Creating a Mathematical Model of a Complicated Situation

Date: 10/09/2005 at 02:37:29
From: Goran
Subject: How to setup a mathematical modell

When riding a bike from point A to point B, I pass a lot of traffic 
lights and they all have a certain probability that they will show red 
and I have to wait for a certain amount of time.  I can take different 
routes for this travel (with different length and number of traffic 
lights), and all this leads to the fairly easy task of calculating the 
expected time it will take for the travel.  Then I can choose the 
quickest one.

Now, to make things more interesting: at some intersections there is
very little traffic, and I can go against the red light without a
problem, or almost without a problem.  There is a certain probability
that there is policeman hiding somewhere, just waiting to catch me
when I go against red.  Let's say that it is 50% of showing red and I
have to wait 30 seconds, and there is a 1% chance of a policeman there
and if I'm caught I will have to pay $200 for going against red.

The problem is how do I introduce the policeman and the fine into my 
model?  If the policeman were just slowing me down (by giving a 
lecture about laws, etc.) 5 minutes, I could handle it, but now I 
suddenly get expected values that contain both time and money!  I 
could also add injuries to the model if going against red...

How do I find the best way now, when I have both time, money, health,
etc. to consider?  Should I try to convert them all into the same
"unit"--let's call it "X"?  Then I let 1s = 1X, $1 = 10X (money is 
more valuable than time) and 1 broken bone = 10000X (health is most
important) and then calculate the expected value in the unit "X" ?

Please tell me, how does one make a model out of this?

Regards, Goran.

Date: 10/09/2005 at 11:50:30
From: Doctor Douglas
Subject: Re: How to setup a mathematical modell

Hi, Goran.

Your problem is very common in real-world applications.  A simple
solution is to "weight" the various factors into a single parameter
that you could optimize (you did this above, calling it X).  Although
you may not know what you should use for the relative weights (I agree 
that health is more important than money or time), this might be an 
advantage in your model, because you can adjust them depending on 
conditions (whether it is raining, whether you are late for an 
important date, whether you are rich or poor, whether or not it is 
more likely to see a policeman at certain days/times, even whether or 
not you actually want the quickest route (perhaps you want a more  
difficult workout, or you want to choose a scenic route on a given 
day).  This type of thinking is reasonable and you can refer to X as a 
"generalized cost".

A more complicated situation occurs if the various factors are
non-independent and can have interactions:  For example, suppose the 
costs of an accident are not only do you have a broken bone, but that 
there is a fine assessed, and your bike riding licence is 
confiscated/suspended for a period of time until you can take a safety 
course.  Maybe these penalties are assessed not on the first at-fault 
accident but on the second or third one.  Or maybe you have medical 
insurance that covers the cost of broken bones up to some fixed 
amount, after which you are responsible for the rest out of your own 
pocket.  Then you can see that your cost function gets complicated - 
it depends not only on the individual factors of time, money, and 
health, but also on the history of what has happened so far.  This is 
not so uncommon - if you are lucky enough to escape with only a
lecture from a policeman on the first offense, he may not be so 
lenient the second time.

Modeling this type of complicated situation can be done, but it is not 
easy.  Sometimes it is referred to as "multifactor analysis".

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 10/09/2005 at 12:28:41
From: Goran
Subject: Thank you (How to setup a mathematical modell)

Thank you Dr. Douglas. 

I think I'll stick to the simpler case with independent factors!