or, The Benefits of Consulting with Mathematicians
Susan recently graduated from the local state university with a degree in Biology. She is twenty-two, bright-eyed, and excited to come home and begin to take over the management of her family's orange farm.
Orange trees, like many fruiting trees, come in a lot varieties, each with its own quirks and specific growth-cycles. Some need to be replanted every dozen years, others can be productive for twice as long.
Susan's family is proud of the few new varietals they have recently planted clones of, and their yields in the past five years have been growing steadily.
But Susan is concerned.
Something about this seems too good to be true.
Could this rapid increase in productivity really continue?
Susan decides to call up a friend from college that studied math to try to answer this question. After about a week or two of back-and-forth communication to build the model of the farm's assets (its trees), it became clear that the growth would come to a peak and then decline.
That means contracts couldn't be met. In as little as ten years.
Susan knows that personal relationships with growers are the lifeblood of a packer's business, and she wanted to do anything in her power to preserve those hard-earned contracts. If she couldn't meet the demand, larger growers might be able to squeeze her family farm out of the business.
Susan consults with her parents, and based on their advice she comes back to her friend with a question:
Would pruning some the trees once they're past their first yields help?
This is a question with series consequences. She has to give up potentially dozens of perfectly good branches in hopes that new ones grow back and increase overall production.
Let's see how we can help Susan.
This one's a little overkill, but we did write a program to actually simulate the growth of the trees over time, run thousands of probabilistic scenarios, and get an expectation of the yield from pruning in order to compare it to the yield if Susan does nothing at all. We also tried to find a good proportion for how many of the trees to prune, as well as the best time to do so.
We can see the yield from inaction with the solid line, and the dotted one represents the expectation of pruning a third of her crop. So, Susan would be making a significant sacrifice from around 2030 to 2035, but there is an overall benefit to the long-term sustainability of production.
We can see from the plot that after the initial decline from pruning (along with a combination of other factors), the yields begin to grow at a much steadier pace.
With this strategy, there will be fewer major variations in Susan's future income.
She can more confidently plan the responsible growth of her business.
With the help of mathematical analysis, Susan can ensure the success and economic sustainability of the family business for future generations to come.
In fact, she can plan for that period of decline with investments now, which we can absolutely begin to factor into our simulations. The first round of analysis uncovered a problem and presented some solutions with supporting evidence. This in turn informed additional logistical questions, that can be addressed with more mathematics. But for the sake of brevity, we conclude this example here.
Mind the Math.