Saturday, January 7, 2023

Your project runs on exponentials!

When you read stuff like this, you have to wonder: what's this all about?!
The greatest shortcoming of the human race is our inability to understand the exponential function
- Albert A. Bartlett, The Essential Exponential! For the Future of Our Planet
Sounds profound! 
And so it is. Exponentials are the way we bend the project around the corner, or move out quickly. Linear straight-line stuff becomes curved.
Think of it: 
  • The future does not have to repeat the past; 
  • We can accelerate from where we are now; and 
  • Phenomena can -- and will -- die off or slip quietly to an asymptotic finish
What is it?
Technically, an exponent is a number that tells you how many times another number, called the 'base', is multiplied by itself. For example, 2 exp 4 tells us to multiply 2 (the base) by itself 4 times, yielding the number 16. The exponent is sometimes referred to as 'the power'. We might say, '2 to the power of 4'.

Fair enough.
Functionally, it's the multiplying-by-itself thing that bends the curve and creates acceleration in values. In linear arithmetic, the first four products are 2, 4, 6, 8. But in exponentials, the first four products are 2, 4, 8, 16 ; clearly the exponential expression is not linear.

It's common to say: "That's increasing (or decreasing) exponentially". 

What that means is that 'as you move away from the present point, things accelerate quickly'.

Now, for the math inclined or curious, see the footnote at the end of this post.

Consider project communications:
The number of ways that N people (or systems or interfaces) can communicate is N*(N - 1), which for large N is approximately N x N, or N-squared (an exponential, N with exponent of 2).  

This the heart of the argument -- made famous by Dr Fred Brooks --  that adding more staff to a late project may make it even later, because ....  Because communications become exponentially more complex, and thus less reliable and predictable, as you add staff.

Consider project finance:
The present value of future business benefits of a project are discounted, exponentially, by the expected risk.

Financiers think of risk in terms of the 'rate of return' they demand in order to agree to commit capital to a risky project.
And, this rate of return, "r", is compounded over time. Compounding means that [1 + r] exponentially decreases with the number of compounding periods. The future value is "discounted" to a present value by that decreasing factor.

Consider risk management:
The idea that risk inversely compounds exponentially with time, and thereby discounts the value of future decisions and outcomes is commonly held concept in risk management. 

Other types of risk are subject to exponential effects. For example, the density of probable failures in the future is inversely and exponentially related to a present value of the failure rate. 

Consider the so-called "bell curve"
The 'bell curve' shows us the effects of natural clustering of random outcomes or observations around the mean value.
The actual formula for the curve is non-linear to be sure, and usually we just look up values we are interested in, which for the most part delineate confidence intervals.

But at the core of the bell curve formula's is an exponential expression which is all about how far from the mean is the confidence interval and the point of observation or measurement, i.e: 'distance'-from-the-mean squared strongly influences the confidence intervals of the 'bell curve'.

Consider schedules at the milestone
There is a 'shift-right' tendency at milestones when two or more tasks have to finish together. If the probability of each finishing is 0.9, then the probability of the milestone finishing on time is 0.9 exp N, where N is the number of tasks finishing together. That is a much lower probability of success than any of the contributing tasks.

Consider the random arrival rate of independent actors (events)
Again, the probability distribution of random arrivals is an exponential, and an important concept in certain elements of risk management (Earthquake prediction, to name one, but other types of failures as well). 

Consider 'utility'
All but the simplest concepts of utility are non-linear, and many ideas of utility can be explained or represented with exponentials.

If you look up Bartlett's book, you'll find most of the chapters are available free in pdf format
Shout-out to herdingcats for the quotation

Footnote for the math inclined or curious:
It's common to think of the number '2' as 2 with an exponent of 1 (any number with an exponent of 1 is equal to itself); and the number '1' is 2 with an exponent of 0 (any number with an exponent of 0 is equal to 1). Other examples: The number '4' is 2 with an exponent of 2; the number '5' is 2 with an exponent of about 2.35, but also 5 is the number 10 with an exponent of about 0.7.

And, a negative exponent is mathematically equivalent to division: 1 divided by the exponential. For example, 0.5 is just 2 with -1 as the exponent, usually written as 1/2.

But here's a limitation of exponent math: there is no exponent that will give us exactly the number '0', although we can get pretty close with an arbitrarily large negative exponent. 

Now the "base" doesn't always have to be '2'. If we change the base to 10, the number 2 is now 10 with exponent of approximately 0.3. (somewhat like there is no exponent that gives us exactly the number 0, there is no exponent of 10 that gives us exactly the number 2. This is the reason that financial reports and other resource reports are not computed with exponential math. Such reports require exact numbers, not approximations)

You may have heard the expression that "things are logarithmic". That's another way of expressing the idea of an exponential. The 'logarithm of 2 (in base 10) is equal to 0.3'. That statement is equivalent to saying '10 with exponent of 0.3 is equal to 2'.

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