Deterministic and Stochastic Extensions of the Jeep Problem

William J Giffen, Purdue University

Abstract

This thesis represents the extension of the Jeep Problem into two previously unexplored areas. The Jeep Problem is a well known logistics problem wherein a jeep must cross a desert wider than it can travel on one tank of fuel. The first extension solves the problem when depots must be purchased with fuel. The second extension adds a stochastic element to the model, in the form of a gamma distributed fuel consumption function. Adding depot costs to the Jeep Problem led to four major results. Firstly, it is proven that all optimal solutions to the modified Jeep Problem require the driver to consume integer amounts of fuel between depots (i.e. no fractional loads of fuel are carried into the desert), and that those amounts are no longer limited to one load of fuel. Secondly, as the number of depots drop, the inter-depot distances tend to equalize. Thirdly, when depot costs depend on the distance from the origin, the earlier distances tend to become smaller, with more fuel “pushed out” to later depots. Finally, higher depot costs result in fewer depots, with more loads of fuel between them. The stochastic extension of the Jeep Problem led to a number of interesting results. The first is that the most appropriate model for modeling fuel mileage was found to be the gamma distribution, as it can model a wide range of mileage functions. However, with the gamma distribution, as the variance increased (relative to the mean), the expected distance traveled dropped. In addition, as the number of loads of fuel are increased, unexpectedly, the number of depots that are used in the optimal solution drops with increased variance. Both behaviors are the result of risk aversion, and in the latter case, a sacrifice of efficiency is made in favor of safety. It is believed that these extensions allow the Jeep Problem, to extend into new classes of problems. These include the manned mission to mars and military supply problems, and other classes of problems where the basic Jeep Problem is not robust enough to be applicable.

Degree

Ph.D.

Advisors

Morin, Purdue University.

Subject Area

Industrial engineering

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