An Inventory System with Postponed Demands

Abstract

In this article we consider a continuous review perishable inventory system in which the demands arrive according to a Markovian arrival process (MAP). The items in the inventory have shelf life times that are assumed to follow an exponential distribution. The inventory is replenished according to an (s, S) policy and the replenishing times are assumed to follow a phase type distribution. The demands that occur during stock out periods either enter a pool which has capacity N (<∞) or leave the system. Any demand that arrives when the pool is full and the inventory level is zero, is also assumed to be lost. The demands in the pool are selected one by one, if the replenished stock is above s, with interval time between any two successive selections is distributed as exponential with parameter depending on the number of customers in the pool. The joint probability distribution of the number of customers in the pool and the inventory level is obtained in the steady state case. The measures of system performance in the steady state are derived and the total expected cost rate is also calculated. The results are illustrated numerically.     

Perishable Inventory System at Service Facilities with N Policy

Abstract

This article presents a perishable stochastic inventory system under continuous review at a service facility in which the waiting hall for customers is of finite size M. The service starts only when the customer level reaches N (< M), once the server has become idle for want of customers. The maximum storage capacity is fixed as S. It is assumed that demand for the commodity is of unit size. The arrivals of customers to the service station form a Poisson process with parameter λ. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. The items of inventory have exponential life times. It is also assumed that lead time for the reorders is distributed as exponential and is independent of the service time distribution. The demands that occur during stock out periods are lost.The joint probability distribution of the number of customers in the system and the inventory levels is obtained in steady state case. Some measures of system performance in the steady state are derived. The results are illustrated with numerical examples.     

Inventory System Subject to Disaster with General Interarrival Times

Abstract

We discuss a single commodity continuous review (s, S) inventory system in which commodities get damaged due to external disaster. Shortages are not permitted and lead time is assumed to be zero. The interarrival times of demands constitute a family of i.i.d. random variables with a common arbitrary distribution. The quantity demanded at a demand epoch is arbitrarily distributed which depends only on the time elapsed since the last demand epoch. Transient and steady state probabilities of the inventory levels are derived by identifying suitable semi-regenerative process. In the case when the demand is for unit item and the disaster affects only an exhibiting item, the steady state probability distribution is obtained as uniform. An optimization problem is discussed and numerical examples are provided.     

A two-echelon base-stock inventory model with Poisson demand and the sequential processing of orders at the upper echelon

We consider a two-echelon inventory system with a number of non-identical, independent ‘retailers’ at the lower echelon and a single ‘supplier’ at the upper echelon. Each retailer experiences Poisson demand and operates a base stock policy with backorders. The supplier manufactures to order and holds no stock. Orders are produced, in first-come first-served sequence, with a fixed production time. The supplier therefore functions as an M/D/1 queue. We are interested in the performance characteristics (average inventory, average backorder level) at each retailer. By finding the distribution of order lead time and hence the distribution of demand during order lead time, we find the steady state inventory and backorder levels based on the assumption that order lead times are independent of demand during order lead time at a retailer. We also propose two alternative approximation procedures based on assumed forms for the order lead time distribution. Finally we provide a derivation of the steady state inventory and backorder levels which will be exact as long as there is no transportation time on orders between the supplier and retailers. A numerical comparison is made between the exact and approximate measures. We conclude by recommending an approach which is intuitive and computationally straightforward.     

A discrete time inventory system with postponed demands

In this article, we consider a discrete-time inventory model in which demands arrive according to a discrete Markovian arrival process. The inventory is replenished according to an (s,S)$(s,S)$ policy and the lead time is assumed to follow a discrete phase-type distribution. The demands that occur during stock-out periods either enter a pool which has a finite capacity N(<∞)$N(<\infty )$ or leave the system with a predefined probability. Any demand that arrives when the pool is full and the inventory level is zero, is assumed to be lost. The demands in the pool are selected one by one, if the on-hand inventory level is above s+1$s+1$, and the interval time between any two successive selections is assumed to have discrete phase-type distribution. The joint probability distribution of the number of customers in the pool and the inventory level is obtained in the steady state case. The measures of system performance in the steady state are derived and the total expected cost rate is also calculated. The results are illustrated numerically.     

Managing an integrated production inventory system with information on the production and demand status and multiple non-unitary demand classes

In this paper, we study a system consisting of a manufacturer or supplier serving several retailers or clients. The manufacturer produces a standard product in a make-to-stock fashion in anticipation of orders emanating from n retailers with different contractual agreements hence ranked/prioritized according to their importance. Orders from the retailers are non-unitary and have sizes that follow a discrete distribution. The total production time is assumed to follow a k₀-Erlang distribution. Order inter-arrival time for class l demand is assumed to follow a k_l-Erlang distribution. Work-in-process as well as the finished product incur a, per unit per unit of time, carrying cost. Unsatisfied units from an order from a particular demand class are assumed lost and incur a class specific lost sale cost. The objective is to determine the optimal production and inventory allocation policies so as to minimize the expected total (discounted or average) cost. We formulate the problem as a Markov decision process and show that the optimal production policy is of the base-stock type with base-stock levels non-decreasing in the demand stages. We also show that the optimal inventory allocation policy is a rationing policy with rationing levels non-decreasing in the demand stages. We also study several important special cases and provide, through numerical experiments, managerial insights including the effect of the different sources of variability on the operating cost and the benefits of such contracts as Vendor Managed Inventory or Collaborative Planning, Forecasting, and Replenishment. Also, we show that a heuristic that ignores the dependence of the base-stock and rationing levels on the demands stages can perform very poorly compared to the optimal policy.     

Stochastic decomposition in production inventory with service time

We study an (s, S) production inventory system where the processing of inventory requires a positive random amount of time. As a consequence a queue of demands is formed. Demand process is assumed to be Poisson, duration of each service and time required to add an item to the inventory when the production is on, are independent, non-identically distributed exponential random variables. We assume that no customer joins the queue when the inventory level is zero. This assumption leads to an explicit product form solution for the steady state probability vector, using a simple approach. This is despite the fact that there is a strong correlation between the lead-time (the time required to add an item into the inventory) and the number of customers waiting in the system. The technique is: combine the steady state vector of the classical M/M/1 queue and the steady state vector of a production inventory system where the service is instantaneous and no backlogs are allowed. Using a similar technique, the expected length of a production cycle is also obtained explicitly. The optimal values of S and the production switching on level s have been studied for a cost function involving the steady state system performance measures. Since we have obtained explicit expressions for the performance measures, analytic expressions have been derived for calculating the optimal values of S and s.     

Optimal EOQ Models for Deteriorating Items with Time-Varying Demand

In this paper, optimal inventory lot-sizing models are developed for deteriorating items with general continuous time-varying demand over a finite planning horizon and under three replenishment policies. The deterioration rate is assumed to be a constant fraction of the on-hand inventory. Shortages are permitted and are completely backordered. The proposed solution procedures are shown to generate global minimum replenishment schedules for both general increasing and decreasing demand patterns. An extensive empirical comparison using randomly generated linear and exponential demands revealed that the replenishment policy which starts with shortages in every cycle is the least cost policy and the replenishment policy which prohibits shortages in the last cycle exhibited the best service level effectiveness. An optimal procedure for the same problem with trended inventory subject to a single constraint on the minimum service level (maximum fraction of time the inventory system is out of stock during the planning horizon) is also proposed in this paper.     

A perishable inventory system with service facility and finite source

In this article, we consider a continuous review (s,S)$(s\text{,}S)$ perishable inventory system with a service facility, wherein the demand of a customer is satisfied only after performing some service on the item which is assumed to be of random duration. We also assume that the demands are generated by a finite homogeneous population. The service time, the lead time are assumed to have Phase type distribution. The life time of the item is assumed to have exponential distributions. The joint distribution of the number of customers in the system and the inventory level is obtained in the steady state case. The Laplace–Stieltjes transform of the waiting time of the tagged customer is derived. Various system performance measures are derived and the total expected cost rate is computed under a suitable cost structure. The results are illustrated numerically.     

Multi-item two-echelon spare parts inventory control problem with batch ordering in the central warehouse under compound Poisson demand

We consider a multi-item two-echelon spare part inventory system in which the central warehouse operates under an (nQ,?R) policy and the local warehouses implement order-up-to S policy, each facing a compound Poisson demand. The objective is to find the policy parameters minimizing expected system-wide inventory holding and fixed ordering costs subject to an aggregate mean response time constraint at each warehouse. In this paper, we propose four alternative approximations for the steady state performance of the system; and extend a heuristic and a lower bound proposed under Poisson demand assumption to the compound Poisson setting. In a computational study, we show that the performances of the approximations, the heuristic, and the lower bound are quite satisfactory; and the relative cost saving of setting an aggregate service level rather than individually for each part is quite high.     

A Continuous Review of (<Emphasis Type="Italic">S — s</Emphasis>) Inventory Systems in which Depletion is Due to Demand and Failure of Units

A stochastic model for an inventory system in which depletion of stock takes place due to random demand as well as random failure of items is studied under the assumption that the intervals between successive unit demands, as well as those between successive unit failures, are independently and identically distributed random variables having negative exponential distribution with different parameters. The transient and steady state probability distributions of the stock level are derived and the optimum decision rules in the long run given.     

Two-Commodity Inventory System with Individual and Joint Ordering Policies and Renewal Demands

Abstract

This article analyzes a two-commodity continuous review inventory system with renewal demands. The ordering policy is a combination of policies namely ordering individual commodities and ordering jointly both commodities. The steady state probability distribution for the joint inventory levels is computed. Various system performance measures in the steady state are derived. The results are illustrated numerically.     

Optimal lateral transshipment policies for a two location inventory problem with multiple demand classes

We consider an inventory model for spare parts with two stockpoints, providing repairable parts for a critical component of advanced technical systems. As downtime costs for these systems are expensive, ready–for–use spare parts are kept in stock to be able to quickly respond to a breakdown of a system. We allow for lateral transshipments of parts between the stockpoints upon a demand arrival. Each stockpoint faces demands from multiple demand classes. We are interested in the optimal lateral transshipment policy. There are three ways in which a demand can by satisfied: from own stock, via a lateral transshipment, or via an emergency procedure. Using stochastic dynamic programming, we characterize and prove the structure of the optimal policy, that is, the policy for satisfying the demands which minimizes the average operating costs of the system. This optimal policy is a threshold type policy, with state-dependent thresholds at each stockpoint for every demand class. We show a partial ordering in these thresholds in the demand classes. In addition, we derive conditions under which the so-called hold back and complete pooling policies are optimal, two policies that are often assumed in the literature. Furthermore, we study several model extensions which fit in the same modeling framework.     

An inventory system with unit demand and varying ordering levels

An inventory system with unit demand, varying ordering levels and random lead times is considered in this paper. Ordering level is determined by the number of demands during last lead time. The ordering quantity will be such as to bring back the inventory level to S at the ordering epoch. No backlog is permitted. The time dependent probability distribution of the inventory level is obtained. Correlation between the number of demands during a lead time and the length of the next inventory dry period is obtained and it is illustrated by an example.     

Analysis of optimal opportunistic replenishment policies for inventory systems by using a (s,S) model with a maximum issue quantity restriction

The analysis of optimal inventory replenishment policies for items having lumpy demand patterns is difficult, and has not been studied extensively although these items constitute an appreciable portion of inventory populations in parts and supplies types of stockholdings. This paper studies the control of an inventory item when the demand is lumpy. A continuous review (s,S) policy with a maximum issue quantity restriction and with the possibility of opportunistic replenishment is proposed to avoid the stock of these items being depleted unduly when all the customer orders are satisfied from the available inventory and to reduce ordering cost by coordinating inventory replenishments. The nature of the customer demands is approximated by a compound Poisson distribution. When a customer order arrives, if the order size is greater than the maximum issue quantity w, the order is satisfied by placing a special replenishment order rather than from the available stock directly. In addition, if the current inventory position is equal to or below a critical level A when such an order arrives, an opportunistic replenishment order which combines the special replenishment order and the regular replenishment order will be placed, in order to satisfy the customer's demand and to bring the inventory position to S. In this paper, the properties of the cost function of such an inventory system with respect to the control parameters s, S and A are analysed in detail. An algorithm is developed to determine the global optimal values of the control parameters. Indeed, the incorporation of the maximum issue quantity and opportunistic replenishment into the (s,S) policy reduces the total operating cost of the inventory system.     

Perishable inventory systems with impatient demands

An inventory system for perishable commodities (PIS) with finite shelf size and finite waiting room for demands is studied; the maximum shelf life and the maximum waiting time of a demand are assumed to be either constant or exponentially distributed, and the arrival rates for items and for demands are state-dependent. We determine the stationary distribution of the system and derive various kinds of cost functionals that are useful to evaluate the efficiency of the PIS.     

On the value of customer information for an independent supplier in a continuous review inventory system

We consider the inventory control problem of an independent supplier in a continuous review system. The supplier faces demand from a single customer who in turn faces Poisson demand and follows a continuous review (R, Q) policy. If no information about the inventory levels at the customer is available, reviews and ordering are usually carried out by the supplier only at points in time when a customer demand occurs. It is common to apply an installation stock reorder point policy. However, as the demand faced by the supplier is not Markovian, this policy can be improved by allowing placement of orders at any point in time. We develop a time delay policy for the supplier, wherein the supplier waits until time t after occurrence of the customer demand to place his next order. If the next customer demand occurs before this time delay, then the supplier places an order immediately. We develop an algorithm to determine the optimal time delay policy. We then evaluate the value of information about the customer’s inventory level. Our numerical study shows that if the supplier were to use the optimal time delay policy instead of the installation stock policy then the value of the customer’s inventory information is not very significant.     

具有两类需求服务的随机库存系统控制策略研究

研究具有两类顾客排队需求服务的随机库存系统.系统采取(s,Q)补货策略且当库存水平下降到安全库存s时,到达的第二类顾客以概率P得到服务.首先,建立库存水平状态转移方程并通过递推算法求解获得库存水平稳态概率分布和系统稳态指标;接下来,构建库存成本函数;最后,采用数值试验的方法研究该库存系统的最优控制策略并考察系统参数的敏感性.     

Inventory Problems with Partially Observed Demands and Lost Sales

This paper considers the case of partially observed demand in the context of a multi-period inventory problem with lost sales. Demand in a period is observed if it is less than the inventory level in that period and the leftover inventory is carried over to the next period. Otherwise, only the event that it is larger than or equal to the inventory level is observed. These observations are used to update the demand distributions over time. The state of the resulting dynamic program consists of the current inventory level and the current demand distribution, which is infinite dimensional. The state evolution equation for the demand distribution becomes linear with the use of unnormalized probabilities. We study two demand cases. First, the demands evolve according to a Markov chain. Second, the demand distribution has an unknown parameter which is updated in the Bayesian manner. In both cases, we prove the existence of an optimal feedback ordering policy. Permanent address of J. Adolfo Minjárez-Sosa: Departamento de Matemáticas, Universidad de Sonora, Hermosillo, Sonora, México. This project was partially supported by NSF Grant 0509278, ARPATP Grant 009741-0019-2006, and CONACYT (Mexico) Grant 46633-F.