Content uploaded by Herbert E. Scarf
Author content
All content in this area was uploaded by Herbert E. Scarf on Aug 18, 2014
Content may be subject to copyright.
Optimal Policies for a Multi-Echelon Inventory Problem
Abstract
In the last several years there have been a number of papers discussing optimal policies for the inventory problem. Almost without exception these papers are devoted to the determination of optimal purchasing quantities at a single installation faced with some pattern of demand. It has been customary to make the assumption that when the installation in question requests a shipment of stock, this shipment will be delivered in a fixed or perhaps random length of time, but at any rate with a time lag which is independent of the size of the order placed. There are, however, a number of situations met in practice in which this assumption is not a tenable one. An important example arises when there are several installations, say 1, 2, …, N, with installation 1 receiving stock from 2, with 2 receiving stock from 3, etc. In this example, if an order is placed by installation 1 for stock from installation 2, the length of time for delivery of this stock is determined not only by the natural lead time between these two sites, but also by the availability of stock at the second installation. In this paper we shall consider the problem of determining optimal purchasing quantities in a multi-installation model of this type.
... Multi-echelon inventory management is a core issue in the supply chain and has a research history of more than 60 years. The study of multi-echelon inventory systems originated from the pioneering work of Clark and Scarf (1960 ...
... Serial Systems: Clark and Scarf (1960) demonstrated that for a "pure" serial inventory system where the fixed order cost is only charged at the highest echelon, an echelon base stock policy is optimal. However, they also noted that in systems with fixed order costs at each echelon, finding an optimal policy, if it exists, can be challenging and intricate to implement. ...
... The Fair Share (FS) policy, introduced by Clark and Scarf (1960), is the most well-known policy in this category. Its objective is to ensure an equal probability of stock-out for all end stocks. ...
Effective inventory management is critical for e-commerce companies to reduce logistics costs and ensure high service levels to customers. One important issue of E-commerce inventory management is effectively distributing multiple items to customers in a multi-echelon distribution system. This thesis studies the optimization of inventory and rationing policies in multi-echelon distribution systems with joint replenishment of multiple items. Firstly, we analyze a periodic-review joint replenishment inventory system controlled by a P(s, S) policy with ordering costs, holding costs and service level constraints. We derive its cost function and service level analytically, and design an efficient algorithm to find the optimal parameters of the policy. Secondly, we investigate a continuous-review, two-echelon distribution system with one central distribution centre (CDC) and multiple regional distribution centres (RDCs). Each stocking location is controlled by a (Q, S) policy. We propose a decomposition and coordination approach for optimizing the inventory policies of the system, and derive a lower bound to evaluate the quality of the policies found. Finally, we study allocation/rationing of the on-hand inventory of the CDC to the RDCs in the system. We propose multiple rationing policies and compare their performance through extensive numerical experiments. Rationing policies considering the inventory positions of the RDCs outperform others.
... The dynamic inventory management problem can be essentially formulated as a Markov Decision Process (MDP). To address this problem, many methods have been developed, dating back to Clark and Scarf (1960) and Sherbrooke (1968). However, these methods are specifically designed for multi-echelon systems with simple structure, e.g., the serial system. ...
... The baseline method is the base-stock policy, which means that for an installation (warehouse or retailer) with a base-stock level s, if the inventory position is less than s, the installation places orders to increase the inventory position to s as close as possible. It should be noted that in a serial system, the base-stock policy is optimal (Clark and Scarf 1960), while in a complex multi-echelon system, the optimal policy is unknown. ...
... Moreover, it is not clear whether inventory productivity decreases or increases upstream. On one hand, information distortion about demand, lead times, and batch sizes all tend to increase upstream, suggesting lower inventory productivity (Clark andScarf 1960, Lee et al. 1997). On the other hand, upstream firms' production is less differentiated and its pace tends to be more steady, suggesting an offsetting positive impact on inventory productivity (Blinder and Maccini 1991). ...
... Therefore, product cost, and consequently, the cost of holding inventory increases downstream. The logic of echelon inventory management (Clark and Scarf 1960) applies here, incentivizing smaller batch sizes downstream, larger batch sizes upstream, and suggesting that inventory productivity, or at least cycle stock turnover, decreases upstream. The larger batches upstream come with a higher ordering cost, further reducing order frequency and inventory productivity. ...
Problem definition: We provide a novel, supply network-based perspective on inventory productivity and incentives for its improvement. Methodology/results: Using data from 2003 to 2019, we find that inventory productivity is lower materially and statistically for firms located upstream in the supply network, and higher for high degree and more central firms. Firms with high inventory productivity show high equity valuations and abnormal returns, with both valuations and abnormal returns amplified for upstream, low degree, and peripheral firms. Moreover, the difference in valuations and abnormal returns between best and worst performing firms is greater upstream, suggesting that financial markets offer outsized rewards for improving inventory productivity to upstream firms. Managerial implications: We show that the information about firm’s upstreamness and centrality in the supply network is a valuable predictor of its inventory productivity and financial performance. Our methods for evaluating upstreamness are useful for that purpose. For operations managers and firm executives, our results highlight strong incentives for inventory productivity improvement upstream in the supply network. For investors, we show that supply network position data can sharpen inventory-based arbitrage opportunities.
Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0229 .
... By considering its demand requirements, every warehouse determines the ideal time and quantity for restocking without taking dependencies and interactions with other warehouses or distribution centers into consideration. Second, Multi-Echelon Inventory Optimization (MEIO) [6] assumes a holistic perspective in which the ideal policies are determined jointly by explicitly taking into consideration interrelationships between warehouses. According to [6], holistic optimization prevents inventory systems from making egoistic decisions at the cost of neighboring inventory systems, thus preventing solutions that may be optimal from a local perspective but not globally. ...
... Second, Multi-Echelon Inventory Optimization (MEIO) [6] assumes a holistic perspective in which the ideal policies are determined jointly by explicitly taking into consideration interrelationships between warehouses. According to [6], holistic optimization prevents inventory systems from making egoistic decisions at the cost of neighboring inventory systems, thus preventing solutions that may be optimal from a local perspective but not globally. However, this comes at the cost of complexity. ...
The operation of inventory systems plays an important role in the success of manufacturing companies, making it a highly relevant domain for optimization. In particular, the domain lends itself to being approached via Deep Reinforcement Learning (DRL) models due to it requiring sequential reorder decisions based on uncertainty to minimize cost. In this paper, we evaluate state-of-the-art optimization approaches to determine whether Deep Reinforcement Learning can be applied to the multi-echelon inventory optimization (MEIO) framework in a practically feasible manner to generate fully dynamic reorder policies. We investigate how it performs in comparison to an optimized static reorder policy, how robust it is when it comes to structural changes in the environment, and whether the use of DRL is safe in terms of risk in real-world applications. Our results show promising performance for DRL with potential for improvement in terms of minimizing risky behavior.
... Multi-echelon systems are typically difficult to analyze because of the dependence between locations. Uncapacitated serial systems under full backlogging form an exception: For those systems base-stock policies are optimal, and the optimal base-stock policy an be identified using dynamic programming, working backwards through the network topology (Clark and Scarf, 1960). Under similar assumptions, pure assembly systems form a second exception; these can be reduced to an equivalent serial system (Rosling, 1989) and analyzed accordingly. ...
... The balance assumption (see Clark and Scarf, 1960;Eppen and Schrage, 1981) is the most common relaxation technique for divergent inventory systems; it works in the context of complete back-ordering. Assuming balanced inventories leads to a complete characterization of the optimal policy, that can even be extended to multi-echelon systems (Diks and De Kok, 1998). ...
... To ensure the use of land, the replenishment cycle of this model should generally not be too short. Therefore, this model is generally applicable to countries or regions with abundant land resources, such as Oregon in the United States [2] , which uses this model for land reserves. China is short of land resources, which is generally not applicable, but this model can be used for reference in areas with small population density such as Xizang and Xinjiang. ...
... The idea of coordinated planning is not new in the mathematical programming area. Since the study from [20], many researchers have investigated multi-echelon systems. Review of mathematical programming planning models in the supply chain context are given by [21]. ...
Due to recent surges in global sourcing and vertical disintegration supply chain optimization has emerged as a major challenge for the automotive industry. Most supply chains in this industry are composed of independent agents with specific preferences. These agents could be distinct firms or even profit centres within a large vehicle manufacturer. In either case, no single agent has control over the entire supply chain, and hence no agent has the power to optimize the whole chain by centralized planning. However, improved supply chain performance is achievable through coordination and collaboration between agents that are direct interacting. While coordination could be achieved with different mechanisms, this paper will focus on supply chain optimization through col-laborative planning with connected advanced planning and scheduling systems (APS). Firstly, the context of planning in the automotive industry will be explained. Followed by a review of different kinds of collaborative planning approaches. Based on a real-life business scenario, a collaborative planning process will be examined using a model of an automotive supply chain. The effects on performance in this particular model will be evaluated using a simulation tool, which has been augmented with an APS master planning component based on finite domain (FD) constraint technology. As benchmark scenarios we will use (1) a current process and (2) a planning process with improved information flow (such as demand and inventory visibility). The final chapter summarizes the key findings of this study and gives recommendations for further research.
... Therefore, these designs should not be seen as a prescription for all African countries; if anything, they can be improved. As Clarke and Scarf argued, 13 the optimal solution (defined in terms of order quantities, output levels, and total costs) for a logistics system with a certain number of tiers or echelons can be replicated by an alternative system with a lower number of tiers or echelons, what is generally referred to as the "system design approach." 14 As Fritz et al. reported, 15 even a simple system-design change, such as switching from manual to electronic logistics management information systems by reducing the frequency of stock-outs of lifesaving health commodities, lowered the mortality rate of newborns and children aged younger than 5 years in Ethiopia, Mozambique, and Tanzania (a reduction of 1.6% to 4.1%). ...
Key Messages
To resolve problems of ensuring secure supplies of all health commodities, health planners in African countries must first identify arrangements that best serve the public interests of promoting choice and competition in ensuring health commodity security.
Investments in inventory management should not be viewed as a one-off exercise but rather as a continuous search for the optimal scale and scope of operations that ensure the availability of essential health commodities most of the time.
Without competing alternatives to manage inventory, public-sector logistics monopolies lack adequate incentives to devise ways of reducing costs and improving output. Further, these monopolies make it more difficult to minimize the impact and duration of catastrophic supply disruptions.
Current efforts to improve public-sector supply logistics must be extended to include the transformation of existing public and private logistics infrastructure for inventory management into a state of prudent multiplicity—one in which there are at least 2 full-line national logistics institutions competing to serve all government, nongovernmental, and private health facilities.
Health planners should consider creating a state of prudent multiplicity in their roadmaps, master plans, and health system strengthening initiatives.
Supply structures are connected nodes of sub-parts of organisations involved in the procurement of resources, their operational transformation and distribution as goods and services. Supply nodes are connected by supply flows and processes. Supply chain managers control resources or assets including buildings, equipment, people involved in supply, materials and services transformed into goods and services, social capital accruing from supply, finance and information for supply and supply knowledge and expertise. All supply nodes exist in their own unique supply structures and have unique vantage points from which they observe and engage with their supply structures upstream and downstream. Supply structures are at multiple systems levels of the internal supply chain, the supply dyad, supply triad, supply base, supply chain, supply network, supply system and supply market. The larger the unit of analysis of supply structure, the less empirical research has been conducted, so our understanding of them is less.
A dynamic, single-item, multi-echelon inventory model. RM-2297, The RAND Corporation
- A J Clark
Clark, A. J. 1958. A dynamic, single-item, multi-echelon inventory model. RM-2297, The RAND Corporation, Santa Monica, Cal-ifornia (December).
A dynamic single-item, multi-echelon inventory model
- A J Clark