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

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 theoritical values. We will also consider the algoriths complexity
and performance, both in resource consumption and in box usage.





\clearpage

\section{Bin packing use cases}


\cite{hofri:1987}



\subsection{Logistics}

\subsection{chepa}

\section{Next Fit Bin Packing algorithm}

% 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}