On properties of discrete (r, q) and (s, T) inventory systems

We consider single-item (r, q) and (s, T) inventory systems with integer-valued demand processes. While most of the inventory literature studies continuous approximations of these models and establishes joint convexity properties of the policy parameters in the continuous space, we show that these properties no longer hold in the discrete space, in the sense of linear interpolation extension and L^?-convexity. This nonconvexity can lead to failure of optimization techniques based on local optimality to obtain the optimal inventory policies. It can also make certain comparative properties established previously using continuous variables invalid. We revise these properties in the discrete space.     

Supply chain inventory optimization with two customer classes in discrete time

In this paper we consider a single-item inventory system where two demand classes with different service requirements are satisfied from a common inventory. A critical level, reorder point, order quantity or (s, q, k) policy is in use. The time axis is divided into discrete time units, which is a common characteristic of many real-life supply-chain processes. The inventory process within the lead time of a replenishment order is modelled as a sequence of (1) an ordinary renewal process and (2) two alternating renewal processes. Approximations are developed for the demand class-specific fill rates and the probability distribution of the waiting time of low priority customer orders. This waiting time distribution is used for the inventory allocation in a two-stage supply chain.     

Periodic review inventory systems with a service level criterion

Full-cost inventory models are mostly studied in the literature, whereas service level constraints are more common to be observed in practical settings. In this paper, we consider periodic review inventory systems with service level restrictions. The control of such inventory systems is limited to (s, S)-type policies in the literature. To the best of our knowledge, we are the first authors to compare such policies with optimal replenishment policies, and illustrate an average cost difference of 0.64%. This justifies the use of these popular (s, S) policies in practice. Furthermore, we propose a new one-dimensional search procedure that is bounded to set the reorder level s and order-up-to level S, whereas the solution space is unbounded and two dimensional. Our heuristic procedure is guaranteed to satisfy the service level constraint and numerical experiments illustrate that it results in an average cost deviation of 1–2% compared with the best (s, S) policy. Consequently, it significantly outperforms all existing procedures from literature, both in service and costs.     

An (s,S) random lifetime inventory model with a positive lead time

Positive lead times substantially complicate the modeling and analysis of inventory systems with finite shelf lifetimes and they have not been sufficiently addressed in the existing literature. In this paper, we analyze an (s, S) continuous review model with a positive lead time. We assume an exponential lifetime and an exponential lead time. Matrix-geometric solutions can be obtained for the steady state probability distribution of the inventory level. We then derive the total expected cost function. We carry out numerical studies and gain insights to the selection of system parameters. The findings on the impact of a positive lead time on the optimal reorder point and reorder batch size will be useful in developing strategies in handling inventory problems with finite lifetimes and positive lead times.     

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.     

Methods for Determining the Re-order Point of an (<Emphasis Type="Italic">s</Emphasis>, <Emphasis Type="Italic">S</Emphasis>) Ordering Policy when a Service Level is Specified

This paper deals with a periodic review inventory system. Methods are discussed for determining the re-order point s of an (s, S) order policy, when a certain service level is required. The results differ from those presented for a (Q, s) model which is usually considered in literature and implemented in practice. Methods are discussed for determining the re-order point of an (s, S) policy when demand is normal or gamma distributed. A numerical investigation demonstrates the applicability of the described methods. In particular, it is shown that these methods are superior to a formula that is implemented in many inventory control systems.     

Nervousness in inventory management: Comparison of basic control rules

Using a rolling horizon planning framework in inventory control leads to nervousness in the planning system caused by instability of order release decisions in successive planning cycles. For a single-stage inventory system with arbitrary stochastic demand it is shown analytically, how planning stability is affected by policy parameters if (s, nQ), (s, S), and (T, S) control rules are applied. It turns out that the reorder point s does not influence stability whereas the lot size determining parameters Q, S − s, and T can have a considerable impact. However, this influence turns out to be quite different for different measures of stability regarding order setup or order quantity deviations, respectively. Using these planning stability results the priority of the different control rules under the aspect of nervousness is discussed.     

On the (R,s, Q) inventory model when demand is modelled as a compound Bernoulli process

In this paper we present an approximation method to compute the reorder point s in an (R, s, Q) inventory model with a service level restriction. Demand is modelled as a compound Bernoulli process, i.e., with a fixed probability there is positive demand during a time unit; otherwise demand is zero. The demand size and the replenishment leadtime are random variables. It is shown that this kind of modelling is especially suitable for intermittent demand. In this paper we will adapt a method presented by Dunsmuir and Snyder such that the undershoot is not neglected. The reason for this is that for compound demand processes the undershoot has a considerable impact on the performance levels, especially when the probability that demand is zero during the leadtime is high, which is the case when demand is lumpy. Furthermore, the adapted method is used to derive an expression for the expected average physical stock. The quality of both the reorder point and the expected average physical stock, calculated with the method presented in this paper, rum out to be excellent, as has been verified by simulation.     

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.     

A stocking policy for spare part provisioning under age based preventive replacement

A new policy, called stocking policy for ease of reference, has been advanced for joint optimization of age replacement and spare provisioning. It combines age replacement policy with continuous review (s, S) type inventory policy, where s is the stock reorder level and S is the maximum stock level. The policy is generally applicable to any operating situation having either a single item or a number of identical items. A simulation model has been developed to determine the optimal values of the decision variables by minimizing the total cost of replacement and inventory. The behaviour of the stocking policy has been studied for a number of case problems specifically constructed by 5-factor second order rotatory design and the effects of different cost elements and item failure characteristics have been highlighted. For all case problems, optimal (s, S) policies to-support the Barlow-Proschan age policy have also been determined. Simulation results clearly indicate that the optimal stocking policy is, in general, more cost-effective than the Barlow-Proschan policy.     

Inventory models with minimal service level constraints

This paper investigates inventory models in which the stockout cost is replaced by a minimal service level constraint (SLC) that requires a certain level of service to be met in every period. The minimal service level approach has the virtue of simplifying the computation of an optimal ordering policy, because the optimal reorder level is solely determined by the minimal SLC and demand distributions. It is found that above a certain “critical” service level, the optimal (s,S) policy “collapses” to a simple base-stock or order-up-to level policy, which is independent on the cost parameters. This shows the minimal SLC models to be qualitatively different from their shortage cost counterparts. We also demonstrate that the “imputed shortage cost” transforming a minimal SLC model to a shortage cost model does not generally exist. The minimal SLC approach is extended to models with negligible set-up costs. The optimality of myopic base-stock policies is established under mild conditions.     

A two-level supply chain with partial backordering and approximated Poisson demand

We consider a two-level supply chain with a number of identical, independent ‘retailers’ at the lower echelon and a single supplier at the upper echelon controlled by continuous review inventory policy (R, Q). Each retailer experiences Poisson demand with constant transportation times. We assume constant lead time for replenishing supplier orders from an external warehouse to the supplier and unsatisfied retailer orders are backordered in the supplier. We assume that the unsatisfied demand is partially backordered in the identical retailers. The partially backordering policy is implemented in the identical retailers using an explicit control parameter ‘b’ which limits the maximum number of backorders allowed to be accumulated during the lead time. We develop an approximate cost function to find optimal reorder points for given batch sizes in all installations, the optimal value of b in the identical retailers and the related accuracy is assessed through simulation.     

The impact of data collection on fill rate performance in the (R,s, Q) inventory model

In this paper we consider the determination of the reorder point s in an (R, s, Q) inventory model subject to a fill rate service level constraint. We assume that the underlying demand process is a compound renewal process. We then derive an approximation method to compute the reorder level such that a target service level is achieved. Restrictions on the input parameters are given, within which this method is applicable. Moreover, we will investigate the effects on the fill rate performance in case the underlying demand process is indeed a compound renewal process, while the demand process is modelled as a discrete-time demand process. That is, the time axis is divided in time units (for example, days) and demands per time unit are independent and identically distributed random variables. It will be shown that smooth and erratic behaviour of the inter-arrival times have different impacts on the performance of the fill rate when demand is modelled as a discrete-time process and in case the underlying demand process is a compound renewal process.     

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.     

An inventory control system for products with optional components under service level and budget constraints

Modularization and customization have made enterprises face the multi-item inventory problems and the interactions among those items. A powerful, affordable information technology system can make the continuous review inventory policy more convenient, efficient, and effective. In this study, a (Q, r) model is developed to find the optimal lot size and reorder point for a multi-item inventory with interactions between necessary and optional components. In order to accurately approximate costs, the service cost is introduced and defined in proportion to the service level. In addition, the service cost and purchasing cost are taken simultaneously, and are treated as a budget constraint for executives to consider because the firm’s strategy could influence the choice of service level. The proposed model is formulated as a nonlinear optimization problem, as the service level is nonlinear. Thus, some known procedures are revised to solve this problem and the results are compared with other models. The results show that the revised procedure performs better than the N–R procedure, leading to important insights about inventory control policy.     

A heuristic for triggering emergency orders in an inventory system

This paper considers a single-echelon inventory system with a warehouse facing compound Poisson customer demand. Normally the warehouse replenishes from an outside supplier according to a continuous review reorder point policy. However, it is also possible to use emergency orders. Such orders incur additional costs but have a much shorter lead time. We consider standard holding and backorder costs as well as ordering costs. A heuristic decision rule for triggering emergency orders is suggested. The decision rule minimizes the expected costs under the assumption that there is only a single possibility for an emergency replenishment, but the rule is used repeatedly as a heuristic. Given a certain reorder point policy for normal replenishments, our decision rule will always reduce the expected costs. A simulation study illustrates that the suggested technique performs well under different conditions.     

Joint pricing and inventory control with a Markovian demand model

We consider the joint pricing and inventory control problem for a single product over a finite horizon and with periodic review. The demand distribution in each period is determined by an exogenous Markov chain. Pricing and ordering decisions are made at the beginning of each period and all shortages are backlogged. The surplus costs as well as fixed and variable costs are state dependent. We show the existence of an optimal (s, S, p)-type feedback policy for the additive demand model. We extend the model to the case of emergency orders. We compute the optimal policy for a class of Markovian demand and illustrate the benefits of dynamic pricing over fixed pricing through numerical examples. The results indicate that it is more beneficial to implement dynamic pricing in a Markovian demand environment with a high fixed ordering cost or with high demand variability.     

Approximate steady-state distribution for a large repairable item inventory system

This paper examines a model for a repairable inventory system where some items are condemned upon failure. These condemnations require an ordering policy which we take to be an (s, S) policy. We model this system as a version of the classical machine repair problem using continuous time Markov processes. We develop solution procedures based on approximations to the steady state probability distribution, which enable us to obtain the operating characteristics of (s, S) policies very easily, with or without an ordering leadtime. The approximate models are found to be very accurate and computationally very efficient.     

Production-inventory systems in stochastic environment and stochastic lead times

We consider a production-inventory system where the production and demand rates are modulated by a finite state Continuous Time Markov Chain (CTMC). When the inventory position (inventory on hand – backorders+inventory on order) falls to a reorder point r, we place an order of size q from an external supplier. We consider the case of stochastic leadtimes, where the leadtimes are i.i.d. exponential(μ) random variables, and orders may or may not be allowed to cross. We derive the distribution of the inventory level, and analyze the long run holding, backlogging, and ordering cost rate per unit time. We use simulation to study the sensitivity of the system to the distribution of the lead times.     

Lost-sales inventory systems with a service level criterion

Competitive retail environments are characterized by service levels and lost sales in case of excess demand. We contribute to research on lost-sales models with a service level criterion in multiple ways. First, we study the optimal replenishment policy for this type of inventory system as well as base-stock policies and (R, s, S) policies. Furthermore, we derive lower and upper bounds on the order-up-to level, and we propose efficient approximation procedures to determine the order-up-to level. The procedures find values of the inventory control variables that are close to the best (R, s, S) policy and comply to the service level restriction for most of the instances, with an average cost increase of 2.3% and 1.2% for the case without and with fixed order costs, respectively.