On segmenting logistical zones for servicing continuously developed consumers

We study the optimal placement problem for several logistical objects. A characteristic feature of this problem is the need for sequential segmentation into servicing zones and accounting for population distributed continuously across the entire region. We reduce this problem to a variational calculus problem in a special form. To study this problem, we develop numerical algorithms that are able to determine the optimal placement of a logistical object inside a given segment. The algorithms are based on constructing wavefronts for a light wave emitted from the boundary of the chosen region. The wave moves inside the region, which lets us account for all inhabitants in this region. In constructing the solution, we have accounted for the loss of smoothness in the wavefront, have developed a software implementation for the computational algorithms, and have conducted a numerical experiment for a number of model problems.