Scheduling, Location, Transportation, and Continuum Mechanics: Some Simple Approximations to Optimization Problems

Many optimization problems involve the determination of a large number of (perhaps vector-valued) parameters which collectively can be represented as a set of points in some space of small dimension. For example, one may wish to determine a set of times \ tⱼ \ at which to dispatch buses (a set of points on the real line), or to locate public facilities in a city (a set of points in the plane). In many of these problems, the objective function is more sensitive to spacings between the points than to their precise location. By elementary methods, one can often find an approximate global optimal by simply choosing the spacings (or approximate density of points) so as to give a “local optimal” everywhere in the space.