Thursday, September 30, 2010

Math can save lives, admiral!

During the Second World War, the British Army adopted a naval convoy system for all their merchant ships. Each convoy consisted of 30-70 freight ships and was surrounded by patrolling ships or airplanes. To minimize the chances of convoy being discovered and bombarded by the enemy submarines, the commandment considered two options: sending each convoy as one big flotilla, or splitting the convoy into a few smaller convoys and sending each separately. Can you help the commandment to make this decision?

Naval convoy usually consist of a few merchant ships surrounded by escort ships or airplanes. Looking from the air, we could approximate a convoy as a large blob or circle on the water. What do you think will have higher chances of safely arriving to the destination, one big blob or few small ones?


Lynnet said...

I thank it would be safer to travel in several smaller convoys then in one large convoy.

Assuming the enemy is taking random shots, they are more likely to hit a larger object or mass of objects then a smaller object or mass of objects.

Tom said...

The Brits were right, and most navies still operate that way, as do merchant convoys in wartime. The way to bet on the most-safe arrival, in this case, is with a medium number of ships (or groups of ships), each strong enough to be seaworthy, fast enough for evasion, large enough to carry a meaningful payload, etc. And the importance of self-defense (protective ships or airplanes) is also critical. Also, redundancy will help the mission.

10,000 rowboats is not the answer of course. And One Big Blob Convoy makes an irresistible target, which, if discovered and badly damaged, will end with unaffordable losses.

Several medium-sized convoys seems right, AND they should pursue different routes to the destination, and probably plan on different arrival times if possible.

Probably there is mathematical way to influence these decisions, but it's not coming to me.


Maria said...

This was my guess as well and in fact splitting a large convoy into few small ones was the initial tactics of the British Army.

However, according to Wikipedia, Operations Research division of the British Army studied the convoy transport problem and in 1943 recommended a strategy switch from a few small to large convoys. This change of tactics is believed to significamtly aid the war. Operational Army Researchers showed that the losses suffered by convoys depended largely on the number of escort vessels present, rather than on the overall size of the convoy. When number of escort vessels is pre-defined, it is therefore better to arrange them around a large convoys than apread them among many small ones.

Additionally, in his fasctinating book "One hundred Essential Things You Didn't Know You Didn't Know" J. D. Barrow describes two other math reasons for large convoy preference:

1) Looking from the air, we could approximate a convoy as a large blob or circle on the water. How much escort would a large blob versus blob split into two or more blobs need? See illustration above.
Let's compare a single convoy (blob) of area A with two blobs of area A/2 each. As escort goes around the convoy, we should compare perimeter of a single blog vs sum of perimeters of smaller blobs. Single blob perimeter turns out to be smaller. Therefore one would need less escort ships for a large convoy than for two small ones assuming same spacing of escort ships.

2)We want to minimize chances of being discovered by the enemy submarines. Submarine scans the water surface seeing a diameter of the convoy blob on the screen. Will diameter of one blob be larger than combined diameters of a few split blobs?
Combined diameters of smaller blobs are 41% larger than a single diameter, making split convoy 41% more detectable by submarines!

I am sure there are few more ways of looking at this problem. Please continue writing your thoughts below. Any ideas count and everyone participating will get a puzzle point! Tom and Lynnet - definitely are. Sorry for the broken link in the newsletter. So gald you found your way to this puzzle.

anne-marie said...

Could we talk of conditional probability when several smaller convoys are present and this probability will be smaller than the probability of a bigger convoy?

Maria said...

Anne-Marie brings a great point. In many situations probability of detecting a large blob of something is higher than probability of detecting a few smaller blobs that add to the same size as the large one. Take - skin mole detection at the dermatologists office. Or berry and mushroom picking. However, one may also say that if one of the small blobs (convoy, mole or mushroom) is detected, then detector becomes more vigilant and chances of finding the rest of the small blobs are increasing.

We should probably also clarify what exactly we mean by detecting several small blobs or convoys: finding all of them and not only a few.

Do you want to elaborate more, Anne-Marie?

Donna said...

The two small blobs would have more patrol vehicles surrounding them, as the circumferences of the two circles would be greater than that of the one big circle, therefore, the two small blobs would be more likely to reach their destination safely.

Maria said...

Donna - you are absolutely right. As British operations research shown, the more patrol flotilla has - the safer it is. However, in the cases when the number of patrol vehicles is pre-defined and can not be increased, it will be wiser to send the boats in one big blob. Splitting them will result in a sparser guarded circle.

Donna said...

Maria, I am trying to post my answers before reading anyone else's comments or answers, and didn't realize the number of patrol vehicles was pre-determined. You are exactly that case it definitely makes more sense to send one big blob. ;)

anne-marie said...

We could have at least two variables.
One has a binary distribution. Y1 being found or not.
Another variable could be being destroyed or not. A big Blob may have a bigger chance to survive several shootings and still arrive at destination.
It is just an idea, I am discovering probabilities right now:)

Thanks for your puzzles and your site.

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