Balanced, capacitated, location-allocation problems on networks with a continuum of demand

Location-allocation problems can be described generically as follows: Given the location or distribution (perhaps, probabilistic) of a set of customers and their associated demands for a given product or service, determine the optimum location of a number of service facilities and the allocation of products or services from facilities to customers, so as to minimize total (expected) location and transportation costs.

This study is concerned with a particular subclass of location-allocation problems involving capacitated facilities and a continuum of demand. Specifically, two minisum, network-based location-allocation problems are analyzed in which facilities having known finite capacities are to be located so as to optimally supply/serve a known continuum of demand.

The first problem considered herein, is an absolute p-median problem in which p > l capacitated facilities are to be located on a chain graph having both nodal and link demands, the latter of which are defined by nonnegative, integrable demand functions. In addition, the problem is balanced, in that it is assumed the total demand equals the total supply. An exact solution procedure is developed, wherein the optimality of a certain location-allocation scheme (for any given ordering of the facilities) is used to effect a branch and bound approach by which one can identify an optimal solution to the problem.

Results from the chain graph analysis are then used to develop an algorithm with which one can solve a dynamic, sequential location-allocation problem in which a single facility per period is required to be located on the chain.

Finally, an exact solution procedure is developed for locating a capacitated, absolute 2-median on a tree graph having both nodal and link demands and for which the total demand is again equal to the total supply. This procedure utilizes an algorithm to construct two subtrees, each of whose ends constitute a set of candidate optimal locations for one of the two elements of an absolute 2-median. Additional localization results are used to further reduce the number of candidate pairs (of ends) that need to be considered, and then a post-localization analysis provides efficient methods of comparing the relative costs of the remaining pairs.