The extraction, transformation, use, and disposal of materials can be represented by directed, weighted networks, known in the material flow analysis (MFA) community as Sankey or flow diagrams. However, the construction of such networks is dependent on data that are often scarce, conflicting, or do not directly map onto a Sankey diagram. By formalizing the forms of data entry, a nonlinear constrained optimization program for data estimation and reconciliation can be formulated for reconciling data sets for MFA problems where data are scarce, in conflict, do not directly map onto a Sankey diagram, and are of variable quality. This method is demonstrated by reanalyzing an existing MFA of global steel flows, and the resulting analytical solution measurably improves upon their manual solution.
