Rectilinear distance to a facility in the presence of a square barrier

This paper derives analytical expressions for the rectilinear distance to a facility in the presence of a square barrier. The distribution of the barrier distance is derived for two regular patterns of facilities: square and diamond lattices. This distribution, which provides all the information about the barrier distance, will be useful for facility location problems with barriers and reliability analysis of facility location. The distribution of the barrier distance demonstrates how the location and the size of the barrier affect the barrier distance. A?numerical example shows that the total barrier distance increases as the barrier gets closer to a facility, whereas the maximum barrier distance increases as the barrier becomes greater in size.