# Name of Math Problem - Iteratively Bringing Supplies from Place A to Place B

I remember hearing / reading about a scenario during WW2 where the US / Western Powers were thinking about attacking Japan via an overland route from India. The problem involved how to leapfrog the supplies from India over to China and there begin the fight with the Japanese. There’s a logic to it as the distance is far less than from California to Japan. But some mathematicians realized that it would take too long to stock up the supplies and so the island hopping strategy was finalized.

The problem, in a nutshell, is Plane A can travel X miles with a full load of supplies. But then it needed to have enough gas to fly back to the home base in order to get more supplies. (Or, there would be some combinations where some planes carry only supplies in the cargo areas while others carry extra gas to allow the planes to return to base to restock up supplies.)

So the questions are:

1. What is the name of this math problem (if such a name exists) where one determines how many trips would it require to bring supplies from place A to place B.
2. Has anyone any information if this calculation ever took place. (This second question may have to be asked at history.stackexchange). I’ve made some perfunctory searches and haven’t found anything - which leads me to question if this story is apocryphal.
• If you want to complicate the practical problem, stop the planes from landing anywhere apart from their take-off point. Then you get things like Operation Black Buck Commented Nov 26, 2019 at 23:42

## 2 Answers

It's called a Transportation problem, studied by Transportation theory. Kantorovich and Koopmans were awarded the Nobel Prize in Economics in 1975 for the theory.

These notes from UCSD mention the WW2 story, but without references.

F. L. Hitchcock formulated a transportation problem in 1941, but it doesn't seem related to WW2.

Ford & Fulkerson in Solving the Transportation Problem mention that Koopmans considered it during WW2, but they don't mention if it was motivated by the needs of the military.

Quoted from Jeep problem - Wikipedia:

The jeep problem, desert crossing problem or exploration problem is a mathematics problem in which a jeep must maximise the distance it can travel into a desert with a given quantity of fuel. The jeep can only carry a fixed and limited amount of fuel, but it can leave fuel and collect fuel at fuel dumps anywhere in the desert.

The problem first appeared in the 9th-century collection Propositiones ad Acuendos Juvenes (Problems to Sharpen the Young), attributed to Alcuin. The De viribus quantitatis (c. 1500) of Luca Pacioli also discusses the problem. A modern treatment was given by N. J. Fine in 1947.

## Problem

There are n units of fuel stored at a fixed base. The jeep can carry at most 1 unit of fuel at any time, and can travel 1 unit of distance on 1 unit of fuel (the jeep's fuel consumption is assumed to be constant). At any point in a trip the jeep may leave any amount of fuel that it is carrying at a fuel dump, or may collect any amount of fuel that was left at a fuel dump on a previous trip, as long as its fuel load never exceeds 1 unit. There are two variants of the problem:

• Exploring the desert – the jeep must return to the base at the end of every trip.
• Crossing the desert – the jeep must return to the base at the end of every trip except for the final trip, when the jeep travels as far as it can before running out of fuel.

In either case the objective is to maximise the distance travelled by the jeep on its final trip. Alternatively, the objective may be to find the least amount of fuel required to produce a final trip of a given distance.

In the classic problem the fuel in the jeep and at fuel dumps is treated as a continuous quantity. More complex variations on the problem have been proposed in which the fuel can only be left or collected in discrete amounts.

## Practical applications

The problem can have a practical application in wartime situations, especially with respect to fuel efficiency. In the context of the bombing of Japan in World War II by B-29s, Robert McNamara says in the film The Fog of War that understanding the fuel efficiency issue caused by having to transport the fuel to forward bases was the main reason why the strategy of launching bombing raids from mainland China was abandoned in favor of the island hopping strategy:

"We had to fly those planes from the bases in Kansas to India. Then we had to fly fuel over the hump into China. [...] We were supposed to take these B-29s—there were no tanker aircraft there. We were to fill them with fuel, fly from India to Chengtu; offload the fuel; fly back to India; make enough missions to build up fuel in Chengtu; fly to Yawata, Japan; bomb the steel mills; and go back to India.

We had so little training on this problem of maximizing [fuel] efficiency, we actually found to get some of the B-29s back instead of offloading fuel, they had to take it on. To make a long story short, it wasn't worth a damn. And it was LeMay who really came to that conclusion, and led the Chiefs to move the whole thing to the Marianas, which devastated Japan."

(The atomic bombing missions at the end of World War II were flown using B-29 Superfortresses from the Pacific island of Tinian in the Northern Marianas Islands.)

See also Operation Black Buck for an application of these ideas. In these missions, conducted during the Falklands War, the Royal Air Force used air to air refuelling by staging tankers to enable the Vulcan bombers based on Ascension Island to bomb targets in the Falkland Islands.

## References

Gardner, Martin (1994). My Best Mathematical and Logic Puzzles. Dover. p. 53. ISBN 0-486-28152-3.

Problems to Sharpen the Young, John Hadley and David Singmaster, The Mathematical Gazette, 76, #475 (March 1992), pp. 102–126.

Optimal Logistics for Expeditions: the Jeep Problem with Complete Refilling (PDF), Gunter Rote and Guochuan Zhang, June 1996

Quoted from Jeep Problem -- from Wolfram MathWorld:

Maximize the distance a Jeep can penetrate into the desert using a given quantity of fuel. The Jeep is allowed to go forward, unload some fuel, and then return to its base using the fuel remaining in its tank. At its base, it may refuel and set out again. When it reaches fuel it has previously stored, it may then use it to partially fill its tank. This problem is also called the exploration problem (Ball and Coxeter 1987).

Alway, G. C. "Crossing the Desert." Math. Gaz. 41, 209, 1957.

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 32, 1987.

Fine, N. J. "The Jeep Problem." Amer. Math. Monthly 54, 24-31, 1947.

Several other references are given, in both articles.

Somewhere I probably still have a clipping of the problem's appearance in D. St. P. Barnard's "Brain-Twister" column in the Observer newspaper in the UK in the late 1960s or early 1970s.

• One statement in the paper by Gale referred to in the MathWorld article is corrected in: David Gale, 'Correction to "The Jeep Once More or Jeeper by the Dozen"', Amer. Math. Monthly 78, No. 6 (Jun. - Jul., 1971), pp. 644-645. Commented Nov 27, 2019 at 20:03