tex: add content for part 1
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Bin packing is the process of packing a set of items of different sizes into
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containers of a fixed capacity in a way that minimizes the number of containers
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used. This has applications in many fields, such as logistics, where we want to
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optimize the storage and transport of items in boxes, containers, trucks, etc. In
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this paper, we will focus on one-dimensional bin packing, where we try to store
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items of different heights in a linear container.
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optimize the storage and transport of items in boxes, containers, trucks, etc.
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Building mathematical models for bin packing is useful in understanding the
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problem and in designing better algorithms, depending on the use case. An
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relibility concerns.
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We will conduct a probabilistic analysis of multiple algorithms and compare
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results to theoritical values. We will also consider the algoriths complexity
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results to theoretical values. We will also consider the algoriths complexity
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and performance, both in resource consumption and in box usage.
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\clearpage
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\section{Bin packing use cases}
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Before studying the mathematics behind bin packing algorithms, we will have a
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look at the motivations behind this project.
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\cite{hofri:1987}
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Bin packing has applications in many fields and allows to automize and optimize
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complex systems. We will illustrate with examples focusing on two use cases:
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logistics and computer science. We will consider examples of multiple dimensions
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to show the versatility of bin packing algorithms.
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\paragraph{} In the modern day, an effective supply chain relies on an automated production
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thanks to sensors and actuators installed along conveyor belts. It is often
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required to implement a packing procedure. All of this is controlled by a
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computer system running continuously.
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\subsection{3D : Containers}
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\subsection{Logistics}
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Storing items in containers can be a prime application of bin packing. These
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tree-dimensional objects of standardized size are used to transport goods.
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While the dimensions of the containers are predictable, those of the transported
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items are not. Storage is furthermore complicated by the fact that there can be
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a void between items, allowing to move around. Multiple types of items can also
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be stored in the same container.
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There are many ways to optimize the storage of items in containers. For
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example, by ensuring items are of an optimal standardized size or by storing a
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specific item in each container, both eliminating the randomness in item size.
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In these settings, it is easy to fill a container by assimilating them to
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rectangular blocks. However, when items come in pseudo-random dimensions, it is
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intuitive to start filling the container with larger items and then filling the
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remaining gaps with smaller items. As containers must be closed, in the event
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of an overflow, the remaining items must be stored in another container.
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\subsection{2D : Cutting stock problem}
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In industries such as woodworking bin packing algorithms are utilized to
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minimize material waste when cutting large planks into smaller pieces of
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desired sizes. Many tools use this two-dimensional cut process. For example, at
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the Fabric'INSA Fablab, the milling machine, laser cutter and many more are
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used to cut large planks of wood into smaller pieces for student projects. In
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this scenario, we try to organize the desired cuts in a way that minimizes the
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unusable excess wood.
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\begin{figure}[ht]
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\centering
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\includegraphics[width=0.65\linewidth]{graphics/fraiseuse.jpg}
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\caption[]{Milling machine at the Fabric'INSA Fablab \footnotemark}
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\label{fig:fraiseuse}
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\end{figure}
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\footnotetext{Photo courtesy of Inés Bafaluy}
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Managing the placement of items of complex shapes can be optimized by using
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by various algorithms minimizing the waste of material.
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\subsection{1D : Networking}
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on which humans
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have decreasing control.
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In this paper, we will focus on one-dimensional bin packing, where we try to
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store items of different heights in a linear container.
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\subsection{chepa}
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\section{Next Fit Bin Packing algorithm}
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\cite{hofri:1987}
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% TODO mettre de l'Histoire
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\section{Next Fit Dual Bin Packing algorithm}
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