This post is a continuation of my previous post about mathematical modeling.
Modeling is a tricky proposition. Good results require a fair amount of judgement on the part of the person using the model, and developing the model usually requires some simplifying assumptions. Let’s jump back to the example about grandmother. In that discussion I said grandmother’s average speed is 45 mph. The wiggle room in that statement is average speed. Grandmother is going to have to stop at a couple of traffic lights before she gets on the interstate, and she’s probably going to have to slow down once or twice for someone else’s grandmother driving even slower. She’ll have to drive extra slow while she squints at the street signs looking for Old Hickory Boulevard. While she’s driving on the interstate she’ll actually be driving a little faster than 40 mph most of the time, but we have to average it out.
That brings you to one of the difficulties of modeling. The basic equation is a given. It’s always the same. But how do we figure out that average speed we’re using? Well, grandmother has driven to our house many times before and we have a pretty good idea of how long it takes her so you we figured it out by observing previous instances of the process we’re modeling. Sadly though, our model only works if she’s driving to our house. If grandmother is driving to our sister’s house in Knoxville, that average speed is going to be different because her route is different. She’ll be stopping at a different number of traffic lights, and she’ll be driving through town instead of mostly interstate driving. So sister will have to figure out her own model (just send her a link to this post). And when grandmother goes to Chattanooga to visit mom and dad, they’ll have to figure out their own model. The basic framework stays the same, but the average speed will be different.
But let’s jump back to our model for grandmother’s trip to our house. What happens if it’s raining and grandmother has to drive slower? We can adjust our average speed downward, but then that messes up our nice simple model because we have to change it every time the weather changes. Instead, we can add a ‘weather adjustment factor’. That way speed is still 40 mph, but the weather adjustment factor will change it as needed. So now our model is:
Distance = Travel time x Average speed x Weather factor
When the weather is nice and sunny grandmother’s cruising speed is 40 mph, the weather factor is 1 so nothing changes. If it rains, well then grandmother drives about 90% of her usual speed. So the weather factor is 0.9. If it’s snowing then grandmother’s going to creep along at half speed so the weather factor is 0.5.
Sunny day => Distance = Travel time x 40 x 1
Rainy day => Distance = Travel time x 40 x 0.9 = Travel time x 32
Snow day => Distance = Travel time x 40 x 0.5 = Travel time x 20
The adjustment factor can be used for other things too. Say, for instance, grandmother has some good gossip about Uncle Ed. She’s going to drive a little bit faster so she can get there and share the juicy stuff with you.
Gossipy grandmother => Distance = Travel time x 40 x 1.25 = Travel time x 50 (grandmother enjoysr gossip)
If you like, you can start stacking the variables and using more than one. What happens if grandmother wants to gossip about Uncle Ed’s first born getting arrested on a rainy day?
Rainy, gossipy grandmother => Travel time x 40 x 0.9 (rain adjustment) x 1.25 (gossip adjustment)
Distance = Travel time x 45
You can get as complex as you want. What if grandmother stops at the grocery on her way? What if your dad is riding with her and she drives slower when she talks to him?
When developing a model the dilemma is always in deciding how complicated to make it. The more variables you add, the more accurate your model will probably be. But there comes a point when adding variables is not worth the extra effort. Does it really matter to you whether it will take grandmother 4 hours or 4 hours and 5 minutes?
So you have to consider whether it’s worth the effort to put extra work into the model just to make it a little more accurate. This is a question most young engineers struggle with and it requires a fair amount of judgement that usually comes with experience. Bridge modeling can tell you how much concrete to pour down to 0.01 cubic ft, but the concrete truck bringing it to the work site has 75 cubic feet in it and the trough that pours it out of the back of the truck doesn’t have a measuring device on it. So you can spend as much time as you want calculating the concrete thickness down to the nearest 0.01 inch, but you’re going to get the closest eyeball estimate the site supervisor or the concrete truck driver can make.
I’ve got one more post coming up about modeling. We’re going to discuss some real world uses and hopefully look at a specific modeling example or two.