\sectionnn{Introduction}

Bin packing is the process of packing a set of items of different sizes into
containers of a fixed capacity in a way that minimizes the number of containers
used. This has applications in many fields, such as logistics, where we want to
optimize the storage and transport of items in boxes, containers, trucks, etc.

Building mathematical models for bin packing is useful in understanding the
problem and in designing better algorithms, depending on the use case. An
algorithm optimized for packing cubes into boxes will not perform as well as
another algorithm for packing long items into trucks. Studying the mathematics
behind algorithms provides us with a better understanding of what works best.
When operating at scale, every small detail can have a huge impact on overall
efficiency and cost. Therefore, carefully developing algorithms based on solid
mathematical models is crucial. As we have seen in our Automatics class, a
small logic breach can be an issue in the long run in systems that are supposed
to run autonomously. This situation can be avoided by using mathematical models
during the design process wich will lead to better choices welding economic and
relibility concerns.

We will conduct a probabilistic analysis of multiple algorithms and compare
results to theoretical values. We will also consider the algoriths complexity
and performance, both in resource consumption and in box usage.

\clearpage

\section{Bin packing use cases}

Before studying the mathematics behind bin packing algorithms, we will have a
look at the motivations behind this project.

Bin packing has applications in many fields and allows to automize and optimize
complex systems. We will illustrate with examples focusing on two use cases:
logistics and computer science. We will consider examples of multiple dimensions
to show the versatility of bin packing algorithms.

\paragraph{} In the modern day, an effective supply chain relies on an automated production
thanks to sensors and actuators installed along conveyor belts. It is often
required to implement a packing procedure. All of this is controlled by a
computer system running continuously.

\subsection{3D : Containers}

Storing items in containers can be a prime application of bin packing. These
tree-dimensional objects of standardized size are used to transport goods.
While the dimensions of the containers are predictable, those of the transported
items are not. Storage is furthermore complicated by the fact that there can be
a void between items, allowing to move around. Multiple types of items can also
be stored in the same container.

There are many ways to optimize the storage of items in containers. For
example, by ensuring items are of an optimal standardized size or by storing a
specific item in each container, both eliminating the randomness in item size.
In these settings, it is easy to fill a container by assimilating them to
rectangular blocks. However, when items come in pseudo-random dimensions, it is
intuitive to start filling the container with larger items and then filling the
remaining gaps with smaller items. As containers must be closed, in the event
of an overflow, the remaining items must be stored in another container.

\subsection{2D : Cutting stock problem}

In industries such as woodworking bin packing algorithms are utilized to
minimize material waste when cutting large planks into smaller pieces of
desired sizes. Many tools use this two-dimensional cut process. For example, at
the Fabric'INSA Fablab, the milling machine, laser cutter and many more are
used to cut large planks of wood into smaller pieces for student projects. In
this scenario, we try to organize the desired cuts in a way that minimizes the
unusable excess wood.

\begin{figure}[ht]
	\centering
	\includegraphics[width=0.65\linewidth]{graphics/fraiseuse.jpg}
	\caption[]{Milling machine at the Fabric'INSA Fablab \footnotemark}
	\label{fig:fraiseuse}
\end{figure}
\footnotetext{Photo courtesy of Inés Bafaluy}

Managing the placement of items of complex shapes can be optimized by using
by various algorithms minimizing the waste of material.

\subsection{1D : Networking}

on which humans
have decreasing control.

In this paper, we will focus on one-dimensional bin packing, where we try to
store items of different heights in a linear container.


\section{Next Fit Bin Packing algorithm}

\cite{hofri:1987}
% TODO mettre de l'Histoire

\section{Next Fit Dual Bin Packing algorithm}

\section{Algorithm comparisons and optimization}

\subsection{NFBP vs NFDBP}

\subsection{Optimal algorithm}

\cite{bin-packing-approximation:2022}

\sectionnn{Conclusion}