The Capacitated p-facility Location Problem on the Real Line

The problem we address involves locating p new facilities to service a set of customers or fixed points on the real line such that a measure of total cost will be minimized. A basic form of this problem was investigated by Love (1976), who observed that the fixed points must be allocated in sequence to the new facilities in an optimal solution, and thus, the problem can be solved by a dynamic programming algorithm. Since then, other forms of the model have been investigated; however, in all cases it is assumed that the new facilities have unlimited capacity so that customer flows are always allocated to the nearest facility. The objective of this paper is to analyze the effect of capacity constraints on the optimal locations of the new facilities. A general fixed-cost function is also included to account for practical considerations such as zoning regulations, and to permit the facilities to be located anywhere on the line instead of only at the fixed vertices. A dynamic programming method is formulated to solve the problem when the variable cost components are increasing convex functions of travel distance. The problem is shown to be NP-hard under more general cost structures.