Single-stage Inventory System Research Articles

Inventory management with advance demand information and flexible shipment consolidation

In this paper, we study a stochastic single-item, single-stage inventory system, in which orders from several production facilities are placed at one warehouse. An (R, Q) policy is applied to control the inventory at the warehouse, and orders arrive according to a Poisson process and include a due date such that some information about future demand is available. This advance demand information (ADI) can be used to adapt a time-based shipment consolidation policy applied to replenish stock at the production facilities. We develop a model to incorporate flexible deliveries, indicating that orders can be shipped before their due date if sufficient reserved transportation capacity is available. We derive analytical, approximate expressions for the expected inventory and shipment costs and therefore enable the evaluation of different inventory and shipment policies including outbound transportation capacities. We additionally show how to compute the optimal policy parameters and conduct a detailed numerical study. Our computational experiments indicate that our approximation works extremely well, with an average total cost deviation of 0.20%, and finds optimal policy parameters in more than 90% of our instances. In line with existing research, we can show that ADI leads to large cost reductions. However, the main cause of the cost reduction is the flexible delivery option. To be able to completely utilize this option, even larger safety stocks are obtained compared to systems without ADI, but savings due to a more efficient transportation policy far exceed the cost increase due to higher safety stocks.

Managing Supply Chains with Expediting and Multiple Demand Classes

We consider a periodic review, single‐stage inventory system with multiple demand classes and a fixed replenishment lead time. Inventory expediting is allowed to alleviate demand–supply mismatches. Priority demands are commonly used in practice to provide differentiated services for customers, and inventory expediting is an effective strategy to improve the service levels of high‐priority demands. However, it is challenging to coordinate the inventory ordering, expediting and allocation decisions in supply chains. We partially characterize the structure of the optimal policy. We also derive various monotone properties of the optimal policy with limited sensitivities. Moreover, we show that a state‐dependent rationing level policy is optimal for inventory allocation and the optimal rationing levels are in fact independent of the backorder quantity of each demand class. We also show when some simple policies are indeed optimal. Numerically, we illustrate the optimal policy and investigate the performances of three proposed simple heuristics. Finally, we extend the results to more general systems, such as serial systems, systems with convex backordering costs, etc.

Technical Note—A Note on the Structure of Joint Inventory-Pricing Control with Leadtimes

We consider a joint inventory-pricing control problem for a periodic-review, single-stage inventory system with a positive order leadtime and a linear order cost. Demands in consecutive periods are independent, but their distributions depend on the price in accordance with a stochastic demand function of additive form. Pricing and ordering decisions are made simultaneously at the beginning of each period. The objective is to maximize the total expected discounted profit over a finite horizon. We partially characterize the structure of the optimal joint ordering and pricing policies. We also show that our structural analysis can be extended to a multistage (or serial) inventory system with constant or stochastic leadtimes and an assemble-to-order system with price-sensitive demand.

Static-dynamic uncertainty strategy for a single-item stochastic inventory control problem

We consider a single-stage inventory system facing non-stationary stochastic demand of the customers in a finite planning horizon. Motivated by the practice, the replenishment times need to be determined and frozen once and for all at the beginning of the horizon while decisions on the exact replenishment quantities can be deferred until the replenishment time. This operating scheme is refereed to as a “static-dynamic uncertainty” strategy in the literature [3]. We consider dynamic fixed-ordering and linear end-of-period holding costs, as well as dynamic penalty costs, or service levels. We prove that the optimal ordering policy is a base stock policy for both penalty cost and service level constrained models. Since an exponential exhaustive search based on dynamic programming yields the optimal ordering periods and the associated base stock levels, it is not possible to compute the optimal policy parameters for longer planning horizons. Thus, we develop two heuristics. Numerical experiments show that both heuristics perform well in terms of solution quality and scale-up efficiently; hence, any practically relevant large instance can be solved in reasonable time. Finally, we discuss how our results and heuristics can be extended to handle capacity limitations and minimum order quantity considerations.

TECHNICAL NOTE—Inventory Systems with a Generalized Cost Model

We study a single-stage inventory system with a generalized shortage penalty cost that includes the following three components: (i) a cost that is an increasing function of the number of backordered units in a period, (ii) a fixed cost incurred for each period in which there is a backorder irrespective of how many units are backordered, and finally (iii) a cost that is an increasing function of the number of periods a customer is backordered. We show the problem can be transformed into one in which the backorder cost depends on the inventory position only. Then we present two sets of conditions; the first one restricts our attention to a special case of the generalized penalty cost model while the second one restricts our attention to stationary demand models with some distributional assumptions. Under the first (resp. second) set of conditions, we show that the expected cost in a period can be expressed as a convex (resp. quasiconvex) function of the after-ordering inventory position. We use this property to prove the optimality of order-up-to policies under both sets of conditions and discuss extensions to the cases where either a fixed ordering cost or a batch ordering constraint is present.

Inventory management with advance supply information

We consider a single stage inventory system with stochastic capacity. The manager receives forecasts from the upstream source about future capacity availability within an information horizon, referred to as “advance supply information”. We study two main questions: (i) How can advance supply information be utilized when making replenishment decisions? (ii) What is the value of such information sharing, compared to a fixed base-stock policy? We show that state-dependent base-stock policies are optimal. We develop easily computable and implementable heuristic policies. We numerically test the accuracy of the approximations and analyze the value of collaboration under various business scenarios.

Now or Later: A Simple Policy for Effective Dual Sourcing in Capacitated Systems

We examine a possibly capacitated, periodically reviewed, single-stage inventory system where replenishment can be obtained either through a regular fixed lead time channel, or, for a premium, via a channel with a smaller fixed lead time. We consider the case when the unsatisfied demands are backordered over an infinite horizon, introducing the easily implementable, yet informationally rich dual-index policy. We show very general separability results for the optimal parameter values, providing a simulation-based optimization procedure that exploits these separability properties to calculate the optimal inventory parameters within seconds. We explore the performance of the dual-index policy under stationary demands as well as capacitated production environments, demonstrating when the dual-sourcing option is most valuable. We find that the optimal dual-index policy mimics the behavior of the complex, globally optimal state-dependent policy found via dynamic programming: the dual-index policy is nearly optimal (within 1% or 2%) for the majority of cases, and significantly outperforms single sourcing (up to 50% better). Our results on optimal dual-index parameters are generic, extending to a variety of complex and realistic scenarios such as nonstationary demand, random yields, demand spikes, and supply disruptions.

(s, S) Optimality in Joint Inventory-Pricing Control: An Alternate Approach

We study a stationary, single-stage inventory system, under periodic review, with fixed ordering costs and multiple sales levers (such as pricing, advertising, etc.). We show the optimality of (s, S)-type policies in these settings under both the backordering and lost-sales assumptions. Our analysis is constructive and is based on a condition that we identify as being key to proving the (s, S) structure. This condition is entirely based on the single-period profit function and the demand model. Our optimality results complement the existing results in this area.

Fill rate of single-stage general periodic review inventory systems

This paper develops an exact formula for the fill rate of a single-stage inventory system that uses a general periodic-review base-stock policy. For normal demand, we present a fill-rate expression that uses the standard normal PDF and CDF, and develop two approximations for the fill rate.

Comparison of ( s, S) and ( s, nQ) inventory control rules with respect to planning stability

The application of a rolling horizon planning framework in inventory control causes different order release decisions in successive planning cycles. These lead to the so-called planning system nervousness. In this paper for a single-stage inventory system with arbitrary stochastic demand it is shown analytically how the setup-oriented planning stability concerning deviations of planned orders in all periods of a given stability horizon is influenced by the use of ( s, S) and ( s, nQ) control rules. It turns out that the reorder point s does not affect stability whereas the lot size determining parameters S− s and Q as well as the level of uncertainty have a considerable impact. The paper is concluded with directions for further research.

Planning stability under (s,S) inventory control rules

The problem of planning instability caused by new and actualized data in material coordination systems with periodic review and replanning activities has been discussed from several viewpoints in literature and it is of great practical importance. The application of a rolling horizon planning framework in forecast-dependent inventory control leads to nervousness in the planning system caused by different order release decisions in successive planning cycles. In this paper a setup-oriented planning stability concerning deviations of planned orders in all periods of a given stability horizon in a single-stage inventory system with arbitrary stochastic demand is analyzed. The influence of an (s, S) control rule on planning stability is derived analytically. It is shown that the reorder points does not affect stability whereas the minimum reorder quantityS-s has a considerable impact. Especially, it is analyzed how planning stability depends on the length of the stability horizon, on period-specific weights, and on the accuracy of demand forecasts.

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.

Nervousness in inventory control: Analytical results

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 stochastic demand it is shown analytically, how planning stability is affected by policy parameters if (s, n Q) and (s, S) control rules are applied. It turns out that the reorder points does not influence stability whereas the lot sizeQ and minimum reorder quantityS-s can have a considerable impact. Under the aspect of nervousness the (s, n Q) policy can be assessed to be less risky than the (s, S) rule.