Today efficient commodity management seems impossible without the support of intelligent systems and computing based on mathematical models of commodity flow assignment since a modern market network is a large-scale system with many elements. The present paper is devoted to the equilibrium commodity flow assignment problem, which solution is valuable from decision-making perspectives. The paper attempts to fill the gap in the analytical relations between the equilibrium commodity assignment pattern and demand-supply parameters. We study commodity assignment in the case of linear demand, supply, and transportation functions. The problem is formulated in a form of nonlinear optimization program with dual variables reflecting supply and demand prices. The equilibrium commodity assignment pattern is obtained explicitly via equilibrium prices. For simple networks of three and four nodes we are lucky to obtain explicit conditions for identification of active commodity flows. In other words, once the demand-supply parameters are given, then active commodity flows can be identified explicitly.