Lecture Video

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So far we have spent a large amount of time and effort learning how to solve very narrow (and occasionally non-specific) problems. It gets tiresome to go through this process for every single question you are posed however, and often you can take many questions and group them together as one single type of problem with minor variations. This is where generalized models will come in handy.

Let’s consider our (earlier) patio example. We were asked to determine the cost of building a patio, presumably by an individual wanting a patio. But what if you worked for a patio-building company? You’d have to go through a similar process many times a week, and doing so would get irritatingly repetitive. But, things that are repetitive are usually susceptible to being generalized, which means you can streamline the process. So, rather than brute forcing the work over and over for many patios, embrace laziness and generalize the process to come up with a “shorthand” version that works for many different patios!

A note on the best way to read the following sections: I will first describe the process in general and then present a short example. It would be ideal to read these things in parallel, but for the sake of continuity I will present one and then the other. I actually recommend reading this content in order first and not worrying too much about fully understanding the general process on the first read through. Then read the example, and come back to reread each section of the “general process” explanation with the corresponding section in the example. This may seem much more time consuming, but it has a much higher chance of helping you learn and understand the process (as oppose to memorizing the content).

First step to generalizing: Determine what can be generalized.

The first step in generalizing a numeric solution may seem “obvious” (in the same way that the first phase of solving the numeric model; “clarifying the problem” seemed like it should be obvious), but it is (again) often deceptively important. A good way to start determining what can be generalized, is to consider what information you both need and already know (or, at least what type of information), and what units that information has. ^[1] (One may restate this to say that a good first step would be to look at what you have as data, and not simply information. Data is often easier to generalize as it is quantifiable, and that quantity is what you are generalizing. It is very important to remember that this is only a start however. Often, in industry, the most pivotal piece of information to generalize isn’t found by doing this, but it at least gives you a place to start.) In our patio example we needed to know what the patio was made of (which we later determined would be cement pavers) as well as what size the patio would be. This first piece of information (what the patio was made of) isn’t in units that we can expect - indeed, it could have been pavers, or boards of wood, or gravel for example. In other words, it isn’t going to be data, and as such may not be a good candidate for generalizing. The size however is going to be some form of area (and thus will be numeric data), thus we can expect some kind of square units (eg square feet or square yards) and so that piece of information may be a good candidate to try and generalize. A good rule of thumb; if the information isn’t quantifiable, then generalizing it will almost always be too difficult to be worthwhile.

Next Step: Determine what should be generalized... and how.

The art of modeling (in the mathematical sense) comes into play when you are trying to decide which aspects to generalize. On the one hand, the more you generalize, the more versatile and useful your model becomes. On the other hand, generalizing takes time and effort, meaning you may spend so much time generalizing that you end up spending more time than you save for having a more general version of your model - something your boss will undoubtedly not appreciate.

The broad rule of thumb is to ask yourself “which of the things that I can generalize are likely to change from project to project.” If a piece of information is likely to change between different variations of projects, then it’s a good candidate for generalizing. For example, not everyone will want the same size or dimension for their patio - so those aspects are likely to change from project to project.

Keep in mind that these things differ by situation. Consider the possibility that you are in a business where all you make are twenty by twenty patios from a variety of materials. In this case, generalizing the size of the patio would be pointless (it’s always the same size after all), but generalizing the materials becomes key; something that might prove very difficult given the previous comments about the difficulty of generalizing non-quantifiable information.

Once you have identified the information that you wish to generalize, the how is “straightforward”. ^[3] (I put straightforward in quotes because, in practice, executing the generalizing step itself might be easy, but keeping track of it as you update/build your model can be very difficult. This is why the advice about keeping a written list of variables and what they mean is absolutely key, especially early in learning this process.) . You generalize a piece of information by replacing its value in your model with the correct type of variable. In order to understand what we mean when we say the “correct type of variable”, you should first understand the role of variables in the model, and what different types exist.