Optimal inventory policy with fixed and proportional transaction costs under a risk constraint

The traditional inventory models focus on characterizing replenishment policies in order to maximize the total expected profit or to minimize the expected total cost over a planned horizon. However, for many companies, total inventory costs could be accounting for a fairly large amount of invested capital. In particular, raw material inventories should be viewed as a type of invested asset for a manufacturer with suitable risk control. This paper is intended to provide this perspective on inventory management that treats inventory problems within a wider context of financial risk management. In view of this, the optimal inventory problem under a VaR constraint is studied. The financial portfolio theory has been used to model the dynamics of inventories. A diverse portfolio consists of raw material inventories, which involve market risk because of price fluctuations as well as a risk-free bank account. The value-at-risk measure is applied thereto to control the inventory portfolio's risk. The objective function is to maximize the utility of total portfolio value. In this model, the ordering cost is assumed to be fixed and the selling cost is proportional to the value. The inventory control problem is thus formulated as a continuous stochastic optimal control problem with fixed and proportional transaction costs under a continuous value-at-risk (VaR) constraint. The optimal inventory policies are derived by using stochastic optimal control theory and the optimal inventory level is reviewed and adjusted continuously. A numerical algorithm is proposed and the results illustrate how the raw material price, inventory level and VaR constraint are interrelated.