To model the distillation or decomposition of products in some manufacturing
processes, a minimum distribution cost problem (MDCP) for a specialized
manufacturing network flow model has been investigated. In an MDCP, a
specialized node called a D-node is used to model a distillation process that
connects with a single incoming arc and several outgoing arcs. The flow
entering a D-node has to be distributed according to a pre-specified ratio
associated with each of its outgoing arcs. This proportional relationship
between arc flows associated with each D-node complicates the problem and
makes the MDCP more difficult to solve than a conventional minimum cost
network flow problem. A network simplex algorithm for an uncapacitated MDCP
has been outlined in the literature. However, its detailed graphical
procedures including the operations to obtain an initial basic feasible
solution, to calculate or update the dual variables, and to pivot flows have
never been reported. In this paper, we resolve these issues and propose a
modified network simplex algorithm including detailed graphical operations in
each elementary procedure. Our method not only deals with a capacitated MDCP,
but also offers more theoretical insights into the mathematical properties of
an MDCP.