Cutting and packing problems are present in many, at first glance
unconnected, areas, therefore it's beneficial to have a good understanding of
their underlying structure, to select proper techniques for finding solutions.
Cutting and packing problems are a class of combinatorial problems in which
there are specified two classes of objects: big and small items and the task is
to place the small items within big items. Even in the 1-dimensional case,
bin-packing is strongly NP-hard (Garey 1978), which suggests, that exact
solutions may not be found in a reasonable time for bigger instances. In the
literature, there are presented many various approaches to packing problems,
e.g. mixed-integer programming, approximation algorithms, heuristic solutions,
and local search algorithms, including metaheuristic approaches like Tabu
Search or Simulated Annealing.
The main goal of this work is to review existing solutions, survey the
variants arising from the industry applications, present a solution based on
constraint programming and compare its performance with the results in the
literature. Optimization with constraint programming is a method searching for
the global optima, hence it may require a higher workload compared to the
heuristic and local search approaches, which may finish in a local optimum. The
performance of the presented model will be measured on test data used in the
literature, which were used in many articles presenting a variety of approaches
to three-dimensional container packing, which will allow us to compare the
efficiency of the constraint programming model with other methods used in the
operational research