An empirical examination of the assumptions of the Square Root Law for inventory centralisation and decentralisation

The Square Root Law (SRL) is a popular formula for assessing inventory levels at varying numbers of warehouses. Its popularity is probably due to its simplicity and the ample opportunities for its application to the managerial dilemma of inventory centralisation vs. decentralisation. However, researchers disagree about which parts of inventory it can be applied to and its underlying assumptions. To address these questions, this study algebraically derives the assumptions necessary for the SRL to apply to regular, safety and total stock. Afterwards, the paper empirically examines to what extent these assumptions hold in practice by analysing four case studies and data from a sample of 280 German manufacturing and trading companies. Most companies do not fulfil the assumptions of the SRL and therefore cannot apply it with accurate results. Trading companies, however, seem to fulfil more assumptions than manufacturing ones, retailers more than wholesalers, industrial goods manufacturers more than consumer goods manufacturers. Although the SRL has traditionally been mainly considered for safety stock, companies, especially trading companies, appear to rather fulfil the assumptions for applying the SRL to cycle stock. Some assumptions seem to be complementary, whereas others appear to be competing, so that it is difficult to fulfil all of them.     

A dynamic multi-plant lot-sizing and distribution problem

In this paper, we investigate a multi-plant, production planning and distribution problem for the simultaneous optimisation of production, inventory control, demand allocation and distribution decisions. The objective of this rich problem is to satisfy the dynamic demand of customers while minimising the total cost of production, inventory and distribution. By solving the problem, we determine when the production needs to occur, how much has to be produced in each of the plants, how much has to be stored in each of the warehouses and how much needs to be delivered to each customer in each period. On a large real data-set inspired by a case obtained from an industrial partner, we show that the proposed integration is highly effective. Moreover, we study several trade-offs in a detailed sensitivity analysis. Our analyses indicate that the proposed scenarios give the company competitive advantage in terms of reduced total logistics cost, and also highlight more possibilities that become available taking advantage of an integrated approach towards logistics planning. These abundant opportunities are to be synergised and exploited in an interconnected open global logistics system.     

Safety Stock Analysis in a Multi-Level Inventory System

In this paper a model is presented for determination of the expected annual total cost of inventory for a multi-level inventory system. The model permits identification of the safety stock levels needed to minimize the expected annual total cost for a multi-level inventory system, such as distributed warehouses. An example of the application of the model is given.     

Effective bullwhip metrics for multi-echelon distribution systems under order batching policies with cyclic demand

A large number of problems in a distribution supply chain require that decisions are made in the presence of the bullwhip effect phenomenon. The impact of the order batching policies on the bullwhip effect is analysed in this paper, when cycle demand on a multi-echelon supply chain operating is considered. While investigating which bullwhip effect metrics are more adequate to measure the bullwhip effect in these type of systems, the optimal reordering plan that minimises the operation costs of the overall system is calculated. A Mixed Integer Linear Programming (MILP) model is developed that takes into account an inventory and distribution system formed by multiple warehouses and retailers with lateral transshipments. The bullwhip effect is measured through four metrics: the echelon average inventory; the echelon inventory variance ratio; the echelon average order; and the echelon order rate variance ratio. As conclusion the inventory metrics suggest that (i) using batching policy reduces instability; (ii) batching may reduce in general order variance if using larger batches and (iii) cycle demand length has no major impact in the bullwhip effect. A motivational example and a real word case study are used and tested.     

Distribution planning for multi-echelon networks considering multiple sourcing and lateral transshipments

Nowadays, due to the increasing complexity of business environment, especially demand uncertainty, supply chain managers need to establish more-effective sourcing and distribution strategies to ensure high customer service and low stock costs. To overcome this challenge multi-echelon network structures and alternative distribution strategies such as lateral transshipments and multiple sourcing should be considered in inventory optimisation models. In this article, we propose a scenario-based modelling approach to solve a two-stage multi-echelon inventory optimisation problem with a non-stationary demand. The model is based on a distribution requirements planning (DRP) approach and minimises the expected total cost that is composed of the fixed allocation, inventory holding, procurement, transportation, and back-ordering costs. Alternative inventory optimisation models, including the lateral transshipment strategy and multiple sourcing, are thus built, and the corresponding stochastic programmes are solved using the sample average approximation method. Through a numerical investigation conducted with several generated instances and an empirical investigation based on the case of a major French retailer’s distribution network, we show the substantial benefit of lateral transshipments and multiple sourcing in reducing the expected total costs of the distribution network.     

Optimal time policy for deteriorating items of two-warehouse inventory system with time and stock dependent demand and partial backlogging

We consider the problem of a two-warehouse inventory system under the effect of stock dependent demand. There are two warehouses to store the goods in which the first is rented warehouse and the second is own warehouse that deteriorates with two different rates. The aim of this study is to determine the optimal order quantity to maximize the profit of the projected model. Finally, some numerical examples and sensitivity analysis of parameters are made to validate the results of the proposed model.     

Simulation of an integrated age replacement and spare provisioning policy using SLAM

This paper presents a SLAM simulation model for determining a jointly optimal age replacement and spare part provisioning policy. The policy, referred to as a stocking policy, is formulated by combining age replacement policy with a continuous review (s, S) type inventory policy, where s is the stock reorder level and S is the maximum stock level. The optimal values of the decision variables are obtained by minimizing the total cost of replacement and inventory. The simulation procedure outlined in the paper can be used to model any operating situation having either a single item or a number of identical items. Results from a number of case problems specifically constructed by 5-factor second order rotatory design have been presented and the effects of different cost elements, item failure characteristics and lead time 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 the separate optimizations of replacement and spare provisioning policies do not ensure global optimality when total system cost has to be minimized.     

An optimal constant level rationing policy under service level constraints

We analyze a periodic review inventory system where inventory of a single item is used to serve high and low priority customers. High priority customers demand a higher service level than low priority customers and both types of customers can have quite general discrete demand distributions. The inventory is controlled by a (S, CL)-policy, where S denotes the base stock level and CL specifies the critical level, i.e., the amount of inventory reserved for high priority demand. We consider a policy, where the critical level is constant over time, which is not optimal, but makes the model analytically tractable. Unlike previous research, we clear backorders optimally and find the optimal solution. We also derive structural results.     

Optimal batch ordering policies for assembly systems with guaranteed service

For a supply chain modelled as a multi-echelon inventory system, effective management of its inventory at each stock is critical to reduce inventory costs while assuring a given service level to customers. In our previous work, we used the guaranteed-service approach (GSA) to design optimal echelon batch ordering policies for continuous-review serial systems with Poisson customer demand and fixed order costs. The approach assumes that the final customer demand is bounded and each stock has a guaranteed service time in the sense that the demand of its downstream stock can always be satisfied in the service time. This paper extends this work by considering more general assembly systems. We first derive an analytical expression for the total cost of the system in the long run. The problem of finding optimal echelon batch ordering policies for the system can then be decomposed into two independent sub-problems: order size decision sub-problem and reorder point decision sub-problem. We develop efficient dynamic programming algorithms for the two sub-problems. Numerical experiments on randomly generated instances show the effectiveness of the proposed approach.     

航空公司飞机航线可更换单元两级库存控制方法

航空公司以维修基地和中央仓库两级库存保障飞机的航线可更换单元需求,同时规定高保障率以提供高航空服务质量,提出了保障率约束下的两级仓库最佳备件数量问题。本文首先通过航线可更换单元的失效数据以极大似然方法估计需求率,然后提出所有仓库总备件成本最小化为目标,备件综合保障率为约束的两级库存控制模型,最后采用边际分析法进行模型求解完成维修基地和中央仓库两级备件量的配置优化,实现航材资金占用量的最小化。通过实例验证表明采用此方法能够有效压缩库存规模,降低库存成本。     

A Stock Location Model for Dual Address Order Picking Systems

A model is formulated to evaluate alternative stock levels and item locations for dedicated storage warehouses where storage and retrieval transactions are interleaved to economize on travel of the order picking vehicle. The model can be used to measure the order picking and inventory cost effects of warehouse layouts representing alternatives to application of economic lot sizing and cube per order index storage assignment. A simple, sequential search heuristic is proposed for generating such alternative warehouse layouts and applied to a sample problem.     

An approach to establishing a method for calculating inventory

In periodic review inventory systems, inventory is classified into cycle stock and safety stock. Cycle stock is defined as inventory that absorbs differences between supply and demand frequencies. It can be calculated without deficiency or excess because a method has been established for ensuring that the minimum on-hand inventory during a periodic review is zero. Safety stock is defined as inventory that absorbs various differences between supply and demand. Unlike for cycle stock, a method for calculating safety stock without deficiency or excess remains to be established. An approach is proposed to establishing a method for calculating inventory in which inventory is classified on the basis of the holding purpose and the calculation factors indicate solutions. This approach was applied to inventory held to absorb, on the basis of fluctuations in demand, the difference in terms of time and quantity between supply and demand. Stock held for this purpose is referred to as ‘fluctuation stock’. The objective is to establish a method for calculating fluctuation stock so that the minimum on-hand inventory during a periodic review is zero and to clarify the relationship between fluctuation stock and safety stock.     

双渠道多级分销网络的预防性库存分拨机制与方法研究

企业通过增加库存量来应对不可延期交货带来的缺货损失的传统运作方式在电子商务催生了线上线下双渠道分销模式后变得捉襟见肘。在双渠道多级分销模式下，传统囤货式的运作模式因多层级、多渠道而产生高额库存成本，这恰恰削弱了双渠道销售的高利润优势。因此，拥有双渠道多级分销网络的企业如何平衡库存成本与缺货损失就成为了本文的研究重点。本文考虑了拥有线上线下双渠道销售的一个由中心仓、区域仓和门店节点构成的三级供应链库存系统，借鉴库存共享思想提出了“预防性库存分拨机制”，即节点内线上线下库存、同层级一对多双向预防性调拨、上下级间一对一单向预防性补货的运作机制，并利用目标级联法(analysis target cascading, ATC)依据预测时间窗内的需求预测结果构建预防性分拨模型。最后以国内某大型进口红酒贸易企业为基础对模型进行仿真和分析，结果表明所提出的预防性库存分拨机制能够有效降低库存成本，为双渠道供应链系统低成本高效率的运作提供了实现的手段。     

The trans-shipment problem in a two-echelon,multi-location inventory system with lost sales

In this paper, we address a two-echelon, multi-location pooling inventory system that consists of an outside supplier, a warehouse and two retailers. To control their inventories, both warehouse and retailers use (R,?s,?S) policy. The retailers face stochastic customer demands for a single product and the warehouse receives only the replenishment orders of retailers. In case of stock-out at retailers, emergency trans-shipments are used to satisfy the unmet demand at one retailer with the surplus from the other retailer. When the stock is insufficient at the warehouse, we propose two rationing policies to allocate on-hand stock between retailers. The demand that cannot be satisfied neither by stock on-hand nor by trans-shipment from retailers is considered lost. Our work has two objectives. First, we propose an inventory model based on three components: the optimisation inventory model, the trans-shipment policy and the rationing policies for determining the best values of (s,?S) at each location that minimise total system cost. Second, we validate this model via an empirical simulation study that allows us to identify the influential parameters on trans-shipment benefits.     

Dynamic planned safety stocks in supply networks

Safety stocks are commonly used in inventory management for tactically planning against uncertainty in demand and/or supply. The usual approach is to plan a single safety stock value for the entire planning horizon. More advanced methods allow for dynamically updating this value. We introduce a new line of research in inventory management: the notion of planning time-phased safety stocks. We assert that planning a time-phased set of safety stocks over a planning horizon makes sense because larger safety stocks are appropriate in times of greater uncertainty while lower safety stocks are more appropriate when demand and/or supply are more predictable. Projecting a vector of safety stock values is necessary to assure upstream members in the supply network have advanced warning of changes. We perform an empirical study of U.S. industry, which demonstrates that significant savings can be achieved by employing dynamic planned safety stocks, confirming recent case study reports. We provide a simple optimisation model for the problem of minimising inventory given a vector of safety stock targets. We propose a computationally efficient solution procedure and demonstrate its implementation in an MRP/ERP system. We then illustrate an MRP/ERP planning system feature, which employs a dynamic planned safety stock module that supports a production planner by showing the inventory implications of safety stock plans.     

On deficiencies of common ordering policies for multi-level inventory control

We consider two common types of ordering policies for multi-level inventory control: installation stock (R,Q)-policies and echelon stock (R,Q)-policies. The batch quantities are assumed to be given, but each policy is optimized with respect to its reorder point R. We demonstrate that there is no bound for the worst case performance ratio of these policies when applied to distribution inventory systems with a central warehouse and a number of retailers.     

On offering economic incentives to backorder

We study a strategy to manage demands that occur when an inventory system is temporarily out of stock: offer the customer facing the unsatisfied demand an economic incentive to backorder. We explore the benefits of this inventory management strategy by analyzing a model of an inventory system with stochastic demand and random supply disruptions, where the probability that a customer facing an unfilled demand will backorder (as opposed to becoming a lost sale) can be influenced by an economic incentive. Our results provide several insights regarding this inventory management strategy and suggest that the benefits of offering backorder incentives can be significant.     

Stock diffusion theory: a dynamic model for inventory control

This paper proposes a dynamic approach for inventory management, which can be used for a definitely non stationary demand whose rate evolves both in mean and in variance. Specifically, the stock consumption is modelled as a Markov process with a slow diffusion term and the Fokker Planck equation is used to derive the probability distribution of the stock consumption and that of the reorder time. The knowledge of these distributions makes it possible to manage the inventory in a dynamical way and to keep the safety stock to a minimum level. To test the model, some typical demand patterns are used: results demonstrate its ability to capture both the evolution of the mean and that of the variance of the demand.     

Efficient heuristics for two-echelon spare parts inventory systems with an aggregate mean waiting time constraint per local warehouse

This paper presents solution procedures for determining close- to-optimal base stock policies in a multi-item two-echelon spare parts inventory system. The system consists of a central warehouse and multiple local warehouses, and there is a target for the aggregate mean waiting time per local warehouse. We develop four different heuristics and derive a lower bound on the optimal total cost. The effectiveness of each heuristic is assessed by the relative gap between the heuristic’s total cost and the lower bound. The results of the computational experiments show that a greedy procedure performs most satisfactorily. It is accurate as indicated by relatively small gaps, easy to implement, and the computational requirements are limited. Its computational efficiency can be increased by using Graves’ approximate evaluation method instead of an exact evaluation method, while the results remain accurate. That results in a feasible algorithm for real-life cases with many items and local warehouses.     

The Inverse Inventory Problem

The classical stochastic inventory problem yields a solution for the optimum amount to stock for a given demand distribution. In the inverse inventory problem the amount stocked is given and the decision maker must determine the optimum value of the parameter of the demand distribution that is under his/her control. The relationship between the solutions of the two problems is explored.