Optimal control and competitive equilibrium of production–inventory systems with application to the petroleum refining industry

An extended abstract of a 2005 winner of the MSOM Society Student Paper Competition, in which a competitive equilibrium production–inventory model with application to the petroleum refining industry is studied. We first analyze a continuous-time optimal control problem of a firm (refinery) making procurement, production, and sales decisions under uncertain raw material (crude oil) price, finished goods (petroleum products) price, and operating costs. The costs include inventory holding cost, production cost, and some other operating costs. The firm maximizes the expected discounted profit over an infinite horizon. We prove the optimality of certain threshold-type control policies. Next, we consider an economy with many raw material suppliers, production firms, and consumers. The source of uncertainties in this economy is the supply and the demand fluctuations. Based on the firm-level optimal control policies, we establish a competitive rational expectations equilibrium and characterize the dynamics of the equilibrium by a set of equations that describes certain operational tradeoffs. We provide a general procedure for computing the equilibrium and derive the equilibrium in explicit form for a special economy. To assess the empirical usefulness of the equilibrium theory, we simulate the industry equilibrium in the special economy under various settings. We find that the simulated equilibrium price and inventory processes exhibit some patterns that are observed in the actual petroleum data. Finally, we fit the theoretical model using the actual data. The results generally support the theoretical model.