Optimal inventory control under stochastic purchase prices

Abstract

Many companies consume a huge amount of market-traded commodities in their daily operations. As the market prices of these commodities fluctuate significantly over time, it is very important for the companies to take into account the volatile spot prices when making procurement decisions of these commodities. In order to address the issue, this thesis studies three multi-period stochastic inventory models for a market-traded commodity with fluctuating purchase price. The first model considers a risk-neutral decision maker, whose objective is to minimize the expected total cost over the planning horizon. We prove that the optimal inventory control policy is a price-dependent base-stock policy, where the base-stock levels depend on the commodity price in that period. We further establish the monotonicity properties of the optimal base-stock level with respect to the commodity price and the uncertain price change, study the relative magnitude of the optimal base-stock level for the model with deterministic demand, and derive the closed-form optimal base-stock level for the single-period model with deterministic demand. The second model assumes that the decision maker is risk-averse, and hence optimizes the expected exponential utility of the present value of the total cost over the planning horizon. The optimal inventory policy remains a base-stock policy depending on the commodity price. The closed-form optimal base-stock level is derived for the single-period model with deterministic demand and binomial-tree price process. The third model incorporates the financial hedging decision in the risk-averse inventory control problem, where the uncertainty in the purchase price can be partially hedged by holding derivatives of the commodity available in the financial market. Again, we establish the optimality of a state-dependent base-stock inventory control policy. It is also proved that a hedging portfolio is optimal if and only if there exists a risk-neutral measure of the stochastic price change satisfying a linear equation of the hedging portfolio. When the financial market is sufficiently diversified, the optimal inventory policy can be obtained independent of the optimal hedging policy, and an optimal hedging portfolio can be constructed by a limited number of derivatives. Building upon these results, we further study several special cases of this model, where the commodity price follows a binomial-tree process or/and the demand of the commodity is deterministic. Various properties are established regarding the monotonicity of the optimal base-stock level with respect to the commodity price, the relative magnitude of the optimal base-stock level, as well as the monotonicity of the optimal hedging policy with respect to the order-up-to level and the risk-aversion parameter, respectively. Based on the results obtained from the three models, we also compare the corresponding optimal base-stock levels through both theoretical analysis and computational studies.