An Expository Analysis
That's academic-speak for
"a basic example to demonstrate a point"
Jim is a proud owner of a small family farm.
Jim's been growing corn for quite some time and had a particularly bad couple harvests the past few years. Jim is wondering if he needs to adjust his planting times.
He is considering a switch to another crop that might net him a more predictable income. His neighbor has had better luck with soybeans, but Jim's never grown them before. He is facing a lot of uncertainty and it is definitely affecting his livelihood and peace of mind.
What do we mean by this? Well, to keep things simple, let's demonstrate by addressing planning Jim’s revenue (we can factor in costs later). How much money can soybeans potentially bring in compared to corn? Should it be just one of the crops or some mix of the two?
Let's see what we can do to help Jim.
So let's say Jim just made a decision, any decision.
What does his predicted revenue look like?
Take a look at the graph below. The orange line represents the full range of revenues Jim might be able to earn if he just made a decision. He can basically expect between $350 and $500 per acre. The chances of him choosing a strategy that results in $500 or more in expected revenue are very low.
If we apply some mathematical analysis, we can provide Jim with a set of possible decisions which make his expected revenue look like the purple curve instead.
Notice how much further right it is. That means Jim can expect more money.
It's also more narrow and peaked, which represents the increased confidence in Jim's expectations.
Jim is more likely to make more money
If he consults with mathematicians before making a decision.
Jim could certainly benefit from a rigorous analysis of hundreds or thousands of “what-if” scenarios. But he simply doesn't have the time to investigate that thoroughly. Computers, however, are perfect for that kind of thing.
What Jim needs is a set of options that result in more certain income.
Done correctly, mathematics can provide him with peace of mind.
Mind the Math
More details (for the curious among you)
To figure out his expected revenue per-acre, Jim would need to multiply the expected yield of each crop by its price per bushel, and sum the contributions of both corn and soy bean production. That's all that is being done here.
There are lots more factors we can take into consideration, but for brevity,
let's focus on how much Jim can make
(and with what confidence).
As of mid-February 2018, the price of corn per bushel is around $3.50 and soy goes for $9.50. We treat these as being certain (for now).
The yields are uncertain, but we expect:
125 bushels/acre for corn
45 bushels/acre for soy
and we take the uncertainties to correspond to the legends in the graphs above (about 25 bushels for corn, 15 for soy, which we use to form the normal distributions chosen to represent the uncertainties in our predicted per-acre yield).
Let's take a look at the distributions again and the relative likelihoods:
The set of decisions that we are then able to provide that result in the purple line are then presented to Jim, and a conversation about his comfort levels of risk and abilities to grow a new crop will inform his ultimate decision.
There are scenarios that lead to high income which correspond to almost every ratio of corn-to-soy that Jim might be willing to try.
The way this works in practice is iterative.
We would start by presenting Jim with the orange line - all the possibilities, then ask what revenues he would like to hit and with what relative confidence.
We then come back to Jim with the target yields he needs to meet for each crop based on that proportion of corn-to-soy, and his desired income/uncertainty.
If he feels that he wants a strategy that has more forgiving target yields, we would go back and re-solve the problem for a lower target revenue, until we come to a happy medium.
This is just the tip of the iceberg of mathematical relevance to Jim's situation.
Mind the Math.
Do note that this is expected revenue, not profit. Please recognize that this is just an example and do not use it as-is for your own planning. The goal was simply to gain an intuition of how data-driven decision making can help us predict our bottom line.