Optimal Dynamic Procurement Policies for a Storable Commodity with L\'evy Prices and Convex Holding Costs

In this paper we study a continuous time stochastic inventory model for a commodity traded in the spot market and whose supply purchase is affected by price and demand uncertainty. A firm aims at meeting a random demand of the commodity at a random time by maximizing total expected profits. We model the firm's optimal procurement problem as a singular stochastic control problem in which controls are nondecreasing processes and represent the cumulative investment made by the firm in the spot market (a so-called stochastic "monotone follower problem"). We assume a general exponential L\'evy process for the commodity's spot price, rather than the commonly used geometric Brownian motion, and general convex holding costs. We obtain necessary and sufficient first order conditions for optimality and we provide the optimal procurement policy in terms of a "base inventory" process; that is, a minimal time-dependent desirable inventory level that the firm's manager must reach at any time. In particular, in the case of linear holding costs and exponentially distributed demand, we are also able to obtain the explicit analytic form of the optimal policy and a probabilistic representation of the optimal revenue. The paper is completed by some computer drawings of the optimal inventory when spot prices are given by a geometric Brownian motion and by an exponential jump-diffusion process. In the first case we also make a numerical comparison between the value function and the revenue associated to the classical static "newsvendor" strategy.Comment: 28 pages, 3 figures; improved presentation, added new results and section

Analysis of purchasing activity with discounted cash flow inventory models = Beszerzési tevékenység elemzése diszkontált pénzáramlású készletmodellel

A klasszikus tételnagyság probléma két fontosabb készletezési költséget ragad meg: rendelési és készlettartási költségek. Ebben a dolgozatban a vállalatok készpénz áramlásának a beszerzési tevékenységre gyakorolt hatását vizsgáljuk. Ebben az elemzésben a készpénzáramlási egyenlőséget használjuk, amely nagyban emlékeztet a készletegyenletekre. Eljárásunkban a beszerzési és rendelési folyamatot diszkontálva vizsgáljuk. A költségfüggvény lineáris készpénztartási, a pénzkiadás haszonlehetőség és lineáris kamatköltségből áll. Bemutatjuk a vizsgált modell optimális megoldását. Az optimális megoldást egy számpéldával illusztráljuk. = The classical economic order quantity model has two types of costs: ordering and inventory holding costs. In this paper we try to investigate the effect of purchasing activity on cash flow of a firm. In the examinations we use a cash flow identity similar to that of in inventory modeling. In our approach we analyze the purchasing and ordering process with discounted costs. The cost function of the model consists of linear cash holding, linear opportunity cost of spending cash, and linear interest costs. We show the optimal solution of the proposed model. The optimal solutions will be presented by numerical examples

On multi-stage production/inventory systems under stochastic demand

This paper was presented at the 1992 Conference of the International Society of Inventory Research in Budapest, as a tribute to professor Andrew C. Clark for his inspiring work on multi-echelon inventory models both in theory and practice. It reviews and extends the work of the authors on periodic review serial and convergent multi-echelon systems under stochastic stationary demand. In particular, we highlight the structure of echelon cost functions which play a central role in the derivation of the decomposition results and the optimality of base stock policies. The resulting optimal base stock policy is then compared with an MRP system in terms of cost effectiveness, given a predefined target customer service level. Another extension concerns an at first glance rather different problem; it is shown that the problem of setting safety leadtimes in a multi-stage production-to-order system with stochastic lead times leads to similar decomposition structures as those derived for multi-stage inventory systems. Finally, a discussion on possible extensions to capacitated models, models with uncertainty in both demand and production lead time as well as models with an aborescent structure concludes the paper