Tapping into Efficiency; How Mathematicians are Saving Our Water

Have your water bills recently increased? Have you ever wondered how we can deal with water inequality? How can we achieve these things you may be asking? Well it is quite simple actually, we can use a tool called Mixed Integer Linear Programming (or MILP for short). Sounds scary? Well it is actually quite simple, all it does is minimise or maximise the desired metric (such as minimise cost of pipes or maximise reliability) while constraining values to be within some acceptable bound. For example, in when minimising the cost of pipes, you can not use any pipes, which doesn’t make sense so we can add constraints to avoid this issue. Thus we can create more efficient and cost effective water distribution systems.

What is MILP

So now, unless you have taken third year applied maths units, you are definitely wondering what “Mixed Integer Linear Programming” is. Well have no fear I completely understand you. I was recently asked to read a research paper for a job. The paper was on scheduling and automated planning [1]; a technique where you make a sequence of decisions intending to optimise some objective. Well, if you are starting to understand how MILP works, you may not be surprised that it was used in the paper. However, at the time, I had no idea what MILP was. Yet after many hours spent, I can now explain it in simple terms.

Let’s first focus on the terms “Linear Programming”. Independently, both these words are mostly likely somewhat familiar to most people, yet, their combination is not. Pretty much, linear programming is a mathematical concept where you are trying to optimise a linear function subject to a set of linear constraints. What does that mean? Let me refresh you. A linear function which we are trying to optimise may look like the following

Where “” and “” are constant values (i.e. we can’t change them) while and are continuous variables (i.e. we can control them and they are allowed to have a decimal value, so etc.). All it means for it to be linear is that we can’t multiply multiple variables together, whilst we can multiply a variable by a constant. You may now be wondering what happens if we have negative pipe lengths or why not only use the cheaper pipe. So with this, we add the second part of linear programming, the constraints.

These constraints ensure that both pipe lengths are positive values and that the combined length is equal to our desired length. Finally, referring back to “Mixed Integer”, all it means that the model can have a mix of continuous and integer (i.e.  etc. Meaning no decimals are allowed) values.

How MILP is being used to save water

So how does all this relate to water distribution? Well, Awwalu et. al. [2] actually applies MILP to improve our water distribution systems. Firstly, they use a variant of MILP which allows for multiple objective functions. These being to minimise the cost of the pipes as well as minimising the energy use at pipe junctions, all whilst maximising the reliability of the system.
Additionally, they add a list of constraints with the goal of - conserving energy inside pipes - providing the population with their water demands - ensuring that the water pressure is above some minimum value - restricting pipe diameter sizes to only those available (e.g. 50cm, 1m, 2m etc.) - conservation of flow

Credit: Gestetner. A. [3]

Where to now?

Since this is only a theoretical model, it is now up to the engineers to use these tools in order to create a more efficient system. However, since MILP provides the most optimal solution to the given problem, its mathemagic powers can be used to vastly improve all our water systems.

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