Uncapacitated and Capacitated Facility Location Problems

The uncapacitated facility location problem ( UFLP) involves locating an undetermined number of facilities to minimize the sum of the (annualized) fixed setup costs and the variable costs of serving the market demand from these facilities. UFLP is also known as the “simple” facility location problem SFLP, where both the alternative facility locations and the customer zones are considered discrete points on a plane or a road network. This assumes that the alternative sites have been predetermined and the demand in each customer zone is concentrated at the point representing that region. UFLP focuses on the production and distribution of a single commodity over a single time period (e.g., one year that is representative of the firm’s long-run demand and cost structure), during which the demand is assumed to be known with certainty. The distinguishing feature of this basic discrete location problem, however, is the decision maker’s ability to determine the size of each facility without any budgetary, technological, or physical restrictions. Krarup and Pruzan (1983) provided a comprehensive survey of the early literature on UFLP, including its solution properties. By demonstrating the relationships between UFLP and the set packing-covering-partitioning problems, they established its NP-completeness. The seminal paper of Erlenkotter (1978), which is reviewed in Sect. 2.2 of this chapter, presents a dual-based algorithm for solving the UFLP that remains as one of the most efficient solution techniques for this problem. Prior to Erlenkotter (1978), the best-known approaches for solving the UFLP were the branch-and-bound algorithm developed by Efroymson and Ray (1966) and the implicit enumeration technique of Spielberg (1969). Efroymson and Ray (1966) use a compact formulation of UFLP to take advantage of the fact that its linear programming relaxation can be solved by inspection. Nonetheless, this linear programming relaxation does not provide tight lower bounds for UFLP; Efroymson and Ray’s model is therefore Chapter 2 Uncapacitated and Capacitated Facility Location Problems