An Optimal Location Problem with an Obstruction

In the problem studied, Corporation A needs to visit the facilities P₁, P₂, P₃ with k₁, k₂, k₃ trips every week, respectively. An obstacle, such as a river, runs across the area from east to west. P₁ and P₂ are north of the river while P₃ is south of the river. There exists only a bridge connecting the regions north and south of the river. The streets in the area run east-west or north-south. The problem posed is to find an office location for Corporation A such that the total distance it travels every week to these three facilities is minimal. The hypothesis is that the optimal location should heavily depend on k₁+k₂ (trips north) and k₃ (trips south). The problem was solved by converting it into a mathematical problem of finding the minimum of a somewhat complicated distance function. By the nature of this problem, the distance travelled from point (x₁, y₁) to point (x₂, y₂) is not given by the typical Euclidean distance, but by |x₂-x₁|+|y₂-y₁| if both points are on the same side of the river and by |x₁|+|y₁|+|x₂|+|y₂| if both points are on opposite sides of the river. The mathematical formulas for the optimal locations were derived and confirmed the hypothesis precisely: if k₁+k₂ > k₃, the optimal location is on the same side as P₁ and P₂; if k₁+k₂ < k₃, P₃ is the optimal location; if k₁+k₂=k₃, a suitable portion of the vertical street through the bridge can be selected for the optimal location. The optimal location minimizes the usage of fuel and manpower for Corporation A.