers to accommodate the cargo or the value of the cargo
loaded is less than the number of containers to accommodate the cargo.

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The problem correlates
with 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. In
this paper, we deal with two types of problem with the maximum output on the
objective. Based on the cargo heterogeneity these problems can be classified
either as three-dimensional, rectangular single large object placement problems
(3D-SLOPP) or as three-dimensional, rectangular single knapsack problems (3D-SKP).

The CLP is eminently
related to the field of transport management. The globalization have resulted
in exchange of products and services by increased numbers and to distances
maximizing the number of origins and destinations and in containerization which
is the definitive method of transporting goods and cargo worldwide. This will
result in increasing the number of challenges required to attain an efficient
transport system and also for upholding the prosperity and economic
development. The problems relevant to the urban freight transportation
(Sánchez-Díaz et al., 2015) or the designs of intermodal networks (Meng and
Wang, 2011) are the new problems that benefits with the decrease of the
congestion of cargo transport units with a resourceful use of the container

The conditions to be
fulfilled while preparing for loading the cargo into the containers: damage to
cargo should not happen during transportation, efficient usage of
transportation space and worker’s safety should not be violated while loading
and unloading of cargo.

However, the solution
to the real-world scenarios will be limited if the real-world constraints such
as cargo stability, container weight-limit and cargo orientation constraints
are not being fulfilled. One of the most important CLP constraint is cargo
stability because it leverages the safety of the cargo, both the workers
associated with loading operations and transportation and the also the vehicle
during transportation.

With respect to the CLP
literature, the splitting of static and dynamic stability is dealt by cargo
stability. The cargo static stability has been assured by the establishment of
full support constraint on the base of the boxes. Although assuring the cargo
static stability, it results in extreme restriction on the use of cargo space
and on meeting the real-world needs when e.g., overhanging cargo is allowed. The
rather oversimplified way static stability way has been treated by the majority
of the authors and also present in existing approaches to dynamic stability,
where the mean number of boxes supporting the items excluding those placed
directly on the floor and the percentage of boxes with insufficient lateral
support defines the stability (Ramos et al., 2015).

The CLP dealt in this
work can be declared as follows: A given set of small items of parallelepiped
shape of type k (k = 1 , . . . , K) (known as boxes), B
= b 1 , b 2 , . . . , b K , where each box type, in
quantity 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 (known
as 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 the
container, while confronting the following geometric loading constraints:

Each face of a box must be parallel to
one of the faces of the container;

No overlap between the boxes must be

All boxes must be placed entirely within
the container;

According to one of the possible
orientations of the box, each box must be sited and each box type can have up
to six possible orientations


The mechanical
properties of the container and the boxes also require the following additional
practical constraints:

Through the container entrance only the
box can be loaded;

Static stability while loading cargo
each box must maintain its loading position undisturbed;

All boxes are rigid;

The geometric centre pretends the centre
of gravity of each box.



The x, y and z
axes is placed parallel to the dimensions ( D, W, H ) of container C,
respectively, of the first octant of a Cartesian coordinates system, with the
back-bottom-left corner lying at the origin of the coordinates system. The
placement of a box b i in the container is set by its minimum and
maximum coordinates, ( x 1 i , y 1 i , z 1 i
) and ( x 2 i , y 2 i , z 2 i ),

The main objective of the
work is to introduce an algorithm for the CLP that deals with the cargo stability
under the realistic framework. The proposed algorithm combines a
multi-population biased random-key genetic algorithm with a constructive
heuristic that imposes a static stability constraint based on the static mechanical
equilibrium conditions applied to rigid bodies derived from Newton’s laws of
motion. Empty spaces are managed by the constructive heuristic using maximal-spaces
representation and maximal-spaces are filled by the layer approach.