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

An Inventory Model for Deteriorating Commodity under Stock Dependent Selling Rate

Economic order quantity (EOQ) is one of the most important inventory policy that have to be decided in managing an inventory system. The problem addressed in this paper concerns with the decision of the optimal replenishment time for ordering an EOQ to a supplier. This Model is captured the affect of stock dependent selling rate and varying price. We developed an inventory model under varying of demand-deterioration-price of commodity when the relationship of supplier-grocery-consumer at stochastic environment. The replenishment assumed instantaneous with zero lead time. The commodity will decay of quality according to the original condition with randomize characteristics. First, the model is addressed to solve a problem phenomenon how long is the optimum length of cycle time. Then, an EOQ of commodity to be ordered by will be determined by model. To solve this problem, the first step is developed a mathematical model based on reference’s model, and then solve the model analytically. Finally, an inventory model for deteriorating commodity under stock dependent selling rate and considering selling price was derived by this research. Keywords: deterioration commodity, expected profit, optimal replenishment time stock dependent selling rate

A Finite Horizon Inventory Model: An Operational Framework

We present a simulation based decision support system to decide the inventory ordering policy in the context of a single commodity, multi pack, and finite horizon situation. The multiple objectives include (a) Minimizing the end of the season inventory, (b) Maximizing the operating profit, (c) Minimizing the peak working capital requirements during the season. Stochastic demand and positive lead time add to the complexity of the problem context. In addition multiple partners in the supply chain with distinct and conflicting set of objectives necessitate the need for a formal approach. The motivation for this model is based on a real life situation. The model addresses the decision choices faced by the distributor in a specific logistics chain. In this chain, a typical distributor has to balance between the stochastic nature of the demand and the attractive nature of financial incentives (order quantity based) proposed by the manufacturer. The problem can be formulated as a multi-period dynamic programming problem with stochastic demand with an objective to optimize the expected operating profit, subject to specific constraints on working capital requirement, service level, order fill rate and end of the season inventory. Such a formulation is hard to solve and does not lend itself to analyze several ordering policies. Based on simulation experiments, we propose an ordering policy which optimizes the overall objectives of supply chain partners and hence demonstrated the possibility of jointly managing the uncertain demand by supply chain partners. The model is simple and easy to use. It is implemented by using spreadsheet. It provides adequate flexibility to conduct what-if analysis. The model has a potential to be useful in a wide range of situations.

Robust Stochastic Lot-Sizing by Means of Histograms

Traditional approaches in inventory control first estimate the demand distribution among a predefined family of distributions based on data fitting of historical demand observations, and then optimize the inventory control using the estimated distributions. These approaches often lead to fragile solutions whenever the preselected family of distributions was inadequate. In this article, we propose a minimax robust model that integrates data fitting and inventory optimization for the single-item multi-period periodic review stochastic lot-sizing problem. In contrast with the standard assumption of given distributions, we assume that histograms are part of the input. The robust model generalizes the Bayesian model, and it can be interpreted as minimizing history-dependent risk measures. We prove that the optimal inventory control policies of the robust model share the same structure as the traditional stochastic dynamic programming counterpart. In particular, we analyze the robust model based on the chi-square goodness-of-fit test. If demand samples are obtained from a known distribution, the robust model converges to the stochastic model with true distribution under generous conditions. Its effectiveness is also validated by numerical experiments.National Science Foundation (U.S.) (Contract CMMI-0758069