Decision Theory Question--Applied Math

[Code Doc]

(Moved from World Events)

Suppose you had the equipment required to rip the length of a large American walnut log straight down the middle using a 42" dia., $9,000 tungsten carbide blade that you just bought for your sawmill and do not wish to destroy. You estimate that blade and the equipment driving it could saw up to $500,000 worth of lumber for your company before needing to be retired.

Unfortunately, your high-tech metal detector says that there is a hard chunk of alloy steel, likely a case hardened drift pin, that is embedded somewhere in the log, but the detector cannot precisely locate it. If you hit that pin while rip sawing with your equipment, the blade will be ruined beyond repair.

Your foreman estimates that the odds are one chance in 50 that you would hit the chunk of steel with the new blade and ruin it. The "free" log was given to you by a homeowner who cleared his land. Would you go ahead and rip saw the log anyway if a buyer offered you $1,000 for both halves of the log, but only after you ripped it in half?

Please advise.

[jemidiah]

This question is very different depending on how much you believe the foreman's estimate.

Suppose you completely believe the foreman--that is, you are certain that, given 5000 identical scenarios, only ~100 (i.e. 1/50th) will ruin the blade. You will make $2,000 (the cost of the wood) 49 out of 50 times. You will lose $9,000 (the cost of the blade) 1 out of 50 times. Over those 5000 identical scenarios, for instance, you'd expect to make 4900/5000*$2,000 - 100/5000*$9,000 = $1960-$180 = $1,780, on average.

Since you'd expect to make a profit if you believe the foreman, you should indeed go ahead and rip saw the log anyway. This analysis assumes the only cost if you hit the pin is replacing the blade--no lost time or collateral damage--since other sources of cost weren't specified.


If you don't believe the foreman, or if you only believe him a little, you might well not want to saw the log. For instance, if you believed the foreman was overly optimistic and the odds were really 1 in 5 of hitting the chunk of steel, the previous analysis would say you should not saw.

[NickThissen]

Why don't you just get a proper metal detector and see where the metal is?

[Code Doc]

Looks like the break even odds are 1 chance in 10.

(0.9)($1,000) + (0.1)(-$9.000) = $0

If the odds are anything less than one chance in 10, I saw the log. If greater, I don't run the risk of ruining the big blade.

Thanks all.

[jemidiah]

The break even odds are actually slightly different since the expected benefit is $2,000: you had said $1,000 per half, and there had better be two halves. But yeah, that's basically it.

p*(2000) + (1-p)*(-9000) = 0
2000p - 9000 + 9000p = 0
11000p = 9000
p= 9/11

=> "break even" point is 2 in 11

[Code Doc]

The break even odds are actually slightly different since the expected benefit is $2,000: you had said $1,000 per half, and there had better be two halves. But yeah, that's basically it.

p*(2000) + (1-p)*(-9000) = 0
2000p - 9000 + 9000p = 0
11000p = 9000
p= 9/11

=> "break even" point is 2 in 11

Note, Jemediah, that we neglected to analyze the potential lost sales if the blade is damaged beyond repair. It is also possible that the blade cannot be replaced for either $9,000 or the MFG of the blade has gone out of business and the blade can thus not be replaced at any cost.

To make the problem even more realistic, a third event could occur, namely that only a nick of the hidden spike could allow repair of the blade for a smaller cost than total replacement. In any damaging event, either a nick or a major hit, down time produces lost sales that needs to be included with both the replacement and/or repair costs.

All of these considerations make the problem more difficult but yet more realistic, and that is what faces many business decision makers. One of these days, I would like to see some Bayesian statistical analysis on this forum, but that is another story.

[jemidiah]

Yup, as I said the above analysis assumed no other sources of cost (or profit) besides the blade breaking (or the log selling), where you also completely believe the foreman's estimate (and where you believe the problem statement saying the customer will indeed buy the log if it's sawed in half). There are a zillion complications you could throw in, including the ones either of us has mentioned.

More realism would be added by giving each eventuality a probability distribution. Eventually, you'd develop a confidence interval--"there's a 97.3% chance you'll make money from sawing the log", for instance, though even that interval should have error bars ideally, since the probability distributions almost certainly aren't known exactly. In the end, statistics is a best guess. Maybe that's why I've never liked it terribly much.... Of course, statistics generates better guesses than humans naturally do, most of the time, wherein lies its value.