A global shooting algorithm for the facility location and capacity acquisition problem on a line with dense demand

• Model a facility location and capacity acquisition problem with dense demand on a line.
• Reformulate the dynamic programming formulation as a two-point boundary value problem.
• Develop an exact shooting algorithm to solve the problem as an initial value problem.
• Effects of demand density, fixed cost distribution, and economies of scale are studied.
• Sensitivity to the number of facilities increases with the level of demand aggregation.

This paper describes the development of an exact allocation-based solution algorithm for the facility location and capacity acquisition problem (LCAP) on a line with dense demand data. Initially, the n-facility problem on a line is studied and formulated as a dynamic programming model in the allocation decision space. Next, we cast this dynamic programming formulation as a two-point boundary value problem and provide conditions for the existence and uniqueness of solutions. We derive sufficient conditions for non-empty service regions and necessary conditions for interior facility locations. We develop an efficient exact shooting algorithm to solve the problem as an initial value problem and illustrate on an example. A computational study is conducted to study the effect of demand density and other problem parameters on the solutions.