Echelon Base-stock Policy Research Articles

Technical Note—Capacitated Serial Inventory Systems: Sample Path and Stability Properties Under Base-Stock Policies

We study a periodically reviewed multiechelon serial inventory system with a capacity constraint on the order quantity at every stage. Under echelon base-stock policies, we demonstrate a simple sample-path result that maps the echelon shortfalls in the serial system to the shortfalls of suitably defined single-stage systems. Because the shortfall processes of single-stage systems are well understood, our result allows us to reinterpret results in the literature on the stability and regeneration times of such multiechelon systems in a simpler fashion with weaker assumptions.

Periodic flexibility, information sharing, and supply chain performance

A two-stage serial supply chain in which a retailer and his supplier are operating in make-to-stock environments and the retailer faces uncertain demands from the end-customers is studied. When this supply chain is centrally managed, the optimal policy is an extension of the Clark–Scarf echelon base stock policy. Since these supply chains are usually operated in a decentralized manner, an operational change is proposed that reduces the inefficiency associated with decentralization. The policy, which is called Periodic Flexibility (PF), provides the retailer with structural flexibility to order any amount in one period of a cycle, while requiring that the retailer receives a fixed-quantity shipment in the other periods. Optimal policies and their associated costs for the non-stationary inventory control problems faced by the retailer and the supplier under PF are characterized. A detailed computational study shows that PF improves the supply chain performance by about 11% on average. This improvement is a 43% (on average) reduction in the efficiency gap between centralized and decentralized control. The improvement of PF is due to information sharing, i.e., the retailer passing her end-customer demand information to the supplier. The PF strategy is compared to the well-known quantity flexibility scheme and it is shown that the PF approach tends to be more efficient. [Supplementary materials are available for this article. Go to the publishers online edition of IIE Transactions for the following free supplemental resources: Additional proofs]

A Decomposition Approach for a Class of Capacitated Serial Systems

We study a class of two-echelon serial systems with identical ordering/production capacities or limits for both echelons. Demands are assumed to be integer valued. For the case where the lead time to the upstream echelon is one period, the optimality of state-dependent modified echelon base-stock policies is proved using a decomposition approach. For the case where the upstream lead time is two periods, we introduce a new class of policies called “two-tier base-stock policies,” and prove their optimality. Some insight about the inventory control problem in N echelon serial systems with identical capacities at all stages and arbitrary lead times everywhere is also provided. We argue that a generalization of two-tier base-stock policies, which we call “multitier base-stock policies,” are optimal for these systems; we also provide a bound on the number of parameters required to specify the optimal policy.

A Single-Unit Decomposition Approach to Multiechelon Inventory Systems

We show the optimality of state-dependent echelon base-stock policies in uncapacitated serial inventory systems with Markov-modulated demand and Markov-modulated stochastic lead times in the absence of order crossing. Our results cover finite-time horizon problems as well as infinite-time horizon formulations, with either a discounted or an average cost criterion. We employ a novel approach, based on a decomposition of the problem into a series of single-unit single-customer problems that are essentially decoupled. Besides providing a simple proof technique, this approach also gives rise to efficient algorithms for the calculation of the base-stock levels.

Echelon base-stock policies are financially sub-optimal

We study a Clark and Scarf multi-echelon inventory model with the objective of optimizing the expected present value of dividends. A counterexample shows that generally there is no optimal echelon base-stock policy if there are financial constraints and two or more echelons.

A Closed-Form Approximation for Serial Inventory Systems and Its Application to System Design

We analyze a serial base-stock inventory model with Poisson demand and a fill-rate constraint. Our objective is to gain insights into the linkage between the stages to facilitate optimal system design and decentralized system control. To this end, we develop a closed-form approximation for the optimal base-stock levels. The development consists of two key steps: (1) convert the service-constrained model into a backorder cost model by imputing an appropriate backorder cost rate, and then adapt the single-stage approximation developed for the latter, and (2) use a logistic distribution to approximate the lead-time demand distribution in the single-stage approximation obtained in (1) to yield closed-form expressions. We then use the closed-form expressions to conduct sensitivity analyses and establish qualitative properties on system design issues, such as optimal total system stock, stock positioning, and internal fill rates. The closed-form approximation and most of the qualitative properties apply equally to the model with a backorder cost, although some differences do exist. Other results of this study include a bottom-up recursive procedure to evaluate any given echelon base-stock policy and lower bounds on the optimal echelon base-stock levels.

Optimal Policy for a Multiechelon Inventory System with Remanufacturing

We analyze a multiechelon inventory system with inventory stages arranged in series. In addition to traditional forward material flows, used products are returned to a recovery facility, where they can be stored, disposed, or remanufactured and shipped to one of the stages to re-enter the forward flow of material. This system combines the key elements of two simpler systems: the series system studied by Clark and Scarf (1960) and the single-stage remanufacturing systems studied by Simpson (1978) and Inderfurth (1997). We focus on identifying the structure of the optimal remanufacturing/ordering/disposal policy for such a system. In particular, we investigate whether the optimal policy inherits the basic structural properties of the simpler systems. We show that if remanufactured items flow into the most upstream stage, then this is the case. Specifically, the system can be solved by decomposition into a sequence of single-stage systems, with each downstream stage following an echelon base-stock policy and the most upstream stage following a three-parameter policy with a simple (and intuitive) structure. We show that similar results hold when remanufactured products flow into a downstream stage; however, in this case some modifications must be made. In particular, the definition of echelon inventory must be adjusted for stages upstream of the remanufacturing stage, and disposal of used items can no longer be allowed. We also compare the information required for managing this system to that required in the Clark and Scarf or Inderfurth settings, and we point out how the requirements are somewhat different depending on whether remanufacturing occurs upstream or downstream.

Inventory Policies in a Decentralized Assembly System

We consider a system in which a single finished good is assembled from two components. Demand for the finished product is stochastic and stationary, and procurement and assembly lead times are constant. Unsatisfied demand is backordered. The inventory of each component or assembly is controlled by a separate firm using a base-stock policy. Each firm is charged holding costs on its own inventory, plus a share of the shortage cost due to backorders of the finished product. We investigate the equilibrium base-stock levels that arise in this system under both echelon and local base-stock policies. In both cases, the component firms’ base-stock levels are economic complements. We then examine the effect on system performance when one firm uses information about other firms’ pipeline inventory. We find that, under echelon base-stock policies, all firms benefit with the use of pipeline information. In contrast, under local policies, using pipeline information may actually increase costs for some firms (including the firm that makes direct use of the information). Also, we compare the behavior of the decentralized system with that of the assembly system under centralized control. Finally, we describe a payment scheme between the final assembler and the suppliers that allows the decentralized system to achieve the centralized solution.

A Series System with Returns: Stationary Analysis

This paper analyzes a series inventory system with stationary costs and stochastic demand over an infinite horizon. A distinctive feature is that demand can be negative, representing returns from customers, as well as zero or positive. We observe that, as in a system with nonnegative demand, a stationary echelon base-stock policy is optimal here. However, the steady-state behavior of the system under such a policy is different from that in systems with nonnegative demands. We present an exact procedure and several approximations for computing the operating characteristics and system costs for any stationary echelon base-stock policy, and also describe an algorithm for computing a good policy. As an alternative to the echelon base-stock policy, we discuss a policy that uses only local information. Finally, we describe how to extend the analysis to the case where returns occur at multiple stages instead of just at the stage closest to demand, and the case where returns require a recovery lead time.

Optimal Policies for a Capacitated Two-Echelon Inventory System

This paper demonstrates optimal policies for capacitated serial multiechelon production/inventory systems. Extending the Clark and Scarf (1960) model to include installations with production capacity limits, we demonstrate that a modified echelon base-stock policy is optimal in a two-stage system when there is a smaller capacity at the downstream facility. This is shown by decomposing the dynamic programming value function into value functions dependent upon individual echelon stock variables. We show that the optimal structure holds for both stationary and nonstationary stochastic customer demand. Finite-horizon and infinite-horizon results are included under discounted-cost and average-cost criteria.

Newsvendor Bounds and Heuristic for Optimal Policies in Serial Supply Chains

We consider the classic N-stage serial supply systems with linear costs and stationary random demands. There are deterministic transportation leadtimes between stages, and unsatisfied demands are backlogged. The optimal inventory policy for this system is known to be an echelon base-stock policy, which can be computed through minimizing N nested convex functions recursively. To identify the key determinants of the optimal policy, we develop a simple and surprisingly good heuristic. This method minimizes 2Nseparate newsvendor-type cost functions, each of which uses the original problem data only. These functions are lower and upper bounds for the echelon cost functions; their minimizers form bounds for the optimal echelon base-stock levels. The heuristic is the simple average of the solution bounds. In extensive numerical experiments, the average relative error of the heuristic is 0.24%, with the maximum error less than 1.5%. The bounds and the heuristic, which can be easily obtained by simple spreadsheet calculations, enhance the accessibility and implementability of the multiechelon inventory theory. More importantly, the closed-form expressions provide an analytical tool for us to gain insights into issues such as system bottlenecks, effects of system parameters, and coordination mechanisms in decentralized systems.

Optimal Replenishment Policies for Multiechelon Inventory Problems Under Advance Demand Information

Customers and downstream supply chain partners often place, or can be induced to place, orders in advance of future requirements. We show how to optimally incorporate advance demand information into periodic-review, multiechelon, inventory systems in series. While the state space for series systems appears to be formidably large, we decompose the problem across locations, as in Clark and Scarf (1960), and reduce the state space at each location by using modified echelon inventory positions (that nets known requirements). We prove the optimality of state-dependent, echelon base-stock policies for finite and infinite horizon problems. We also show that myopic policies are optimal and very easy to compute when costs and demands are stationary. We provide managerial insights into the value of advance demand information through a numerical study.

Optimal Policies for Multiechelon Inventory Problems with Markov-Modulated Demand

This paper considers a multistage serial inventory system with Markov-modulated demand. Random demand arises at Stage 1, Stage 1 orders from Stage 2, etc., and Stage N orders from an outside supplier with unlimited stock. The demand distribution in each period is determined by the current state of an exogenous Markov chain. Excess demand is backlogged. Linear holding costs are incurred at every stage, and linear backorder costs are incurred at Stage 1. The ordering costs are also linear. The objective is to minimize the long-run average costs in the system. The paper shows that the optimal policy is an echelon base-stock policy with state-dependent order-up-to levels. An efficient algorithm is also provided for determining the optimal base-stock levels. The results can be extended to serial systems in which there is a fixed ordering cost at stage N and to assembly systems with linear ordering costs.