# The unintended consequeces of a small bit of math

In 2012 I wrote a paper with Roger Warburton (Boston University) concerned with the Lambert W Function and how it could be used to minimise the net present value (NPV) of the cash flows in the Economic Order Quantity (EOQ) decision; the EOQ is a classic operations management problem. We showed the order quantity that minimises the NPV of the cash flows in the EOQ model is much smaller than the order quantity advocated by the traditional EOQ model. I thought this was interesting as the result offers some theoretical support for reducing batch sizes, an improvement action often advocated by the Lean Production movement.

We had a problem publishing the idea in a journal at first; the referees rejected it saying the idea could not be taught in an MBA class and could not be calculated in an Excel spreadsheet. This motivited us to consider how it could be taught in class (and I have subsequently taught it in my MBA class, as well as several MSc classes), and we also developed a User Defined Function, via an Excel Add-in, that is available here. These additions were duly incorporated into our paper; eventually the paper got published (but not in the original journal).

I thought that was the end of the story. But, over time I have come know about how other people are finding our Excel Lambert W Function Add-in to be really useful. Robert Ventura and Stephen Samuel used the Add-in to aid the design of a fuel injector for a diesel engine. It turns out that the size of the droplet of fuel that is injected into the engine, and the way it wets the cylinder wall, has a large effect on fuel efficiency (of up to 9%). That engine design problem was also solved with the Lambert W Function Add-in. You can download Robert’s and Stephen’s paper here.

In another use case, Christian Stefano Schuster used our Lambert W Function to analyze the energy yield of solar cells. Previously, these calculations have only been possible numerically. The explicit solution, enabled by the Lambert W Function, will enable scientists to better consider the impact of different weather/climate effects on the energy yield when designing the next generation of solar cells. You can download Christian’s paper here.

Who would have a thought that a small bit of math and computer programming would have such a nice impact on the world? If you have another use case for our Lambert W Function Add-in, I would be very interested to hear about it, please let me know.

##### Stephen Disney

My research interests involve the application of control theory and statistical techniques to operations management and supply chain scenarios to investigate their dynamic, stochastic, and economic performance.