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 channel 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. Technologic and economic factors very often dictate hierarchical structure for distribution networks -high-level distribution channels from which low-level channels branch. Further, due to high costs and/or special features, high-level channels are frequently built in straight lines. This structure has, on one hand, a price in constraining the design of the remaining of the network by restricting branching facilities to be located along these lines, while the customers, to whom the goods or services are provided, are located anywhere in the plane. Costs, which may be high, are associated with other components of the distribution network, too. Therefore, the number of branching facilities, their locations along the lines, and allocation of destinations to them are important components in system design. On the other hand, under the hierarchical structure, the problem is decomposed into several location-allocation problems along a line- one for each high-level channel. The analysis of these effects is the purpose of this work. To this aim, location-allocation problems along a line are studied. Problems' properties are, first, examined, mathematical optimization models are formulated, and their properties and complexity are discussed. Following these discussions, lower bounds on objective values and heuristic solution schemes are developed. Finally, a computational study is performed, through which the merit of the proposed solution schemes is evaluated, the effects of various parameters are examined, and more insight into the problem is gained, in particular, into the consequences of the hierarchical structure.