.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/optimization/dynamic_programming/plot_01_knapsack.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_optimization_dynamic_programming_plot_01_knapsack.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_optimization_dynamic_programming_plot_01_knapsack.py:


Bellman's principle of optimality: the 0/1 knapsack
=========================================================

Fills a table of optimal values for every prefix of the items and every
capacity, then traces back the chosen items. The table is the
dynamic-programming "value function".

.. GENERATED FROM PYTHON SOURCE LINES 11-15

.. code-block:: Python

    import matplotlib.pyplot as plt

    from mathematicskit.optimization import knapsack








.. GENERATED FROM PYTHON SOURCE LINES 16-18

Six items, capacity 15
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 18-25

.. code-block:: Python


    values = [10, 13, 7, 8, 15, 4]
    weights = [5, 6, 3, 4, 7, 2]
    capacity = 15
    result = knapsack(values, weights, capacity)
    print(f"best value {result.value:.0f} using items {result.items.tolist()} (total weight {result.weight})")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    best value 32 using items [0, 2, 4] (total weight 15)




.. GENERATED FROM PYTHON SOURCE LINES 26-28

The value table V(i, c)
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-35

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(9, 4))
    im = ax.imshow(result.table, cmap="viridis", aspect="auto", origin="lower")
    fig.colorbar(im, ax=ax, label="best value")
    ax.set_xlabel("capacity c")
    ax.set_ylabel("items considered i")
    ax.set_title("Bellman's value table for the knapsack problem (1957)")



.. image-sg:: /api/gallery/optimization/dynamic_programming/images/sphx_glr_plot_01_knapsack_001.png
   :alt: Bellman's value table for the knapsack problem (1957)
   :srcset: /api/gallery/optimization/dynamic_programming/images/sphx_glr_plot_01_knapsack_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    Text(0.5, 1.0, "Bellman's value table for the knapsack problem (1957)")




.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.037 seconds)


.. _sphx_glr_download_api_gallery_optimization_dynamic_programming_plot_01_knapsack.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_knapsack.ipynb <plot_01_knapsack.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_knapsack.py <plot_01_knapsack.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_knapsack.zip <plot_01_knapsack.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_