Analyzing the effect of inventory policies on the nonstationary performance with transfer functions

In this paper we apply transfer function methods to analyze the performance of supply chains in response to nonstationary demand and, in particular, we investigate how various inventory policies and demand forecasting parameters affect supply chain responsiveness. In a single-echelon inventory system we investigate the performance of a common base stock policy. Specifically, we describe the order and inventory trajectories using discrete transfer functions, and we derive closed-form analytical expressions for the transient behavior in response to a step change in demand. We introduce performance measures commonly used to analyze nonstationary performance and derive closed-form expressions for these measures. Next, we study the performance of a two-echelon supply chain under installation stock and echelon stock policies. We explicate the performance tradeoff in response to stationary versus nonstationary demand, and show that the transient response of orders and inventory levels can be either underdamped or overdamped depending on the exponential smoothing parameter. We show that the echelon stock policy is more responsive than the installation stock policy when both policies have similar stationary performances.     

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.     

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.     

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.     

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.     

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.     

Inventory performance of some supply chain inventory policies under impulse demands

This paper attempts to study the impact of impulsive demand disturbances on the inventory-based performance of some inventory control policies. The supply chain is modelled as a network of autonomous supply chain nodes. The customer places a constant demand except for a brief period of sudden and steep change in demand (called demand impulse). Under this setting, the behaviour of each inventory policy is analysed for inventory performance of each node. It is found that the independent decision-making by each node leads to a bullwhip effect in the supply chain whereby demand information is amplified and distorted. However, under a scenario where the retailer places a constant order irrespective of the end customer demand, the inventory variance was actually found to decrease along the supply chain. The variance of the inventory remained constant along the chain when only the actual demands are transmitted by each node. The results also showed that the inventory policy which is best for one supply chain node is generally less efficient from a supply chain perspective. Moreover, the policy which performs poorly for one node can be most efficient for the supply chain. In a way, our results also provide a case for coordinated inventory management in the supply chain where all members prepare a joint inventory management policy that is beneficial for all the supply chain nodes. The results have significant industrial implications.     

具有自消化引擎的库存模型初探

具有自消化引擎的库存模型是与传统库存完全不同的库存模型.自消化引擎库存是指库存系统具有内循环需求引擎、库存物资具有自修复能力以及寿命递减等生命库存的特征,库存的保有量与物资的用途、使用方法、技术定义、管理模式以及外部变化或扰动等密切相关.本文提出自消化引擎这一概念来描述这类库存的模型特征,并对这类库存模型的相关特征进行了讨论和初步研究.     

The impact of the number of parallel warehouses on total inventory

In the strategic design of a distribution system, the right number of stock points for the various products is an important question. In the past decade, a strong trend in the consumer goods industry led to centralizing the inventory in a single echelon consisting of a few parallel warehouses or even a single distribution center for a Europe-wide distribution system. Centralizing inventory is justified by the reduction in total stock which mostly overcompensates the increasing transportation cost. The effect of centralization is usually described by the “Square Root Law”, stating that the total stock increases with the square root of the number of stock points. However, in the usual case where the warehouses are replenished in full truck loads and where a given fill rate has to be satisfied, the Square Root Law is not valid. This paper explores that case. It establishes functional relationships between the demand to be served by a warehouse and the necessary safety and cycle stock for various demand settings and control policies, using an approximation of the normal loss function and its inverse. As a consequence, the impact of the number of parallel warehouses on the total stock can be derived. The results can be used as tools in network design models.     

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.     

Computing optimal ( s,S ) policies for inventory systems with a cut-off transaction size and the option of joint replenishment

This paper presents a mathematical model that is developed for the synthesis of optimal replenishment policies for items exhibiting lumpy demand patterns. In order to avoid the disruption of the inventory system, a maximum issue quantity restriction is incorporated into the inventory control policy. In doing so, customer demands with a size exceeding w units will be filtered out of the inventory system, but satisfied by placing a special replenishment order. The rest of the customer demands will be met from stock. The control discipline is the continuous review ( s, S ) inventory policy and the nature of the demands is approximated by a discrete compound Poisson distribution. To further reduce the total annual ordering cost, specification is made such that when the current inventory position is below a critical level, A , at the time when a customer demand with a size exceeding w units arrives, a joint replenishment is placed that will not only initiate a direct shipment to the customer, but will also raise the inventory position to S . An algorithm is developed to determine the global optimal values of the control parameters, s and S for given w and A . A numerical example is used to illustrate the methodology, and the results obtained clearly show that a better cost performance can be obtained with the incorporation of both a maximum issue quantity restriction and the option of joint replenishment into the inventory control policy.     

Lead-time analysis and a control policy for production lines

We address a control problem for a production line that produces one product to stock and faces random demand. During stockouts, the system quotes a fixed response time for each arriving order, and the customers place their orders only if the response time promised meets their deadlines. Customer orders are filled on a first come, first served basis. A penalty cost is incurred whenever a customer is served later than promised. A two-parameter admission/inventory control policy is implemented that maintains a bounded backlog and a constant inventory position (total inventory minus backlog) in the system. For production lines with exponential processing times and Poisson demand, the mean profit rate of the system is computed analytically using closed queueing network formulae. For systems with general processing or interarrival time distributions, the profit rate is estimated via simulation. Simple properties are established that ensure that the profit maximising control parameters can be determined in finite time using exhaustive search. Numerical results show that the proposed policy performs better than the make-to-order/zero-inventory and the lost-sales/make-to-stock policies.     

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.     

Effect of limited capacity on optimal planning parameters for a multi-item production system with setup times and advance demand information

An analytical production/inventory model to optimise the planning parameters lot-size, safety stock and planned lead time is developed for a stochastic single-stage production system with multiple items and limited capacity. Based on queuing analysis, the influence of item specific lot-sizes on the production lead time distribution is modelled. Applying stochastic advance demand information, the expected values for finished-goods-inventory, backorders and service level are explicitly stated. Numerical optimisation is applied to solve the respective cost minimisation problem and a solution heuristic is developed to support this approach. A numerical study provides managerial insights concerning capacity limitation effects on the optimal planning parameters. Higher shop loads, i.e. tighter capacity constraints, are found to significantly increase optimal lot-size and optimal safety stock. Safety stock and planned lead time are substitutes, an increase of both leads to higher FGI and lower backorders, however, the specific trade-off depends on the demand information quality. A sensitivity analysis investigating other (non-) financial system parameters is conducted as well. The main contribution of this paper is that the interaction of different planning parameters, i.e. lot-size, safety stock and planned lead time, for different items is simultaneously studied for a capacity constrained production/inventory system.     

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.     

Planning stability in a product recovery system

Recovery of used products is an issue of growing importance due to customer expectations and environmental regulation. As a consequence, companies need to adapt their material management taking into account inbound flows of used products. Corresponding inventory control models have been proposed in literature. In this paper we address the issue of planning stability in a product recovery context. To this end, we consider rolling horizon planning for a stock point facing stochastic demand and product returns. We analyze the impact of the return flow on planning stability and compare the system behaviour with a traditional production environment. We show that structural results derived for traditional inventory models remain valid in a product recovery context. Moreover we discuss counterintuitive effects resulting from interaction between planning stability and stock levels.     

Analysis of investments in autonomous maintenance activities

In this paper, we model the situation where operator maintenance activities improve the failure process of equipment. We analyze the business decision to reduce both the mean and variance of the production cycle time and the overall inventory level through an investment in planned autonomous maintenance. We answer: (i) when do optimal autonomous maintenance decisions most improve inventory levels?; and (ii) how do capacity restrictions, equipment characteristics the maintenance response function, and product characteristics impact the autonomous maintenance investment decision? Extensive numerical analyses are performed to develop an approximation to the optimal response for both inventory and autonomous maintenance investments over a wide range of problem parameters. Our solutions provide guidelines on how much time should be invested in autonomous maintenance activities and describe when companies can most benefit from autonomous maintenance programs that increase equipment reliability. We determine the investment in autonomous maintenance activities as a function of available capacity, equipment reliability and demand characteristics.     

（T，S）策略下含模糊需求及缺陷率的安全库存

基于模糊可信性理论,探讨了在需求及到货产品质量不确定环境中,周期盘点(T,S)库存策略下的安全库存问题,建立了确定安全库存水平的数学模型,并分析了预定服务水平及需求和缺陷率的不确定性对安全库存的影响。     

Bullwhip and inventory variance in a closed loop supply chain

A simple dynamic model of a hybrid manufacturing/remanufacturing system is investigated. In particular we study an infinite horizon, continuous time, APIOBPCS (Automatic Pipeline Inventory and Order Based Production Control System) model. We use Åström’s method to quantify variance ratios in the closed loop supply chain. Specifically we highlight the effect of a combined “in-use” and remanufacturing lead-time and the return rate on the inventory variance and bullwhip produced by the ordering policy. Our results clearly show that a larger return rate leads to less bullwhip and less inventory variance in the plant producing new components. Thus returns can be used to absorb demand fluctuations to some extent. Longer remanufacturing and “in-use” lead-times have less impact at reducing inventory variance and bullwhip than shorter lead-times. We find that, within our specified system, that inventory variance and bullwhip is always less in supply chains with returns than supply chains without returns. We conclude by separating out the remanufacturing lead-time from the “in-use” lead-time and investigating its impact on our findings. We find that short remanufacturing lead-times do not qualitatively change our results.     

Base stock versus WIP cap in single-stage make-to-stock production–inventory systems

A single-stage production-inventory system produces parts in a make-to-stock mode, and unsatisfied demand is backordered. The system operates under a so-called base stock with WIP cap replenishment policy, which works as follows. Whenever the Work-In-Process (WIP) plus finished goods inventory falls below a specified level, called base stock, a replenishment order for the production of a new part is issued. If the WIP inventory is below a different specified level, called WIP cap, the order goes through and a new part is released for production; otherwise, the order is put on hold until the WIP inventory drops below the WIP cap. First, it is shown that the optimal base stock that minimizes the long-run, average, inventory holding cost for a given minimum customer service level, is a non-increasing function of the WIP cap that reaches a minimum value, called minimum optimal base stock, at a finite WIP cap value, called critical WIP cap. Then, it is shown that the optimal WIP cap is less than or equal to the critical WIP cap and therefore the optima! base stock is greater than or equal to the minimum optimal base stock. More interestingly, however, it is conjectured that the optimal WIP cap is in fact exactly equal to the critical WIP cap and therefore the optimal base stock is exactly equal to the minimum optimal base stock. The minimum optimal base stock is none other than the optimal base slock of the same system operating under a classical base stock policy (with no WIP cap). Finally, the optimal parameters of a system operating under a base stock with WIP cap policy are related to the optimal parameter of the same system operating under a make-to-stock CONWIP policy.