Periodic-review System Research Articles

Effective and simple EOQ-like solutions for stochastic demand periodic review systems

In this paper we examine a periodic review system under stochastic demand with variable stockout costs. The optimal values for cycle length and amount of safety stock are difficult to obtain because one of the First Order Conditions does not have a closed form solution. However, by using a Taylor series expansion to approximate part of the cost function, we produce a simple cost function structure which is similar to that of deterministic models. We argue that this simple structure is also beneficial to promote the solution in other problems where coordination of cycles is required. To illustrate, we use the joint replenishment problem for multiple items under stochastic demand and suggest simple and efficient solution procedures.

Some insights regarding the optimal reorder period in periodic review inventory systems

The periodic review inventory system is not only pervasive, but also has an extensive literature dealing with various aspects, from its theoretical underpinnings through to its performance. However, the selection of the best review period and its behaviour with respect to basic inventory parameters such as demand and supply variability appears to be poorly understood. In this paper we treat the normal and gamma distributions which are probably the two most commonly used distributions of demand over the key (protection) time interval in a periodic review system. We show that the total cost function is not necessarily well behaved as a function of the review period; in particular, it can be non-convex, which complicates the determination of the best value of the review interval. Also we demonstrate and explain how the best review period changes as the values of various parameters are modified. Some of these changes are initially counterintuitive in nature. Managerially oriented interpretations should also be helpful to organisations seeking improvement in inventory system performance.

Optimal ordering policies for periodic-review systems with a refined intra-cycle time scale

Chiang [C. Chiang, Optimal ordering policies for periodic-review systems with replenishment cycles, European Journal of Operational Research 170 (2006) 44–56] recently proposed a dynamic programming model for periodic-review systems in which a replenishment cycle consists of a number of small periods (each of identical but arbitrary length) and holding and shortage costs are charged based on the ending inventory of small periods. The current paper presents an alternative (and concise) dynamic programming model. Moreover, we allow the possibility of a positive fixed cost of ordering. The optimal policy is of the familiar ( s, S) type because of the convexity of the one-cycle cost function. As in the periodic-review inventory literature, we extend this result to the lost-sales periodic problem with zero lead-time. Computation shows that the long-run average cost is rather insensitive to the choice of the period length. In addition, we show how the proposed model is modified to handle the backorder problem where shortage is charged on a per-unit basis irrespective of its duration. Finally, we also investigate the lost-sales problem with positive lead-time, and provide some computational results.

An exact calculation of the cycle service level in a generalized periodic review system

This paper is focused on the exact calculation of the cycle service level for a generalized periodic review system accepting any discrete demand with a known probability function and any cycle service target. Previously, it was necessary to modify the definition of the cycle service level metric (also known as non stock out probability) so that it could be applied to these cases. The purpose of the proposed (R, S) system is to handle simultaneously fast-moving items, slow-moving items and intermittent or sporadic demand items any of them with high, medium or low cycle service targets.

Changes in performance under various lengths of review periods in a periodic review inventory control system with lost sales

PurposeThis study investigates the impacts of changing the length of review period on the performance of a periodic review system with lost sales under varying demand patterns by using simulation method.Design/methodology/approachThe simulation model has been built using a spreadsheet program and depicts a special case of several retailers supplied by a central warehouse. Simulation scenarios are created based on two factors: length of the review period, and product type in terms of the average and standard deviation of the corresponding product's demand distribution.FindingsResults obtained from the simulation model show that inventory performance is quite sensitive to the duration of review periods, and selecting the appropriate period length is largely dependent on the variability of demand. Specifically, relatively shorter review periods are required for products with high variable demand, while the inventory system can tolerate longer review periods if the product demand is less fluctuating.Originality/valueThe study provides some practical guidelines for the inventory managers regarding the selection of an appropriate length for a review period in a periodic review system with lost sales.

Optimal ordering policies for periodic-review systems with replenishment cycles

In this paper, we consider inventory models for periodic-review systems with replenishment cycles, which consist of a number of periods. By replenishment cycles, we mean that an order is always placed at the beginning of a cycle. We use dynamic programming to formulate both the backorder and lost-sales models, and propose to charge the holding and shortage costs based on the ending inventory of periods (rather than only on the ending inventory of cycles). Since periods can be made any time units to suit the needs of an application, this approach in fact computes the holding cost based on the average inventory of a cycle and the shortage cost in proportion to the duration of shortage (for the backorder model), and remedies the shortcomings of the heuristic or approximate treatment of such systems (Hadley and Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, NJ, 1963). We show that a base-stock policy is optimal for the backorder model, while the optimal order quantity is a function of the on-hand inventory for the lost-sales model. Moreover, for the backorder model, we develop a simple expression for computing the optimal base-stock level; for the lost-sales model, we derive convergence conditions for obtaining the optimal operational parameters.

Managing Inventories: The Periodic Review System

This note addresses inventory-planning and -control activities using periodic-review systems. The reorder point system is addressed in a separate technical note (UVA-OM-0936).

Developing behavioural services to meet defined standards within a national system of specialist education services

In response to increasing demand for more accountability and improved outcomes in the provision of behavioural services, Specialist Education Services (SES) in New Zealand employed the trainer of trainers programme developed by the Institute for Applied Behaviour Analysis (IABA). The goal was to develop a national training team capable of training SES staff to carry out assessments and develop support plans that could meet defined standards. The purpose of this study was to evaluate the outcomes of this trainer of trainers project. The primary methods of evaluation included the pre-post training comparisons of trainees' functional assessments and positive behaviour support plans, against 140 defined criteria. The results of this project indicated that the SES national training team was able to train SES staff to meet the same standards of service delivery as the external IABA trainers. Further, a Periodic Service Review (PSR) system was implemented to insure that service standards could be maintained at a high level. A major conclusion reached in this study was that a trainer of trainer approach appears to be effective in preparing large numbers of educational personnel to meet the increasing demands by schools for professionals to meet a high standard of service delivery.

Inventory control based on advanced probability theory, an application

Whenever stock is placed as a buffer between consumption and supply the decision when to replenish the stock is based on uncertain values of future demand and supply variables. Uncertainty exists about the replenishment lead time, about the number of demands and the quantities demanded during this period. We develop a new analytical expression for the reorder point, which is based on the desired service level and three distributions: the distribution of the quantity of single demands during lead time, the distribution of the lengths of time intervals between successive demands, and the distribution of the lead time itself. The distribution of lead time demand is derived from the distributions of individual demand quantities and not from the demand per period. It is not surprising that the resulting formulae for the mean and variance are different from those currently used. The theory developed is also applicable to periodic review systems. The system has been implemented at CERN and enables a significant enhancement of the service level, while reducing the average stock.

Quantifying the Bullwhip Effect in a Supply Chain with a Periodic Review System

Quantifying the Bullwhip Effect in a Supply Chain with a Periodic Review System

Musictoday, LLC: Implementing a Periodic Review Inventory System

This case is used in Darden’s first-year core operations class as part of a module on supply chain management. The Musictoday company provided online retailing services for such clients as the Dave Matthews Band and the Rolling Stones. But the lack of a formal inventory-management process had Musictoday concerned about future stockouts that would result in lost sales. This case covers the basics of safety stock within the context of a periodic review system. It introduces students to the periodic review system and provides them with an opportunity to link the optimal review period with the EOQ concept.

Regulatory strategy for effective application of periodic safety review system

In less than 10 years, the first commercial pressurized water reactor (PWR) plant in Korea will reach its official design life. As part of safety activities, developed countries have already implemented periodic safety review (PSR) or equivalent programs to check and improve the safety of operating nuclear power plants (NPP) during their plant life. At the end of 1999, it was decided by the Korean Atomic Energy Safety Committee to adopt the PSR program and to apply it to Korean operating NPP. Since Kori Unit 1 started the review for the first tentative application of PSR as a model case in May 2000, it is now progressing well. Management of aging is one of the major factors to be considered in PSR and life extension of a nuclear plant. This paper is intended to introduce the regulatory aspect and strategy of Korean PSR. The background and scope of basic PSR guidelines are described, and a summary of technical criteria for aging management, which shows a regulatory direction for PSR, is also presented.

Inventory control with a modified Croston procedure and Erlang distribution

This paper considers an inventory control system, primarily for a finished goods inventory. The purpose is to create a procedure that can handle both fast-moving items with regular demand and slow-moving items. The suggested procedure should be easy to implement in a modern computerized ERP-system. Essentially, the system is a periodic review system built around a Croston forecasting procedure. An Erlang distribution is fitted to the observed data using the mean and variance of the forecasted demand rate. According to probabilities for stock shortages, derived from the probability distribution, the system decides if it is time to place a new order or not. The Croston forecasting method is theoretically more accurate than ordinary exponential smoothing for slow-moving items. However, it is not evident that a Croston forecasting procedure (with assumed Erlang distribution) outperforms ordinary exponential smoothing (with assumed normal distribution) applied in a “practical” inventory control system with varying demand, automatically generated replenishment, etc. Our simulation study shows that the system in focus will present fewer shortages at lower inventory levels than a system based on exponential smoothing and the normal distribution.

Uniform distribution of inventory positions in two-echelon periodic review systems with batch-ordering policies and interdependent demands

We consider a two-level inventory system in which there are one supplier and multiple retailers. The retailers face stochastic, interdependent customer demands. Each location employs a periodic-review ( R, nQ), or lot-size reorder point, inventory policy. We show that each location's inventory positions are stationary and the stationary distribution is uniform and independent of any other's.

Order splitting under periodic review inventory systems

Most research on order splitting have focused on the reduction of safety stock in the multiple sourcing setting. Moreover, all works study the use of order splitting for the continuous review inventory systems. In this paper, we investigate the possibility of the multiple-delivery arrangement in the sole sourcing environment. In addition, we concentrate on the reduction of cycle stock for periodic review systems. We show that splitting an order into multiple deliveries can significantly reduce the total cost especially if the cost of despatching an order for an item is not small. Although the use of information technology such as EDI decreases the ordering cost and thus shortens the period length, order splitting remains a cost-effective approach as long as the cost of despatching an order is not close to zero.

Three different operations research models for the same (s,S) policy

Operations Research techniques are usually presented as distinct models. Difficult as it may often be, achieving linkage between these models could reveal their interdependency and make them easier for the user to understand. In this article three different models, namely Markov Chain, Dynamic Programming, and Markov Sequential Decision Processes, are used to solve an inventory problem based on the periodic review system. We show how the three models converge to the same (s,S) policy and we provide a numerical example to illustrate such a convergence.

Multi-level control and asymptotic behaviour of some inventory systems

Periodic-review systems with demand and supply in batches of a fixed size are investigated. It is proved that all multi-level controls satisfying certain conditions provide the same asymptotic behaviour of the system as its capacity tends to infinity. Optimization problems are also considered.

(s, S) Continuous Review Models for Products with Fixed Lifetimes

We analyze an (s, S) continuous review perishable inventory system with a general renewal demand process and instantaneous replenishments. Though continuous review systems seem more amenable to optimization analysis than do periodic review systems, the existing literature addressing this type of model is rather limited. This limitation motivated us to seek greater understanding of this important class of inventory models. Using a Markov renewal approach, we obtain closed-form solutions for the steady state probability distribution of the inventory level and system performance measures. We then construct a closed-form expected cost function. Useful analytical properties for the cost function are identified and extensive computations are conducted to examine the impact of different parameters. The numerical results illustrate the system behavior and lead to managerial insights into controlling such inventory systems.

Periodic review and joint replenishment in stochastic demand environments

In this paper we consider periodic review systems under stochastic demands. First, we study the single product periodic review problem and propose a simple solution procedure which is near optimal. Then, given the existence of this simple procedure, we study the joint replenishment problem for multiple items under stochastic demands and suggest simple heuristics which provide very good results. We find the simple procedures combined with the robustness of the cost function to be very attractive in other applications which require coordination of cycle times under stochastic demands.

Inventory management with periodic ordering and minimum order quantities

Periodic review systems are commonly employed by distributors and retailers to replenish their inventories (for example, to co-ordinate in-bound transportation). It is also often the case that vendors specify minimum purchase quantities for physical (for example, packaging) or strategic reasons. When inventory systems recommend order quantities below the prescribed minimum, a decision must be made on whether or not to order. We describe how this environment can be modelled, and employ an extensive factorial experiment involving a simulation to demonstrate situations in which a common simple ‘rounding up’ decision rule performs poorly. In this paper, we discuss financial implications and implementation issues.