An investigation of setup instability in non-stationary stochastic inventory systems

In stochastic inventory systems unfolding uncertainties in demand lead to the revision of earlier replenishment plans which in turn results in an instability or so-called system nervousness. In this paper, we provide the grounds for measuring system nervousness in non-stationary demand environments, and gauge the stability and the cost performances of (R,S) and (s,S) inventory policies. Our results reveal that, both the stability and the cost performance of inventory policies are affected by the demand pattern as well as the cost parameters, and the (R,S) policy has the potential to replace the cost-optimal (s,S) policy for systems with limited flexibility.     

A robustness study of a multi-echelon inventory model via simulation

This paper examines the robustness of a standard model of multi-echelon inventory systems, specifically the models discussed in Axsäter (Oper. Res. 48(5) (2000) 686). A simulation model was developed to explore the model's ability to predict system performance for a two-echelon one-warehouse, multiple retailer system using (R,Q) inventory policies under conditions that violate the model's fundamental modeling assumptions. In particular, the impact of non-stationary demand on this stationary demand inventory model was examined. The model performs well at the low demand and large retailer order batch size situations, but our testing of the model indicated that care must be taken when applying this model to situations that violate its fundamental assumption. These results should help practitioners to better understand the assumptions of these models and to determine when or when not to apply these models in practice.     

Lead time reduction strategies in a single-vendor-single-buyer integrated inventory model with lot size-dependent lead times and stochastic demand

This paper studies alternative methods for reducing lead time and their impact on the safety stock and the expected total costs of a (Q,s) continuous review inventory control system. We focus on a single-vendor-single-buyer integrated inventory model with stochastic demand and variable, lot size-dependent lead time and assume that lead time consists of production and setup and transportation time. As a consequence, lead time may be reduced by crashing setup and transportation time, by increasing the production rate, or by reducing the lot size. We illustrate the benefits of reducing lead time in numerical examples and show that lead time reduction is especially beneficial in case of high demand uncertainty. Further, our studies indicate that a mixture of setup time and production time reduction is appropriate to lower expected total costs.     

Base stock policies with degraded service to larger orders

We study an inventory system controlled by a base stock policy assuming a compound renewal demand process. We extend the base stock policy by incorporating rules for degrading the service of larger orders. Two specific rules are considered, denoted as Postpone(q,t) and Split(q), respectively. The parameter q distinguishes between regular orders (of size less than or equal to q) and larger orders. We develop mathematical expressions for the performance measures: order fill rate of the regular orders and average on-hand inventory level. We make numerical experiments where the postpone parameter t and the base stock levels of each rule are such that all customers (of both order types) are indifferent between the two rules. When comparing the difference in the average on-hand inventory levels, we can then make an assessment of the threshold value of the cost of splitting an order (which may otherwise be hard to quantify) in the rule Split(q). Our numerical results indicate that this threshold value is increasing in the variance of the order sizes. Based on the numerical experiment our conclusion is therefore that when the variance of the order sizes is low, then Postpone(q,t) seems to be a good option, while when the variance is high, then Split(q) is more competitive.     

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.     

Reducing the mean supply delay of spare parts using lateral transshipments policies

Lateral transshipment has been studied lately as a promising policy for increasing the performances of multi-echelon spare parts inventory system. By lateral transshipment spare parts can be moved from one location with excess inventory to another location, at the same echelon, in shortage, with the aim of reducing supply delays of spare parts. This paper will examine the relative effectiveness of two lateral shipments approaches in reducing the mean supply delay (MSD) of a non-repairable item, with respect to a classical policy of no lateral shipments. A simulation model of a two echelon supply network has been implemented and an experiment has been performed by varying different parameters of the supply network, such as the number of warehouses (locations at the lower echelon), the supply lead time from the central depot, the spare parts demand uncertainty, and the size variability of the warehouses. Results show appreciable reductions of MSD when lateral shipments are allowed with respect to the classical policy, in almost every network configuration.     

Inventory model for an inventory system with time-varying demand rate

The standard inventory problems of the multi-period have been modeled under different situations. Specifically we have considered the demand subjects of a continuous distribution and a discrete distribution, and whether the demand of each period is unchanged or not. A method to get an economic order quantity in inventory systems with discrete and unchanged demand was presented in a previous paper, and this method has been generalized to an inventory model with varying continuous demand. However, it was not achieved due to there being many classified cases in the general situations. In this article the above method is discussed in the case discrete demand to determine whether it increases or decreases from period to period. A theoretical method is presented by using previous results and some examples are given which suggest how the concept can handle on inventory system. In order to make the decision, an algorithm is also presented under some conditions, and examples are shown by using the computer software program, Mathematica, which helps to explain the findings. In general cases, we view the optimal policy in the inventory problems in only a few periods.     

Optimal inventory control of empty containers in inland transportation system

In this paper, we deal with an inventory control problem of empty containers in an inland transportation system. In inland container transportation, freights (containers) are transported between terminal and the customer’s location by trucks, trains and barges. Empty containers are an important logistic resource and shipping companies try to operate and manage empty containers efficiently. Because of the trade imbalance between hub ports, empty containers should be periodically repositioned from surplus areas to shortage areas. However, it is not easy to exactly forecast the demand of empty containers, and we therefore need to build an efficient way to reposition the empty containers. In this paper, we consider a shortage area and propose an efficient inventory policy to control empty containers. We assume that demands per unit time are independent and identically distributed random variables. To satisfy the demand of empty containers, we reposition empty containers from other hubs based on the (s, S) inventory policy, and also consider the lease of empty containers with zero lead time. For the leased containers, we should return the number of empty containers leased to the leaser after the specified period. For a given policy, simulation is used to estimate the expected cost rate and we use the optimization tool, OptQuest^® in Arena to obtain the near optimal (s, S) policy in numerical examples.     

Impact of frequency of alignment of physical and information system inventories on out of stocks: A simulation study

Inaccuracy in the information system inventory as compared to the physical inventory may lead to out of stocks. Inaccuracy may occur for many reasons, a principal one being random losses such as theft. One way to reduce this inaccuracy is to adjust the inventory information in the systems at some regular frequency. Such alignments are quite expensive in practice. Thus how often to align the two inventories is the focus of this research. A simulation model is employed to investigate the effect of such loss defined by the stock loss parameter (λ) and the frequent alignment of physical and information system inventories on the stockout (S_out) and average inventory (I). A term to be called the effective value of stock loss parameter is introduced to signify the effect of frequency of alignment (f) on S_out. The results derived in this study provide a powerful tool in the hands of an inventory manager. It has been noted that, so far as stockout is concerned, by selecting a moderate value of alignment frequency (f), the effective value of stock loss parameter (λ_e) can be reduced to∼ λ/f. The accuracy of S_out and I values across a number of runs in the simulation studies, sensitivity of S_out and I on various parameters and the nature of stochastic demand distribution, and application of these results with or without deployment of RFID to reduce the loss due to stockout are also discussed. The results, verified under various scenarios, indicate that there is a significant reduction in stockout loss when the alignment is done monthly vs. annually, but it does not add much value beyond a monthly check.     

A binary solution method for the multi-product newsboy problem with budget constraint

Multi-product newsboy problem (MPNP) with budget constraint is a classical inventory control/management problem. However, solution methods for MPNP under general demand distributions are limited in the current literature. In this paper, by analyzing properties of the optimal solution to the MPNP with a budget constraint, we develop a solution algorithm for the constrained MPNP. The proposed algorithm is binary in nature, and is applicable to general types of demand distribution functions, discrete as well as continuous. For continuous demand distribution function, our approach can obtain the optimal or near optimal solution to the constrained MPNP with polynomial computation complexity of the o(n) order. On the other hand, for discrete demand distribution functions, it can effectively provide good approximate solution. Numerical experiments are presented to show the performance of our method.     

Near-optimal (r,Q) policies for a two-stage serial inventory system with Poisson demand

We consider a two-stage serial inventory system whose cost structure exhibits economies of scale in both stages. In the system, stage 1 faces Poisson demand and replenishes its inventory from stage 2, and the latter stage in turn orders from an outside supplier with unlimited stock. Each shipment, either to stage 2 or to stage 1, incurs a fixed setup cost. We derive important properties for a given echelon-stock (r, Q) policy for an approximation of the problem where all states are continuous. Based on these properties, we design a simple heuristic algorithm that can be used to find a near-optimal (r, Q) policy for the original problem. Numerical examples are given to demonstrate the effectiveness of the algorithm.     

The uniform distribution as a first practical approach to new product inventory management

It is common sense that the premises usually considered in inventory models have little applicability to new product inventory management. This paper develops a first practical approach to deal with this issue: the solution to the (Q, r) inventory model for uniform demand forecasts and lead-times. Based on the fact that the uniform distribution is defined by two parameters that are easy to estimate—maximum and minimum—this paper shows that such a premise may comprise a helpful and accurate decision support tool for managers until they begin to learn about the distribution characteristics of the demand during the lead-time.     

Integrated inventory models with controllable lead time and backorder discount considerations

Many companies use time as a means of differentiating themselves in the marketplace. In many literatures, the controllable lead time is regarded as a decision variable and decomposed into several components, each having a crashing cost function for the respective reduced lead time. When an item is out of stock, the loyal, patient and captive customers will wait until the outstanding orders arrive and are served from them. To compensate for the inconvenience of backordering and to secure orders, the supplier may offer a price discount on the stockout item. In this paper, an integrated inventory system in which shortage is allowed and both lead time and backordering are negotiable is investigated. The lead time crashing cost is represented as a function of reduced lead time and the quantities in the orders. There are two inventory models proposed in the paper, one with normally distributed demand, and another with generally distributed demand.     

A stochastic inventory model of dual sourced supply chain with lead-time reduction

Recently, the amount of literature on analyzing the effects of investment strategies to control lead times has been increasing. Issues on investment to reduce lead times are important because variability in lead time between successive stages often has a great effect on the coordination of supply chain.This paper considers dual-sourcing models with stochastic lead times and constant unit demand in which lead times are reduced at a cost that can be viewed as an investment. In order to obtain an analytically tractable model, the distributions of lead times for two suppliers are assumed to be exponential. In our two-supplier model, we will concentrate on lead times as random variables, which are made endogenous in the stochastic lead-time model through “expediting factors”, the constants of proportionality between the expedited lead times and ordinary lead times, as was done by Bookbinder and Çakanyildirim (Eur. J. Oper. Res. 115 (1999) 300).Firstly, we determine the order quantity (Q), reorder point level (r), and order splitting proportion (k₁) in the case of no lead-time reduction. Using the (Q,r) found, we decide the expediting factors and new k₁ in the case of lead-time reduction. We compare the expected total cost per unit time for the two models and investigate savings. Additionally, sensitivity analyses are conducted with respect to the various cost parameter ranges, and remarks are made for further research.     

Exact and approximate calculation of the cycle service level in periodic review inventory policies

The parameters of stock policies are usually determined to minimize costs while satisfying a target service level. In a periodic review policy the time between reviews can be selected to minimize costs while the order-up-to-level is based on the fulfilment of a target service level. Generally, the calculation of this service measurement is obtained using approximations based on an additional hypothesis related to the demand pattern. Previous research has shown that there is a substantial difference between exact and approximate calculations in some general circumstances, so in these cases the service level is not accomplished or the stock level is overestimated. Although an exact calculation of CSL was developed in previous work, the computational effort required to apply it in practical environments leads to the proposal of two approximate methods (PI and PII) that, with the classic approximation, are analysed and evaluated in this paper. This analysis points out the risks of using the classic approximation and leads one to suggest PII as the most suitable and accurate enough procedure to compute the CSL straightforwardly in practice. Additionally, a heuristic approach based on PII is proposed to accept or reject an inventory policy in terms of fulfilling a given target CSL. This paper focuses on uncorrelated, discrete and stationary demand with a known distribution pattern and without backlog.     

Single item inventory control under periodic review and a minimum order quantity

In this paper we study a periodic review single item single stage inventory system with stochastic demand. In each time period the system must order none or at least as much as a minimum order quantity Q_min. Since the optimal structure of an ordering policy with a minimum order quantity is complicated, we propose an easy-to-use policy, which we call (R, S, Q_min) policy. Assuming linear holding and backorder costs we determine the optimal numerical value of the level S using a Markov Chain approach. In addition, we derive simple news-vendor-type inequalities for near-optimal policy parameters, which can easily be implemented within spreadsheet applications. In a numerical study we compare our policy with others and test the performance of the approximation for three different demand distributions: Poisson, negative binomial, and a discretized version of the gamma distribution. Given the simplicity of the policy and its cost performance as well as the excellent performance of the approximation we advocate the application of the (R, S, Q_min) policy in practice.     

An integrated product planning model for pricing and bundle selection using Markov decision processes and data envelope analysis

When bundling products during the product planning stage, there are a number of possible combinations that can be offered to customers. Consider a firm that offers N distinctive products, then there are 2^N−(N+1) possible bundling combinations. Now, if we wish to make pricing and bundle selection decisions, keeping in my mind limited inventory and a finite time horizon, then the size of the state space could be very large and finding an optimal solution could be impossible. To tackle this issue, we formulate an integrated model that utilizes a Markov decision process and data envelope analysis. Bundle selections are made using data envelope analysis in each decision epoch. Once the efficient bundles are selected they are priced by solving a Markov decision process using dynamic programming. Numerical examples are solved to exhibit the model's potential in solving real-world problems.     

Integer-ratio policies for distribution/inventory systems

We study a multistage distribution/inventory system with a central warehouse and N retailers. Customer demand arrives at each retailer at a constant rate. The retailers replenish their inventories from the warehouse, which in turn orders from an outside supplier. It is assumed that shortages are not allowed and lead times are negligible. The goal is to determine policies which minimize the overall cost in the system, that is, the sum of the costs at each facility consisting of a fixed charge per order and a holding unit cost. We propose a heuristic procedure to compute near-optimal policies. Computational results on several randomly generated problems are reported.     

A perishable inventory model with Markovian renewal demands

In the inventory model, people usually assume that the inter-demand time is independently identical distributed which may not be true in reality. Here we study an (s,S) continuous review model for items with an exponential random lifetime and a general Markovian renewal demand process. By constructing Markovian renewal equations, we derive the mean and the variance of the reorder cycle time and lead to a simple expression for the total expected long run cost rate. The numerical results illustrate the system behavior and lead to managerial insights into controlling such inventory systems.     

Inventory performance under pack size constraints and spatially-correlated demand

Motivated by a problem facing a large retailer, we consider the impact of pack size on the performance of a periodic review inventory system in the presence of spatial (i.e., between retailers) correlation of demand—which we model using an equicorrelated multivariate Poisson distribution. Employing simulation, we utilise a full factorial experiment to provide support for decisions on product and supplier selection, and whether or not packs should be split during distribution. We consider variables such as pack size, correlation, and the number of branches, and discuss how they and their interactions impact performance metrics such as inventory and shortage levels and the bullwhip effect.