A Leap of Rationality

Stepping off a high dive for the first time can be a frightening experience. It’s something many of us never forget. I well remember my first plunge. It must have taken a dozen trips up the ladder – and back down – before I gathered the courage to jump. Fear kept me from my goal, even though I knew there wasn’t any real danger. I’d seen many people do it before. I certainly knew the water would break my fall. Yet my apprehension kept me from taking the final leap.

The use of analytics can require a similar leap. The tools of analytics are often mathematically sophisticated. How can decision-makers be expected to rely on numbers if they don’t fully understand where those numbers come from? Consider the following example:

The CFO Problem

The CFO of a division of a large corporation is evaluating five different investments. If undertaken, each investment requires a capital outlay now and a capital outlay in six months, with revenue realized 12 months in the future. For example, Project 1 requires a capital investment of $11 million now and $3 million six months from now, and will generate $16 million in 12 months.

The investments along with their capital outlays are shown in Table 1. The amounts of capital the CFO has available to spend now ($26 million) and in six months ($12 million) are also shown in this table. Capital that is not used now may not be carried forward and invested in six months.

A project may be undertaken in a fractional amount at a fractional cost, but it also generates a fractional return. For example, the CFO may choose to take half a position in Project 1 at a cost of $5.5 million today and $1.5 million in six months, but will only realize a return of $8 million. Of the five potential projects, what positions should the CFO take in order to maximize revenue one year hence?

The problem is straightforward as stated, and open to whatever logic the CFO might want to use to solve it. He might, for example, recognize that Project 5 has the highest return and invest in that project alone, generating $40 million (it would exhaust all available investment capital in the sixth month, so no other projects could be undertaken). When he arrives at a solution, he can trace the logic and defend his position.

Analytics professionals approach the problem differently, expressing the problem in a slightly more mathematical form, as shown in Table 2.

Any solution to the mathematical constraints in Table 2 corresponds in a natural way to a solution of the original problem. A key advantage to expressing the problem mathematically is that it’s very precise. The mathematical formulation can be provided to an optimization algorithm with a request to return an optimal solution – values x1, x2, x3, x4, and x5 that, among all feasible solutions, do so with the highest revenue. Analytics professionals develop algorithms for finding such solutions, and understanding these algorithms requires advanced mathematical training. But analytics professionals also devote considerable time to modeling: taking business problems like those faced by the CFO and converting them to mathematics.

Analytics professionals don’t concern themselves with optimal solutions when modeling. Instead, they focus on constraints. How much capital is available? What positions can be taken? If the constraints are correctly expressed, they accept the answer supplied by the optimization algorithm. They take a leap of rationality.

And a leap of rationality is often required. It’s not difficult to generate some good solutions to the CFO Problem, and the reader is encouraged to try. The optimal solution is given at the end of this article. However, it’s not intuitive, and even those trained in optimization rarely find it by trial and error. Analytics professionals have no trouble accepting the solution because they understand the underlying mathematics. Without such an understanding, however, acceptance is understandably more difficult.

How can non-mathematicians gain the confidence to use solutions from mathematical optimization? First and foremost, it requires help from an analytics professional who understands more than mathematics. Not surprisingly, people who enjoy analytics enjoy mathematics. But seasoned analytics professionals are acutely aware that success is the most important aspect of an analytics project, and that success requires far more than good mathematics. Such professionals understand how to communicate with examples and straightforward language, not Greek characters.

And that level of communication is vital. Decision-makers get where they are because they make decisions. That means finding solutions to problems, not modeling them and handing them to a computer. Decision-makers who have no familiarity with optimization will reason out a solution. But it can be extremely costly when better solutions exist. Decision-makers don’t need to learn the mathematics of optimization, but they must understand the concepts well enough to overcome their fear. Only then are they freed to take the final leap of rationality and use analytics with confidence.