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Variational Problems for Determining Optimal Paths of a Moving Facility

This paper is concerned with some variational calculus/optimization problems involving the determination of constrained optimal path functions or routes of a single moving facility, which traverses a region encompassing a set of existing facilities. Situations of this type may involve a reconnaisance plane maintaining a surveillance of certain locations, a patrol car traveling in radio contact with several stations with the possibility of being dispatched to answer service calls originating over several designated regions, or an aircraft determining a flight plan over a region containing enemy missile sites. We formulate the general problem of maximizing total benefits, or maximizing the benefits per unit travel time, subject to constraints on the travel path function. The specific problems analyzed involve a direct distance related benefit (or cost) function, employing rectilinear or squared-Euclidean distance measures, and they constrain the paths to be straight lines between parallel boundaries or across rectangular regions, with both ends free or with one end fixed, or to be general paths between fixed end-points. In each case, the problem is reduced to a finite dimensional optimization problem. Illustrative examples and computational results on a personal computer are provided, and several extensions are proposed for future research.