.. Created by Adam Cunnningham on Fri Jun 3 2016.

**Linear Programming**
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Linear programming problems are a special class of optimization problem. They involve finding the maximum (or minimum) of some linear objective function :math:`f(x)` of a vector :math:`x = (x_1 , x_2 , \ldots, x_n )`, subject to a set of linear equality constraints :math:`A_{eq} \cdot x = b_{eq}` and inequality constraints :math:`A_{ub} \cdot x \leq b_{ub}`. The online class notes give a more detailed description on page 66.

Report Description
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- Model the pipelines problem in Python using **linprog** from the **scipy.optimize** package.
- Use linprog to find the maximum possible flow through the pipelines using the "simplex" algorithm. This is the default method used by linprog.
- Give full details of your solution, clearly identifying the flow through each pipeline.
- Suggest a way to increase the maximum flow by at least 3 units for as little cost as possible.
- Confirm that the suggested change is successful.

.. tip::

    Describing the pipelines problem as a linear programming problem means first answering the following key questions before starting to code:

    - What are the variables?
    - What are the bounds on each variable?
    - What is the objective function?
    - What are the inequality constraints?
    - What are the equality constraints?