Capacitated location-allocation problems on a line

For economic and other reasons, distribution networks are often constructed in hierarchies, where, due to high costs, high-level distribution channels are built in straight lines from which low-level channels branch. Branching facilities, too, may incur high costs and thus, their number, locations along the lines, and allocation of destinations to them are important components in system design, which can be accomplished by solving a series of capacitated location-allocation problems on a line. This problem is studied here and its properties are discussed, and appropriate mathematical optimization models formulated and their properties and complexity are also considered. Heuristic solution schemes and lower bounds on objective values are proposed for the models. The efficiency and effectiveness of these schemes are evaluated and more insight into the models is gained through a numerical study. The major conclusion is that the hierarchical structure considered is highly restrictive and imposes high costs. Therefore, its use must be properly justified.