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.

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

space.

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:

1.

Each face of a box must be parallel to

one of the faces of the container;

2.

No overlap between the boxes must be

present;

3.

All boxes must be placed entirely within

the container;

4.

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:

1.

Through the container entrance only the

box can be loaded;

2.

Static stability while loading cargo

each box must maintain its loading position undisturbed;

3.

All boxes are rigid;

4.

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 ),

respectively.

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.