Bounded and Unbounded Problems

Dividing complex problems into manageable ones

Bounded and Unbounded Problems

Inventor Charles Kettering once wrote, “A problem well stated is a problem half-solved.” And he was speaking about some pretty heavy problems. This was after all the man who invented modern refrigeration, air conditioning, and electric car ignitions (just to name a few).

Most of the very hard things we all face (in life, in business, and beyond) are initially presented in ways that make them seem daunting, if not impossible. A tremendous amount of effort is therefore wasted on inefficiently (and ineffectively) attempting to solve problems that could be tackled much more easily with the right framing. In other words, we need to “well state” the problem to render it more manageable.

Here are a few examples of such hopeless problems on the business side that I’ve heard from teams I’ve worked with or advised:

  • “We need to improve customer satisfaction!”

  • “We need to expand to more markets globally!”

  • “We need to increase brand awareness!”

  • “We need to grow our sales!”

I call all of the above Unbounded Problems. The term comes from math, where a function can be described in one of two ways. An unbounded one can grow infinitely in any direction. It cannot be contained. And as you follow it, it may lead you astray because it is open ended.

Compare that to a bounded function. This is one that is always contained within some band: No matter how far you follow it, it will always lie within a maximum and a minimum. Therefore, you know the solution will always lie within that range.

An unbounded function (blue) and a bounded one (red)

All of those business problems listed above with exclamation marks? They’re unbounded problems. Trying to solve them as phrased can (and likely will) send you off in a myriad different directions. The problems themselves aren’t well stated; they’re open ended, unquantified, and their solutions could be anywhere. Throw in the difficulty of then getting a team of people to rally around solving these unbounded problems, and now you’ve got a real challenge on your hands.

Going back to Charles Kettering’s words, how does one “well state” an unbounded problem? To phrase it differently, how can you turn an unbounded problem into a bounded one?

The short answer is: You can’t. Unbounded problems are, by definition, sprawling and unrestrained.

But that doesn’t mean all hope is lost… because unbounded problems may be broken up, and those parts can be bounded.

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Break It Down Again

To solve any unbounded problem, first break it into as many bounded smaller problems as possible. It’s the combination of these two approaches (segmenting and bounding) that makes the impossible achievable.

So let’s find a way to break a problem down into its components and tackle each one separately. And taking it further, we need to do so in a way that allows as many of the sub-problems to be bounded (i.e. to have their solutions exist in a more manageable solution space).

We can return to one of the examples I cited above. “We need to improve customer satisfaction!” A hypothetical furniture company may approach this segmenting and bounding strategy by instead framing its problem as the union of the following smaller problems:

  • “There’s a 15% return rate on our products due to manufacturing defects.”

  • “40% of our customer complaints last month cited long waiting times for support.”

  • “20% of customer reviews mention delivery taking longer than what we promised.”

Why are these specific issues bounded? Because every single percentage point of product returns is attributable to a specific solvable issue. The same is true of the delays in product deliveries. The solution space for each one of these problems is finite. Each of these problems are, in fact, symptoms. And the symptoms are all causally linked back to something specific. That something, in the language of mathematics, is the “solution” we’re looking for.

In the unbounded case, the same cannot be said. Broadly stating at a company all-hands, “We need to improve customer satisfaction!” is a problem that has no causal link to a specific set of solutions. Its solutions are therefore unbounded and sprawling; it would be highly inefficient (or impossible) to send a team off in search of them.

Let’s look at another example: Learning a foreign language.

The unbounded approach? “I want to learn French.” Anyone who’s attempted to learn a foreign language (particularly via self-teaching) might relate to this daunting task to which few people ever commit. Why? Because the open-ended nature of the problem yields it seemingly insurmountable.

A segmented and bounded approach:

  • “I can’t describe all the items in my house in French.”

  • “I don’t know how to conjugate the 100 most common French verbs.”

  • “I can’t read Les Misérables in the original French.”

Once more, the issues are segmented. The pieces are bounded. The solutions are finite. The person tasked with tackling these problems is much more likely to quickly and correctly find the right path to reach a solution. And, just as importantly, they are more likely to stick with it until the solution is reached, motivating them to then take on the next segmented and bounded challenge on the list.

On OKRs and Bicycles

Some people may find similarities between what I’ve written here and some common frameworks. For instance, the OKR framework (Objectives and Key Results) favors setting quantifiable goals and breaking each of those into subgoals that are also quantifiable. But there is a significant difference: OKRs assume that you already know how to solve your problem. You just need to build the steps to get there. In other words, OKRs are top-down. Here, we’re talking about the problems that are so broad that, before defining any “objectives” at all, you need to “well state” which problems you’re trying to solve in the first place. Only then will you be able to find a solution. Unbounded problems require a bottoms-up approach.

I’m reminded of my daughter, who recently learned how to ride a bike. It’s a fascinating experience teaching that skill, which as a grownup I know so well despite not knowing at all how to explain it. A classic unbounded problem. My approach? Instead of stating it as “Learn to ride a bike”, break it down into bounded problems:

  • “Learn to stop without falling” (to build confidence)

  • “Learn to evenly distribute your weight” (to stay balanced)

  • “Find a softer surface to learn on” (to reduce the risk of injury)

  • “Figure out whether this bike is even the right size” (spoiler: it wasn’t)

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