A lost sales inventory model with a compound poisson demand pattern (original) (raw)
Related papers
Annals of Operations Research, 2015
Supply contracts are designed to minimize inventory costs or to hedge against undesirable events (e.g., shortages) in the face of demand or supply uncertainty. In particular, replenishment terms stipulated by supply contracts need to be optimized with respect to overall costs, profits, service levels, etc. In this paper, we shall be primarily interested in minimizing an inventory cost function with respect to a constant replenishment rate. Consider a single-product inventory system under continuous review with constant replenishment and compound Poisson demands subject to lost-sales. The system incurs inventory carrying costs and lost-sales penalties, where the carrying cost is a linear function of on-hand inventory and a lost-sales penalty is incurred per lost sale occurrence as a function of lost-sale size. We first derive an integro-differential equation for the expected cumulative cost until and including the first lost-sale occurrence. From this equation, we obtain a closed form expression for the time-average inventory cost, and provide an algorithm for a numerical computation of the optimal replenishment rate that minimizes the aforementioned time-average cost function. In particular, we consider two special cases of lost-sales penalty functions: constant penalty and loss-proportional penalty. We further consider special demand size distributions, such as constant, uniform and Gamma, and take advantage of their functional form to further simplify the optimization algorithm. In particular, for the special case of exponential demand sizes,
In this paper we extend earlier work that analyzes a single echelon single item base-stock inventory system where Demand is modeled as a compound Poisson process and the lead-time is stochastic. The extension consists in considering a cost oriented system where unfilled demands are lost. The case of partial lost sales is assumed. We first model the inventory system as a Makovian M/G/∞ queue then we propose a method to calculate numerically the optimal base-stock level. A preliminary numerical investigation is also conducted to show the performance of our solution.
Base Stock Inventory Systems with Compound Poisson Demand: Case of Partial Lost Sales
IFIP Advances in Information and Communication Technology, 2013
In this paper we extend earlier work that analyzes a single echelon single item base-stock inventory system where Demand is modeled as a compound Poisson process and the lead-time is stochastic. The extension consists in considering a cost oriented system where unfilled demands are lost. The case of partial lost sales is assumed. We first model the inventory system as a Makovian M/G/∞ queue then we propose a method to calculate numerically the optimal base-stock level. A preliminary numerical investigation is also conducted to show the performance of our solution.
Production-Inventory Systems with Lost Sales and Compound Poisson Demands
Operations Research, 2014
This paper considers a continuous-review, single-product, production-inventory system with a constant replenishment rate, compound Poisson demands, and lost sales. Two objective functions that represent metrics of operational costs are considered: (1) the sum of the expected discounted inventory holding costs and lost-sales penalties, both over an infinite time horizon, given an initial inventory level; and (2) the long-run time average of the same costs. The goal is to minimize these cost metrics with respect to the replenishment rate. It is, however, not possible to obtain closed-form expressions for the aforementioned cost functions directly in terms of positive replenishment rate (PRR). To overcome this difficulty, we construct a bijection from the PRR space to the space of positive roots of Lundberg's fundamental equation, to be referred to as the Lundberg positive root (LPR) space. This transformation allows us to derive closed-form expressions for the aforementioned cost metrics with respect to the LPR variable, in lieu of the PRR variable. We then proceed to solve the optimization problem in the LPR space and, finally, recover the optimal replenishment rate from the optimal LPR variable via the inverse bijection. For the special cases of constant or loss-proportional penalty and exponentially distributed demand sizes, we obtain simpler explicit formulas for the optimal replenishment rate.
A two-echelon inventory model with lost sales
International Journal of Production Economics, 2001
This paper considers a single-item, two-echelon, continuous-review inventory model. A number of retailers have their stock replenished from a central warehouse. The warehouse in turn replenishes stock from an external supplier. The demand processes on the retailers are independent Poisson. Demand not met at a retailer is lost. The order quantity from each retailer on the warehouse and from the warehouse on the supplier takes the same fixed value Q, an exogenous variable determined by packaging and handling constraints. Retailer i follows a (Q, R i ) control policy. The warehouse operates an (SQ, (S À 1)Q) policy, with non-negative integer S. If the warehouse is in stock then the lead time for retailer i is the fixed transportation time L i from the warehouse to that retailer. Otherwise retailer orders are met, after a delay, on a first-come first-served basis. The lead time on a warehouse order is fixed. Two further assumptions are made: that each retailer may only have one order outstanding at any time and that the transportation time from the warehouse to a retailer is not less than the warehouse lead time. The performance measures of interest are the average total stock in the system and the fraction of demand met in the retailers. Procedures for determining these performance measures and optimising the behaviour of the system are developed.
A decision model for an inventory system with two compound Poisson demands
Uncertain Supply Chain Management, 2020
A real inventory system for single item with specific demand characteristics motivates this works. The demand can be seen as two types of independent demand, where compound Poisson process describes the characteristics of each demand. The first type of demand is rarely occurred with relatively large size, while the second type of demand is often happened with relatively small size. In order to maintain inventory level, every time the first type of demand occurs a replenishment of stock is conducted which follows order-up-to-level inventory policy. In order to find the optimal inventory decision for that system, a mathematical model of the system is developed with the objective to minimize expected total inventory cost. Some of model assumptions are infinite replenishment, deterministic lead time, and completely backlogged shortages. To solve the model, it is then divided into two sub-problems and classical optimization technique is employed to help find the solution of each sub problem. .
On the (S ? 1,S) lost sales inventory model with priority demand classes
Naval Research Logistics, 2002
In this paper an inventory model with several demand classes, prioritised according to importance, is analysed. We consider a lot-for-lot or (S − 1, S) inventory model with lost sales. For each demand class there is a critical stock level at and below which demand from that class is not satisfied from stock on hand. In this way stock is retained to meet demand from higher priority demand classes. A set of such critical levels determines the stocking policy. For Poisson demand and a generally distributed lead time, we derive expressions for the service levels for each demand class and the average total cost per unit time. Efficient solution methods for obtaining optimal policies, with and without service level constraints, are presented. Numerical experiments in which the solution methods are tested demonstrate that significant cost reductions can be achieved by distinguishing between demand classes. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 593–610, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10032
Base-stock policy for the lost-sales inventory system with periodic review
Unpublished manuscript, 2005
Periodic review systems in which unfilled demands are treated as lost sales are of importance as building blocks for inventory control and coordination, particularly in the retail sector. When replenishments are provided by a single supplier, it is also natural to assume that orders do not cross in time, i.e., lead times are sequential. We suggest an approximate cost model in order to find good base-stock policies for such a setting, when lead times are gamma distributed, demand is compound Poisson, and a standard cost structure is used. The approximation is based upon a distribution for the inventory on order specified in analogy with the continuous-review case. Little's formula is then used to find the approximate average inventory on hand and the average lost sales. The base-stock level prescribed by our approximate cost model is easy to find. Several alternative approximate models are suggested or obtained from the literature. In a numerical study we evaluate the approximate models by means of simulation. It is shown that the standard approximations might perform quite badly in some cases, and that our suggested approximation outperforms the alternative approximations for the set of parameter variations tested. Surprisingly, the base-stock levels obtained from our approximation also work well when orders are allowed to cross in time because lead times are independent.
An inventory model where customer demand is dependent on a stochastic price process
International Journal of Production Economics, 2019
We investigate the optimal inventory operations of a firm selling an item whose price is driven by an exogenous stochastic price process which consequently impacts customer arrivals between ordering cycles. This case is typical for retailers that operate in different currencies, or trade products consisting of commodities or components whose prices are subject to market fluctuations. We assume that there is a stochastic input price process for the inventory item which determines purchase and selling prices according to a general selling price function. Customers arrive according to a doubly-stochastic Poisson process that is modulated by stochastic input prices. We analyze optimal ordering decisions for both backorder and lost-sale cases. We show that under certain conditions, a price-dependent base stock policy is optimal. Our analysis is then extended to a price-modulated compound Poisson demand case, and the case with fixed ordering cost where a price-dependent (s, S) policy is optimal. We present a numerical study on the sensitivity of optimal profit to various parameters of the operational setting and stochastic price process such as price volatility, customer sensitivity to price changes etc. We then make a comparison with a corresponding discrete-time benchmark model that ignores within-period price fluctuations and present the optimality gap when using the benchmark model as an approximation.
Exact Analysis of a Lost Sales Model under Stuttering Poisson Demand
2009
We investigate the (S-1,S) inventory policy under stuttering Poisson demand and generally distributed lead time when the excess demand is lost. We correct results presented in Feeney and Sherbrooke's seminal paper (1966). We also prove that the distribution of ordered unit delivery times becomes increasingly concentrated as the variance-to-mean ratio of demand increases.