Unbalanced, capacitated p-median problems on a chain graph with a continuum of link demands

This article is concerned with the problem of locating p capacitated facilities on a chain graph and simultaneously determining the allocation of their supplies in order to satisfy a continuum of demand. The demand is characterized by some weighted probability density functions defined on the chain graph. The objective is to minimize the total (expected) trnasportation cost. This location-allocation problem is also referred to as the capacitated p-median problem on a chain graph. Two unbalanced cases of this problem are considered, namely, the overcapacitated case when total supply exceeds total demand and the deficit capacity case when total supply is less than total demand. Both these problems are nonconvex and are shown to be NP-hard even if the demand density function is piecewise uniform and positive. We provide a first-order characterization of optimality for these two problems and prescribe an enumerative algorithm based on a partitioning of the dual space in order to optimally solve these problems. An extension of these algorithms for solving the unbalanced, capacitated 2-median problem on a tree graph is also given.