کنترل موجودی و سیاست های سرمایه گذاری

The idea underlying this paper is to consider inventory control as part of a wider class of economic problems, namely financial risk management. Only discrete-time models are investigated here. In the framework of the Cox–Ross–Rubinstein model of a financial market a combined inventory replenishment and investment policy is obtained. Company solvency is studied as well.

The aim of this paper is to draw attention of inventory researchers to new perspectives opened by treating inventory problems within the wider scope of financial risk management. In the year 2001 we celebrate the half century anniversary of the pioneering work of Arrow et al. (1951). Along with the seminal papers of Dvoretzky 1952 and Dvoretzky 1953 it formed the foundation of modern inventory theory (or, more strictly, its cost approach). It is interesting to recall that the 1971 Nobel Prize winner, the Honorary President of ISIR Kenneth Arrow, has also greatly contributed to the development of financial economics. The classical (AHM) inventory model and its various modifications (see, e.g. Chikán, 1986) usually took into account inventory replenishment and holding costs, as well as a shortage penalty. Although some authors (see, e.g., Rustenburg et al., 1999) considered budget restrictions, for the most part the cash amount available for inventories was assumed to be unlimited and the decisions were aimed at the minimization of expected accumulated costs (since demand was supposed to be random). There was a lot of discussion about capital tied up in inventories (see, e.g., Pujawan and Kingsman, 1999). We mention in passing that, in an attempt to avoid the undesirable effects of inventory holding, a new just-in-time concept has been introduced in inventory theory and practice (see, e.g., Groenevelt, 1993). One could also say that implicitly the existence of a financial market was taken into account from the outset by discounting future expenses. Now the time has come to use explicitly and in full the powerful tools of financial mathematics by combining inventory control with investment policy. For simplicity we consider below only discrete-time models. Section 2 contains some preliminary results. First we point out the effect of a budget constraint in the classical inventory model. Then we drop the nonnegativity restriction on inventory order, allowing for seasonal sales. The one-period demand is assumed to be a random variable with a finite mean (taking negative values as well in order to incorporate the possibility of customers returning bought product or remanufacturing). In Section 3 we introduce the Cox–Ross–Rubinstein model of a financial market (for more details see the fundamental book by Shiryaev (1999)). Assuming an inventory replenishment policy to be fixed we establish the optimal investment strategy. Using a reliability approach we treat the company solvency problem. In the framework of the cost approach we obtain the optimal inventory replenishment and investment policy. Almost all proofs are omitted due to lack of space. In Section 4 we draw conclusions and outline further research directions.

The results of the previous section justify the commonly used present value calculation of future cash-flow streams for risk averse investors (in the framework of the Cox–Ross–Rubinstein model). In a general (B,S)-model the dynamics of assets prices is described by two discrete stochastic differential equations View the MathML source here we use the notation ΔAn=An−An−1. The corresponding one-period interest rates rn are View the MathML source-measurable and ρn are View the MathML source-measurable for all n⩾1. In this case View the MathML source is a filtration, that is a nondecreasing sequence of σ-fields in a probability space View the MathML source and View the MathML source. Hence, we get instead of (11) View the MathML source where View the MathML source. Therefore a result similar to Theorem 5 is valid if, for each View the MathML source and rn are ordered with probability 1. The choice of an appropriate objective function in the general case is one of the standing problems. Comparing Theorem 3, for n=1, and the result of Remark 3, explicitly taking into account investment, one can see that in the latter case the optimal replenishment policy is still specified by critical levels. However these parameters depend on asset profitability. A multiperiod version of such models is treated in Bulinskaya (2000), which also incorporates investment in the models described by Eq. (5). Here we considered discrete-time models. Another interesting research direction is an investigation of continuous-time models. However, this involves intricate mathematical tools such as continuous-time stochastic processes, Ito's calculus, Girsanov's theorem and so on.