سیاست های موجودی مطلوب با هزینه های ثابت و معاملات متناسب تحت خطر محدودیت

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.

Inventories are stocks of raw materials, components and finished goods that are stored in warehouses, the instrumentalities of transportation, and retail stores. Raw material inventories are necessary to manufacturers because they create buffers against irregular supplies and demand shifts, guarantying product availability. Yet, according to Ballou [1], stockpiling inventory may result in costs in the range of 20%–40% of annual invested capital. Thus, good inventory control will provide lower costs and promote overall company performance. Recognizing the importance of inventory management, copious literature investigating optimal inventory strategies exists. Arrow et al. [2] laid the foundation of modern inventory theory, in which expected costs are chosen as an objective function. 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. Examples include the popular (EOQ) model and the (s,Ss,S) model [3], [1] and [4]. The conventional inventory control methods are appropriate for risk-neutral managers, but in reality most managers are risk averse [5], and in the field of the supply chain, risk analysis and risk control have become more and more important [6] and [7]. Therefore, techniques that consider both risks and returns of holding inventory are crucial.

This paper presented optimal inventory policies under a VaR constraint. By considering the raw material inventories as a kind of risky investment, we optimized the portfolio consisting of the risk free bank account and the raw material inventory. The problem is formulated as a continuous time stochastic control problem with fixed ordering costs and proportional selling and holding costs. We have proven that the optimal problem is reduced to solve the Hamilton–Jacobi–Bellman equation by applying the dynamic programming technique and the stochastic control theory. The VaR constraint is imposed continuously to control the risk of the portfolio. The optimal inventory strategy under a VaR constraint is summarized as a continually reviewed policy (rB,RB,rS,RS)(rB,RB,rS,RS). In the end, we proposed a numerical algorithm to solve the constrained optimal stochastic problem with a power-law utility function. From the numerical results, we find that risk is effectively reduced where holdings in raw material inventories are optimally decreased.