A two-level supply chain with partial backordering and approximated Poisson demand

We consider a two-level supply chain with a number of identical, independent ‘retailers’ at the lower echelon and a single supplier at the upper echelon controlled by continuous review inventory policy (R, Q). Each retailer experiences Poisson demand with constant transportation times. We assume constant lead time for replenishing supplier orders from an external warehouse to the supplier and unsatisfied retailer orders are backordered in the supplier. We assume that the unsatisfied demand is partially backordered in the identical retailers. The partially backordering policy is implemented in the identical retailers using an explicit control parameter ‘b’ which limits the maximum number of backorders allowed to be accumulated during the lead time. We develop an approximate cost function to find optimal reorder points for given batch sizes in all installations, the optimal value of b in the identical retailers and the related accuracy is assessed through simulation.     

A two-echelon inventory model with lost sales

This paper considers a single-item, two-echelon, continuous-review inventory model. A number of retailers have their stock replenished from a central warehouse. The warehouse in turn replenishes stock from an external supplier. The demand processes on the retailers are independent Poisson. Demand not met at a retailer is lost. The order quantity from each retailer on the warehouse and from the warehouse on the supplier takes the same fixed value Q, an exogenous variable determined by packaging and handling constraints. Retailer i follows a (Q, R_i) control policy. The warehouse operates an (SQ, (S − 1)Q) policy, with non-negative integer S. If the warehouse is in stock then the lead time for retailer i is the fixed transportation time L_i from the warehouse to that retailer. Otherwise retailer orders are met, after a delay, on a first-come first-served basis. The lead time on a warehouse order is fixed. Two further assumptions are made: that each retailer may only have one order outstanding at any time and that the transportation time from the warehouse to a retailer is not less than the warehouse lead time. The performance measures of interest are the average total stock in the system and the fraction of demand met in the retailers. Procedures for determining these performance measures and optimising the behaviour of the system are developed.     

On the value of customer information for an independent supplier in a continuous review inventory system

We consider the inventory control problem of an independent supplier in a continuous review system. The supplier faces demand from a single customer who in turn faces Poisson demand and follows a continuous review (R, Q) policy. If no information about the inventory levels at the customer is available, reviews and ordering are usually carried out by the supplier only at points in time when a customer demand occurs. It is common to apply an installation stock reorder point policy. However, as the demand faced by the supplier is not Markovian, this policy can be improved by allowing placement of orders at any point in time. We develop a time delay policy for the supplier, wherein the supplier waits until time t after occurrence of the customer demand to place his next order. If the next customer demand occurs before this time delay, then the supplier places an order immediately. We develop an algorithm to determine the optimal time delay policy. We then evaluate the value of information about the customer’s inventory level. Our numerical study shows that if the supplier were to use the optimal time delay policy instead of the installation stock policy then the value of the customer’s inventory information is not very significant.     

Order splitting in continuous review (Q,r) inventory models

A number of recent articles in the literature have argued the case, when lead time is variable, for splitting a replenishment order for Q between n suppliers by comparing this with the alternative of placing a single order for Q on one supplier. The split order compares favourably on the grounds that the arrival of the first component of a split order cannot be later than the arrival of an order from any one specified supplier. This note argues that an alternative comparison could be made with a policy of ordering Q/n from a single supplier (n times as often). It makes this comparison in the context of a continuous review (Q, r) inventory model but does so not by comparing aggregate costs but by fixing Q and the customer stock service level and comparing the average stock — an approach which is more appropriate to how many companies manage inventory in practice. We consider Poisson and deterministic demand processes, a general lead time distribution and both lost sales and backorder models.     

Stock Rationing in a Continuous Review Two-Echelon Inventory Model

In this paper we consider a 1-warehouse, N-retailer inventory system where demand occurs at all locations. We introduce an inventory model which allows us to set different service levels for retailers and direct customer demand at the warehouse. For each retailer a critical level is defined, such that a retailer replenishment order is delivered from warehouse stock if and only if the stock level exceeds this critical level. It is assumed that retailer replenishment orders, which are not satisfied from warehouse stock, are delivered directly from the outside supplier, instead of being backlogged. We present an analytical upper bound on the total cost of the system, and develop a heuristic method to optimize the policy parameters. Numerical experiments indicate that our technique provides a very close approximation of the exact cost. Also, we show that differentiating among the retailers and direct customer demand can yield significant cost reductions.     

A stochastic periodic review integrated inventory model involving defective items, backorder price discount, and variable lead time

The purpose of this article is to investigate a stochastic integrated supplier-retailer inventory problem. The model analyzed in this article explores the problem of the protection interval, the backorder price discount, the lead time, and the numbers of shipments from the supplier to the retailer in one production run as control variables to widen applications for an integrated periodic review inventory model. We consider the situation in which the supplier and the retailer establish a long-term strategic partnership and contract to jointly determine the best strategy. We assume that the protection interval demand follows a normal distribution. Our objective is to determine the optimal review period, the optimal backorder price discount, the optimal lead time, and the optimal number of shipments from the supplier to the retailer in one production run, so that the joint expected annual total cost incurred has the minimum value. Furthermore, an algorithm of finding the optimal solution is developed. Also, the sensitivity analysis included and a numerical example is given to illustrate the results of the proposed model.     

Coordinating lead times and safety stocks under autocorrelated demand

We consider a supply chain in which orders and lead times are linked endogenously, as opposed to assuming lead times are exogenous. This assumption is relevant when a retailer’s orders are produced by a supplier with finite capacity and replenished when the order is completed. The retailer faces demands that are correlated over time – either positively or negatively – which may, for example, be induced by a pricing or promotion policy. The auto-correlation in demand affects the order stream placed by the retailer onto the supplier, and this in turn influences the resulting lead times seen by the retailer. Since these lead times also determine the retailer’s orders and its safety stocks (which the retailer must set to cover lead time demand), there is a mutual dependency between orders and lead times. The inclusion of endogenous lead times and autocorrelated demand represents a better fit with real-life situations. However, it poses some additional methodological issues, compared to assuming exogenous lead times or stationary demand processes that are independent over time. By means of a Markov chain analysis and matrix analytic methods, we develop a procedure to determine the distribution of lead times and inventories, that takes into account the correlation between orders and lead times. Our analysis shows that negative autocorrelation in demand, although more erratic, improves both lead time and inventory performance relative to IID demand. Positive correlation makes matters worse than IID demand. Due to the endogeneity of lead times, these effects are much more pronounced and substantial error may be incurred if this endogeneity is ignored.     

历史订单信息对牛鞭效应的影响分析

客户需求信息的失真是导致牛鞭效应存在的原因,基于零售商的历史订单数据对其需求进行预测可以部分消除牛鞭效应。论文基于零售商-分销商二级供应链视角,分析了在零售商的需求为线性自回归模式的二级供应链中,分销商利用零售商历史订单数据和现有订单数据进行需求预测时自身库存成本的变更以及整个供应链的牛鞭效应的缓解程度。结果表明:分销商利用历史订单数据进行库存的决策可以显著地降低自己的平均库存和需求的波动,这种降低程度在零售商的订货提前期较大的情况下比较明显,但是零售商的需求预测相关系数对它影响不大。     

An integrated production and inventory model to dampen upstream demand variability in the supply chain

We consider a two-echelon supply chain: a single retailer holds a finished goods inventory to meet an i.i.d. customer demand, and a single manufacturer produces the retailer’s replenishment orders on a make-to-order basis. In this setting the retailer’s order decision has a direct impact on the manufacturer’s production. It is a well known phenomenon that inventory control policies at the retailer level often propagate customer demand variability towards the manufacturer, sometimes even in an amplified form (known as the bullwhip effect). The manufacturer, however, prefers to smooth production, and thus he prefers a smooth order pattern from the retailer. At first sight a decrease in order variability comes at the cost of an increased variance of the retailer’s inventory levels, inflating the retailer’s safety stock requirements. However, integrating the impact of the retailer’s order decision on the manufacturer’s production leads to new insights. A smooth order pattern generates shorter and less variable (production/replenishment) lead times, introducing a compensating effect on the retailer’s safety stock. We show that by including the impact of the order decision on lead times, the order pattern can be smoothed to a considerable extent without increasing stock levels. This leads to a situation where both parties are better off.     

Approximate analysis and optimization of batch ordering policies in capacitated supply chains

Devising manufacturing/distribution strategies for supply chains and determining their parameter values have been challenging problems. Linking production management to stock keeping processes improves the planning of the supply chain activities, including material management, culminating in improved customer service levels. In this study, we investigate a multi-echelon supply chain consisting of a supplier, a plant, a distribution center and a retailer. Material flow between stages is driven by reorder point/order quantity inventory control policies. We develop a model to analyze supply chain behavior using some key performance metrics such as the time averages of inventory and backorder levels, as well as customer service levels at each echelon. The model is validated against simulation, yielding good agreement of robust performance metrics. The metrics are then used within an optimization framework to design the supply chain so as to minimize expected total system costs. The outcome of the optimization framework specifies how to move inventory throughout the supply chain and how to set inventory control parameters, i.e., reorder levels and replenishment batch sizes.     

When supplier’s availability affects the replenishment lead time—An extension of the supply-interruption problem

This paper presents an exact treatment of a continuous-review inventory system with compound Poisson demand, Erlang-distributed lead times and random supply interruptions. In contrast with the existing models in the literature, we take into account the supplier’s availability in characterizing the lead time of a replenishment order. Assuming that the supplier’s availability can be described by a continuous-time homogeneous Markov chain with two states (on and off) and that stockouts are lost, we derive the stationary distribution of the inventory level (stock-on-hand) under an (s, Q)-type control policy. This probability distribution is then used to formulate an exact expression for the long-run average cost per unit time of operating the inventory system. Some numerical results are also provided.     

A measure of bullwhip effect in supply chains with a mixed autoregressive-moving average demand process

In this paper, we quantify the impact of the bullwhip effect – the phenomenon in which information on demand is distorted as moving up a supply chain – for a simple two-stage supply chain with one supplier and one retailer. Assuming that the retailer employs a base stock inventory policy, and that the demand forecast is performed via a mixed autoregressive-moving average model, ARMA(1, 1), we investigate the effects of the autoregressive coefficient, the moving average parameter, and the lead time on the bullwhip effect.     

Consignment contracting: Who should control inventory in the supply chain?

Consignment is a popular form of business arrangement where supplier retains ownership of the inventory and gets paid from the retailer based on actual units sold. The popularity of such an arrangement has come with some continued debates on who should control the supply chain inventory, the supplier or retailer. This paper aims at shedding light on these debated issues. We consider a single period supply chain model where a supplier contracts with a retailer. Market demand for the product is price-sensitive and uncertain. The supplier decides his consignment price charged to the retailer for each unit sold, and the retailer then chooses her retail price for selling the product. We study and compare two different consignment arrangements: The first allows the retailer to choose the supply chain inventory, together with her retail price, and is labeled as a Retailer Managed Consignment Inventory (RMCI) program; and the second calls for the supplier to decide the inventory, together with his consignment price, and is labeled as a Vendor Managed Consignment Inventory (VMCI) program. We show that with an RMCI program, the supply chain loses at least 26.4% of its first-best (expected) profit, while with VMCI, it loses just or no more than 26.4% of the first-best profit. Second, we demonstrate that both programs lead to an equal split of the corresponding channel profit between the supplier and the retailer. These results indicate that it is beneficial both to the supplier and to the retailer when delegating the inventory decision to the supplier rather than to the retailer in the channel.     

Controlling distribution inventory systems with shipment consolidation and compound Poisson demand

We consider a one-warehouse-multiple-retailer inventory system where the retailers face stochastic customer demand, modelled as compound Poisson processes. Deliveries from the central warehouse to groups of retailers are consolidated using a time based shipment consolidation policy. This means that replenishment orders have to wait until a vehicle departures, which increases the lead time for the retailers and therefore also the safety stock. Thus, a trade-off exists between expected shipment costs and holding costs. Our aim is to determine the shipment intervals and the required amount of safety stock for each retailer and the warehouse to minimize total cost, both for backorder costs and fill rate constraints. Previous work has focused on exact solutions which are computationally demanding and not applicable for larger real world problems. The focus of our present work is on the development of computationally attractive heuristics that can be applied in practice. A numerical study shows that the proposed heuristics perform well compared to the exact cost minimizing solutions. We also illustrate that the approaches are appropriate for solving real world problems using data from a large European company.     

A real-time decision rule for an inventory system with committed service time and emergency orders

In this paper, we study the inventory system of an online retailer with compound Poisson demand. The retailer normally replenishes its inventory according to a continuous review (nQ, R) policy with a constant lead time. Usually demands that cannot be satisfied immediately are backordered. We also assume that the customers will accept a reasonable waiting time after they have placed their orders because of the purchasing convenience of the online system. This means that a sufficiently short waiting time incurs no shortage costs. We call this allowed waiting time “committed service time”. After this committed service time, if the retailer is still in shortage, the customer demand must either be satisfied with an emergency supply that takes no time (which is financially equivalent to a lost sale) or continue to be backordered with a time-dependent backorder cost. The committed service time gives an online retailer a buffer period to handle excess demands. Based on real-time information concerning the outstanding orders of an online retailer and the waiting times of its customers, we provide a decision rule for emergency orders that minimizes the expected costs under the assumption that no further emergency orders will occur. This decision rule is then used repeatedly as a heuristic. Numerical examples are presented to illustrate the model, together with a discussion of the conditions under which the real-time decision rule provides considerable cost savings compared to traditional systems.     

Inventory allocation and shipping when demand temporarily exceeds production capacity

We address the concept of an integrated inventory allocation and shipping model for a manufacturer with limited production capacity and multiple types of retailers with different backorder/waiting and delivery costs. The problem is to decide how to allocate and deliver produced items when the total retailer demand exceeds the production capacity, so that total retailer backorder and delivery costs are minimized. Our analytical model provides optimal allocation and shipping policies from the manufacturer’s viewpoint. We also investigate the allocation strategy of a manufacturer competing with other retailers to directly sell to end consumers.     

Impact of information sharing and lead time on bullwhip effect and on-hand inventory

This work analyzes a two echelon (warehouse–retailer) serial supply chain to study the impact of information sharing (IS) and lead time on bullwhip effect and on-hand inventory. The customer demand at the retailer is assumed to be an autoregressive (AR(1)) process. Both the echelons use a minimum mean squared error (MMSE) model for forecasting lead time demand (LTD), and follow an adaptive base-stock inventory policy to determine their respective order quantities. For the cases of without IS and inter as well as intra echelon IS, expressions for the bullwhip effect and on-hand inventory for the warehouse are obtained, considering deterministic lead-time. The results are compared with the previous research work and an easy analysis of the various bullwhip effect expressions under different scenarios, is done to understand the impact of IS on the bullwhip effect phenomenon. It is shown that some part of bullwhip effect will always remain even after sharing both inter as well as intra echelon information. Further, with the help of a numerical example it is shown that the lead time reduction is more beneficial in comparison to the sharing of information in terms of reduction in the bullwhip effect phenomenon.     

A periodic-review inventory system with a capacitated backup supplier for mitigating supply disruptions

We consider a periodic-review inventory system with two suppliers: an unreliable regular supplier that may be disrupted for a random duration, and a reliable backup supplier that can be used during a disruption. The backup supplier charges higher unit purchasing cost and fixed order cost when compared to the regular supplier. Because the backup supplier is used at unplanned moments, its capacity to replenish inventory is considered limited. Analytical results partially characterize the structure of the optimal order policy: a state-dependent (X(i), Y(i)) band structure (with corresponding bounds of X(i) and Y(i) to be given), where i represents the status of the regular supplier. Numerical studies illustrate the structure of the optimal policy and investigate the impacts of major parameters on optimal order decisions and system costs.     

An inventory-transportation system with stochastic demand

We study a logistic system in which a supplier has to deliver a set of products to a set of retailers to face a stochastic demand over a given time horizon. The transportation from the supplier to each retailer can be performed either directly, by expensive and fast vehicles, or through an intermediate depot, by less expensive but slower vehicles. At most one time period is required in the former case, while two time periods are needed in the latter case. A variable transportation cost is charged in the former case, while a fixed transportation cost per journey is charged in the latter case. An inventory cost is charged at the intermediate depot. The problem is to determine, for each time period and for each product, the quantity to send from the supplier to the depot, from the depot to each retailer and from the supplier to each retailer, in order to minimize the total expected cost. We first show that the classical benchmark policy, in which the demand of each product at each retailer is set equal to the average demand, can give a solution which is infinitely worse with respect to the optimal solution. Then, we propose two classes of policies to solve this problem. The first class, referred to as Horizon Policies, is composed of policies which require the solution of the overall problem over the time horizon. The second class, referred to as Reoptimization Policies, is composed of a myopic policy and several rolling-horizon policies in which the problem is reoptimized at each time period, once the demand of the time period is revealed. We evaluate the performance of each policy dynamically, by using Monte Carlo Simulation.     

Optimal ordering and transfer policy for an inventory with stock dependent demand

This paper deals with an ordering-transfer inventory model to determine the retailer’s optimal order quantity and the number of transfers per order from the warehouse to the display area. It is assumed that the amount of display space is limited and the demand rate depends on the display stock level. The objective is to maximize the average profit per unit time yielded by the retailer. The proposed models and algorithms are developed to find the optimal strategy by retailer. Numerical examples are presented to illustrate the models developed and the sensitivity analysis is also reported.