This paper presents a scheduling model of multi-stage manufacturing systems with multiple job classes and variable production capacity constraints at workstations. The model aims to minimize the total cost which is the sum of all costs incurred in holding, processing and schedule delay over the whole study horizon. On the basis of analyzing optimality conditions, the prices of production capacities or resources are derived and explained in economic meaning. It is shown that the model can be transferred into an equivalent static minimum cost network flow problem over a space-time network which is a temporal expansion of the base workstation network. An adapted network approach based on the Frank-Wolfe algorithm is proposed to solve the equivalent problem and tested on a numerical example.