Determining the reorder point and order-up-to-level in a periodic review system so as to achieve a desired fill rate and a desired average time between replenishments (original) (raw)
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Fill rate in a periodic review Order-Up-To policy under correlated normally distributed demand
2014
We investigate the inventory customer service metric known as the fill rate. The fill rate is defined as the proportion of demand that is immediately fulfilled from inventory. However, the task of finding analytical solutions for general cases is difficult. In the literature two approximate approaches are proposed. The first of these approximations is the traditional fill rate (or p2 service measure) that is exact in the Order-Up-To replenishment policy with Minimum Mean Squared Error forecasting, zero lead-time and independent and identically distributed (i.i.d.) demand. However, when any of these assumptions is relaxed then the traditional fill rate measure is only a lower bound. A second approximation in the literature has been proposed by Sobel (2004) that is better able to cope with non-zero leadtimes as it manages the double accounting of accumulated backlogs. Sobel’s approach still requires positive i.i.d. demands implying there is no correlation between demand and net stock....
International Journal of Production Economics, 2015
We investigate the inventory service metric known as the fill rate-the proportion of demand that is immediately fulfilled from inventory. The task of finding analytical solutions for general cases is complicated by a range of factors including; correlation in demand, double counting of backlogs, and proper treatment of negative demand. In the literature, two approximate approaches are often proposed. Our contribution is to present a new fill rate measure for normally distributed, autocorrelated, and possibly negative demand. We treat negative demand as returns. Our approach also accounts for accumulated backlogs. The problem reduces to identifying the minimum of correlated normally distributed bivariate random variables. There exists an exact solution, but it has no closed form. However, the solution is amenable to numerical techniques, and we present a custom Microsoft Excel function for practical use. Numerical investigations reveal that the new fill rate is more robust than previous measures. Existing fill rate measures are likely to cause excessive inventory investment, especially when fill rate targets are modest, a strongly positive or negative autocorrelation in demand is present, or negative demands exist. Our fill rate calculation ensures that the target fill rate is achieved without excessive inventory investments.
2015
Background: Due to random changes in demand, inventory management is still despite the development of alternative goods flow management concepts an important issue both in terms of costs of maintenance and replenishment as well as the level of service measured by inventory availability levels. There are a number of replenishment systems to be used in such conditions, but they are most often formed on the basis of two basic ones: a system based on the reorder point and based on periodic inspection. This paper refers to the former system, the BS system (min-max), in which an order is placed after reaching inventory level B (information level, reorder point) for a quantity allowing to reach level S. This system is very often used in business practice. Observations conducted under realistic conditions indicate the need to improve the classical models describing the system. This results, among other things, from the fact that the actual level of available inventory at the start of the re...
Model of Inventory Replenishment in Periodic Review Accounting for the Occurrence of Shortages
2014
Background: Despite the development of alternative concepts of goods flow management, the inventory management under conditions of random variations of demand is still an important issue, both from the point of view of inventory keeping and replenishment costs and the service level measured as the level of inventory availability. There is a number of inventory replenishment systems used in these conditions, but they are mostly developments of two basic systems: reorder point-based and periodic review-based. The paper deals with the latter system. Numerous researches indicate the need to improve the classical models describing that system, the reason being mainly the necessity to adapt the model better to the actual conditions. This allows a correct selection of parameters that control the used inventory replenishment system and as a result to obtain expected economic effects. Methods: This research aimed at building a model of the periodic review system to reflect the relations (obs...
Analysis and Management of Periodic Review, Order-Up-To Level Inventory Systems with Order Crossover
Production and Operations Management, 2013
In this paper we investigate the (R, S) periodic review, order-up-to level inventory control system with stochastic demand and variable leadtimes. Variable leadtimes can lead to order crossover, in which some orders arrive out of sequence. Most theoretical studies of order-up-to inventory systems under variable leadtimes assume that crossovers do not occur and, in so doing, overestimate the standard deviation of the realized leadtime distribution and prescribe policies that can inflate inventory costs. We develop a new analytic model of the expected costs associated with this system, making use of a novel approximation of the realized (reduced) leadtime standard deviation resulting from order crossovers. Extensive experimentation through simulation shows that our model closely approximates the true expected cost and can be used to find values of R and S that provide an expected cost close to the minimum cost. Taking account of, as opposed to ignoring, crossovers leads, on average, to substantial improvements in accuracy and significant cost reductions. Our results are particularly useful for managers seeking to reduce inventory costs in supply chains with variable leadtimes.
European Journal of Operational Research, 2012
The primary goal of this paper is the development of a generalized method to compute the fill rate for any discrete demand distribution in a periodic review policy. The fill rate is defined as the fraction of demand that is satisfied directly from shelf. In the majority of related work, this service metric is computed by using what is known as the traditional approximation, which calculates the fill rate as the complement of the quotient between the expected unfulfilled demand and the expected demand per replenishment cycle, instead of focusing on the expected fraction of fulfilled demand. This paper shows the systematic underestimation of the fill rate when the traditional approximation is used, and revises both the foundations of the traditional approach and the definition of fill rate itself. As a result, this paper presents the following main contributions: (i) a new exact procedure to compute the traditional approximation for any discrete demand distribution; (ii) a more suitable definition of the fill rate in order to ignore those cycles without demand; and (iii) a new standard procedure to compute the fill rate that outperforms previous approaches, especially when the probability of zero demand is substantial. This paper focuses on the traditional periodic review, order up to level system under any uncorrelated, discrete and stationary demand pattern for the lost sales scenario.
The exact fill rate in a periodic review base stock system under normally distributed demand
Omega, 2011
In this paper we consider a periodic review order-up-to-level (or base stock) inventory control system under normally distributed demand. For such circumstances an expression for the exact fill rate (fraction of demand satisfied without backordering) has been available in the literature but has not been widely known, let alone used by practitioners. In this paper we redevelop the expression and contrast our derivation with the earlier published one. The paper has two purposes. First, we hope that the reappearance of the exact result in this journal will lead to its wider adoption. Second, showing two contrasting approaches to obtaining the same result may be useful for both research and pedagogical purposes.
A Continuous Review Inventory System with Bulk Demand and Markov Dependent Replenishment Quantities
Calcutta Statistical Association Bulletin, 1991
The paper deals with a continuous review bulk demand (positive-integer valued) ( s, S) inventory system where the interarrival times of demands are independent and identically distributed random variables. We assume that the successive quantities demallded lie between a and b (0< a⩽ b; s- b+l⩾0) with pk, k= a, a+l, ... , b-1, b as the probability that k items are demanded by an arriving unit. The maximum capacity of the system is S units and as soon as the inventory level falls to the set A= { s- b + 1, s- b + 2, ... , s -1, s}, order is placed for a quantity S- i if the ordering level is i, i ε A. Our model assumes that the quantity replenished forms a Markov chain defined over thestatespace E={ c, c + l, ... , S - s} with c⩾ b. Lead time is zero and no shortage is permitted. The distribution of the on band inventory at arbitrary time point and also the limiting distributions are obtained. A numerical illustration associated with the model is also provided.
Mathematical and Computer Modelling, 2013
In this article we analyze a lost sales (s, Q) inventory system with two types of customers and stochastic lead times. Demands from each type of customer arrive according to two independent Poisson processes. A comparative study of the average cost rate for the cases where there is differentiation between service to customers based on the type of customer and the case where there is no differentiation between customers is carried out by using the concepts of rationing. We provide numerical examples where differentiation between customers yield lower cost and lower shortage rates for both types of customers.