Gale's Round-Trip Jeep Problem

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- . In 1947 Fine [Fin] introduced and solved a problem of maximizing the distance a jeep can travel into the desert using n drums of fuel. Subsequently, Phipps [Phi], Alway [Alw], and Gale [Gal] gave other solutions to the original problem or considered related problems. As mentioned in [Fin], the original problem is similar to one which arose in air transport operations in the China theater during World War II, and it has been suggested that may be applications to Arctic expeditions and interplanetary travel. Near the end of [Gal], the author states, An apparently simple question is the round trip problem in which fuel is available at both ends of the desert, but I must confess . . . that I have not been able to find the solution. It is not hard to see that one can do at least as well in this case as in the case of two jeeps making one-way trips, but it may be possible to do better. The difficulty here as with many optimization problems is that does not appear to be any simple way to determine whether or not a given solution is optimal. Gale's problem can be interpreted in two equivalent ways. (i) Given unlimited fuel at each end of a desert of given length, find a round trip across the desert which uses as little fuel as possible. (ii) Given a fixed amount of fuel which can be distributed between the two ends of a desert, find the maximum length desert which can be crossed in a round trip using the available fuel. We find it convenient to consider (ii) and give an optimal solution for it. We also describe a solution for the analogous round trip problem where the two allowed depots may be placed anywhere in the desert. In each of the above problems the jeep can carry exactly 1 drum. It is implicit that the jeep can store whatever fraction of a drum is desired at any point in the desert. (Perhaps the driver carries large plastic bags for fuel storage.) In [Dew], Dewdney proposed an interesting variation of the one-way problem. Although Dewdney's problem was given in terms of drums, gallons, and miles, it can be rephrased as follows: Find the maximum distance a jeep can travel into the desert using n drums of fuel where the jeep can carry 1 drum plus 1/5 of a drum in its tank, but only drums can be stored. That is the jeep can dump at most 5/6 of its fuel capacity in the desert. It is interesting to note that Dewdney's problem has been solved as a linear programming problem; an optimal algorithm for Dewdney's problem appears in [Jac]. But the problems solved in this paper apparently are not easily posed as either linear or dynamic programming problems. In [Gal], Gale also points out that there is a feeling among many people that the original jeep problem can be solved by the functional equation method of dynamic programming. . . I know of no way of solving the problem by this method. 1. A BRIEF DESCRIPTION OF THE SOLUTION TO GALE'S PROBLEM. In solving Gale's problem we will start by considering the longest desert which can be crossed in a round trip if are m drums of fuel at the start S and k drums of