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!
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