A linear model for compound multicommodity network flow problems

This work proposes a network flow linear program model to solve the problem of minimizing costs of production and distribution of compound multicommodities. In our proposed model, coupling constraints are considered in order to treat the existing proportionality among several flows of different commodities that are necessary to mixture for composing new commodities. The coupling constraint matrix for this type of problem is very large in general. Our formulation reduces the number of proportionality constraints enabling the use of a solution technique based on a specialization of the primal-dual simplex algorithm applied in two distinct phases. As initial solution it is used a basis built through the heuristic method that allocates flows in low cost lanes. To perform the change of basis operation, the working matrix is stored as a product form of the inverse to keep constant its dimension and to preserve sparsity. Experimental results containing around 200,000 constraints and 75,000 arcs applicable to the distribution of multicommodities of a petrochemical industry were accomplished with success. The results obtained show computer efficiency of the developed algorithm and the applicability of the formulated model.