Lost-sales inventory theory: A review

In classic inventory models it is common to assume that excess demand is backordered. However, studies analyzing customer behavior in practice show that most unfulfilled demand is lost or an alternative item/location is looked for in many retail environments. Inventory systems that include this lost-sales characteristic appear to be more difficult to analyze and to solve. Furthermore, lost-sales inventory systems require different replenishment policies to minimize costs compared to backorder systems. In this paper, we classify the models in the literature based on the characteristics of the inventory system and review the proposed replenishment policies. For each classification and type of replenishment policy we discuss the available models and their performance. Furthermore, directions for future research are proposed.     

Analysis of optimal opportunistic replenishment policies for inventory systems by using a (s,S) model with a maximum issue quantity restriction

The analysis of optimal inventory replenishment policies for items having lumpy demand patterns is difficult, and has not been studied extensively although these items constitute an appreciable portion of inventory populations in parts and supplies types of stockholdings. This paper studies the control of an inventory item when the demand is lumpy. A continuous review (s,S) policy with a maximum issue quantity restriction and with the possibility of opportunistic replenishment is proposed to avoid the stock of these items being depleted unduly when all the customer orders are satisfied from the available inventory and to reduce ordering cost by coordinating inventory replenishments. The nature of the customer demands is approximated by a compound Poisson distribution. When a customer order arrives, if the order size is greater than the maximum issue quantity w, the order is satisfied by placing a special replenishment order rather than from the available stock directly. In addition, if the current inventory position is equal to or below a critical level A when such an order arrives, an opportunistic replenishment order which combines the special replenishment order and the regular replenishment order will be placed, in order to satisfy the customer's demand and to bring the inventory position to S. In this paper, the properties of the cost function of such an inventory system with respect to the control parameters s, S and A are analysed in detail. An algorithm is developed to determine the global optimal values of the control parameters. Indeed, the incorporation of the maximum issue quantity and opportunistic replenishment into the (s,S) policy reduces the total operating cost of the inventory system.     

考虑顾客退货和固定成本的联合补货与定价策略

为了吸引更多顾客,许多电子商务零售商允许顾客在一定时间内退货,导致其利润明显减少。同时,在补货时不仅产生依赖补货量的变动成本,而且会产生与补货量无关的固定成本。基于此,以最大化电子商务零售商的利润为目标,建立考虑顾客退货和固定成本的联合补货与定价模型,其中顾客的退货量与满足的需求呈正比。在一般需求情形下,部分刻画多期问题的最优策略;在特殊需求情形下,证明(s,S,p)策略对单期问题最优,并对多期问题的最优策略进行严格刻画。根据已有刻画为多期问题构造启发式策略。数值结果表明启发式策略近似最优;当初始库存水平足够高/低时,最优补货水平和定价随退货率与固定成本单调变化。关键词:联合补货与定价模型;顾客退货;固定成本;随机动态规划;最优策略     

Optimal pricing and ordering policy for non-instantaneous deteriorating items under inflation and customer returns

This paper deals with an economic production quantity inventory model for non-instantaneous deteriorating items under inflationary conditions considering customer returns. We adopt a price- and time-dependent demand function. Also, the customer returns are considered as a function of both price and demand. The effects of time value of money are studied using the Discounted Cash Flow approach. The main objective is to determine the optimal selling price, the optimal replenishment cycles, and the optimal production quantity simultaneously such that the present value of total profit is maximized. An efficient algorithm is presented to find the optimal solution. Finally, numerical examples are provided to solve the presented inventory model using our proposed algorithm, which is further clarified through a sensitivity analysis. The results of analysing customer returns provide important suggestions to financial managers who use price as a control to match the quantity sold to inventory while maximizing revenues. The paper ends with a conclusion and an outlook to future studies.     

The effect of information sharing on supply chain stability and the bullwhip effect

This paper analyzes the bullwhip effect in multi-stage supply chains operated with linear and time-invariant inventory management policies and shared supply chain information. Such information includes past order sequences and inventory records at all supplier stages. The paper characterizes the stream of orders placed at any stage of the chain when the customer demand process is known and ergodic, and gives an exact formula for the variance of the orders placed. The paper also derives robust analytical conditions, based only on inventory management policies, to predict the presence of the bullwhip effect and bound its magnitude. These results hold independently of the customer demand. The general framework proposed in this paper allows for any inventory replenishment policies, any ways of sharing and utilizing information, and any customer demand processes. It is also shown as a special case that sharing customer demand information across the chain significantly reduces, but does not completely eliminate, the bullwhip effect.     

An inventory model with returns and lateral transshipments

In this paper, we propose a single-item inventory model with returns. The model allows lateral transshipment of returns from one inventory system to another. Each inventory system is under continuous review and an (r, Q) policy is employed as the inventory control. An approximated closed-form solution of the system steady-state probability distribution is derived when Q is large. The approximated inventory cost and replenishment cost can be written in terms of this distribution. We show that the rejection rate of returns is reduced significantly when transshipment of returns is allowed between the inventory systems.     

Capacitated replenishment and disposal planning for multiple products with resalable returns

We consider a replenishment and disposal planning problem (RDPP) that arises in settings where customer returns are in as-good-as-new condition. These returns can be placed into inventory to satisfy future demand or can be disposed of, in case they lead to excess inventory. Our focus is on a multi-product setting with dynamic demands and returns over a finite planning horizon with explicit replenishment and disposal capacities. The problem is to determine the timing of replenishment and disposal setups, along with the associated quantities for the products, so as to minimize the total costs of replenishment, disposal, and inventory holding throughout the planning horizon. We examine two variants of the RDPP of interest both of which are specifically motivated by a spare part kitting application. In one variant, the replenishment capacity is shared among multiple products while the disposal capacity is product specific. In the other variant, both the replenishment and disposal capacities are shared among the products. We propose a Lagrangian Relaxation approach that relies on the relaxation of the capacity constraints and develop a smoothing heuristic that uses the solution of the Lagrangian problem to obtain near-optimal solutions. Our computational results demonstrate that the proposed approach is very effective in obtaining high-quality solutions with a reasonable computational effort.     

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.     

Coordinating pricing and inventory control in a fluctuating environment

This paper addresses the simultaneous determination of pricing and inventory replenishment strate- gies under a fluctuating environment. Specifically, we analyze the single item, periodic review model. The demand consists of two parts： the deterministic component, which is influenced by the price, and the stochastic component （perturbation）. The distribution of the stochastic component is determined by the current state of an exogenous Markov chain. The price that is charged in any given period can be specified dynamically. A replenishment order may be placed at the beginning of some or all of the periods, and stockouts are fully backlogged. Ordering costs that are lower semicontinuous, and inventory/backlog （or surplus） costs that are continuous with polynomial growth. Finite-horizon and infinite-horizon problems are addressed. Existence of optimal policies is established. Furthermore, optimality of （s,S,p）-type policies is proved when the ordering cost consists of fixed and proportional cost components and the surplus cost （these costs are all state-dependent） is convex.     

Optimal ordering policies for periodic-review systems with a refined intra-cycle time scale

Chiang [C. Chiang, Optimal ordering policies for periodic-review systems with replenishment cycles, European Journal of Operational Research 170 (2006) 44–56] recently proposed a dynamic programming model for periodic-review systems in which a replenishment cycle consists of a number of small periods (each of identical but arbitrary length) and holding and shortage costs are charged based on the ending inventory of small periods. The current paper presents an alternative (and concise) dynamic programming model. Moreover, we allow the possibility of a positive fixed cost of ordering. The optimal policy is of the familiar (s, S) type because of the convexity of the one-cycle cost function. As in the periodic-review inventory literature, we extend this result to the lost-sales periodic problem with zero lead-time. Computation shows that the long-run average cost is rather insensitive to the choice of the period length. In addition, we show how the proposed model is modified to handle the backorder problem where shortage is charged on a per-unit basis irrespective of its duration. Finally, we also investigate the lost-sales problem with positive lead-time, and provide some computational results.     

Constraint programming for stochastic inventory systems under shortage cost

One of the most important policies adopted in inventory control is the replenishment cycle policy. Such a policy provides an effective means of damping planning instability and coping with demand uncertainty. In this paper we develop a constraint programming approach able to compute optimal replenishment cycle policy parameters under non-stationary stochastic demand, ordering, holding and shortage costs. We show how in our model it is possible to exploit the convexity of the cost-function during the search to dynamically compute bounds and perform cost-based filtering. Our computational experience show the effectiveness of our approach. Furthermore, we use the optimal solutions to analyze the quality of the solutions provided by an existing approximate mixed integer programming approach that exploits a piecewise linear approximation for the cost function.     

Optimal transshipments and reassignments under periodic or cyclic holding cost accounting

In a centrally managed system, inventory at a retailer can be transshipped to a stocked-out retailer to meet demand. As the inventory at the former retailer may be demanded by future customers of that retailer and transshipment time/cost is non-negligible, it can be more profitable to not transship in some situations. When unsatisfied demand is backordered, reassignment of inventory to a previously backordered demand can perhaps become profitable as demand uncertainty resolves over time. Despite this intuition, we prove that no reassignments are necessary for cost optimality under periodic holding cost accounting in a two-retailer system. This remains valid for multi-retailer systems according to numerical analyses. When holding costs are accounted for only at the end of each replenishment cycle, reassignments are necessary for optimality but insignificant in reducing the total cost. In most instances tested, the decrease in total cost from reassignments is below 2% for end of cycle holding cost accounting. These results simplify transshipment policies and facilitate finding good policies in both implementation and future studies, as reassignments can be omitted from consideration in optimization models under periodic holding cost accounting and in approximation models under cyclical cost accounting.     

A note on inventory policies for products with residual-life-dependent demand

We study inventory ordering policies for products that attract demand at a decreasing rate as they approach the end of their usable lifetime, for example, perishable items nearing expiration. We consider the “product freshness’’, or equivalently, the time until expiration (“residual life”) as a factor influencing the customer demand. In a profit-maximizing framework, we build on the Economic Order Quantity (EOQ) replenishment model and formulate the inventory ordering problem using a deterministic demand function that is concave decreasing in the the age of the product. We provide analytical results on the optimal ordering policy, including an explicit characterization of the decisions in the linear-demand case, and we develop an easy-to-implement adaptive heuristic policy for the general case. Numerical examples show that the optimal policy generates significant profit gains compared to the traditional cost-based policies and the adaptive heuristic policy performs highly satisfactorily in the tested instances.     

Can-order policy for the periodic-review joint replenishment problem

In this paper we study the stochastic joint replenishment problem. We compare the class of periodic replenishment policies and the class of can-order policies for this problem. We present a method, based on Markov decision theory, to calculate near-optimal can-order policies for a periodic-review inventory system. Our numerical study shows that the can-order policy behaves as well as, if not better than, the periodic replenishment policies. In particular, for examples where the demand is irregular, we find cost differences up to 15% in favour of the can-order policy.     

Inventory control using a service constraint on the expected customer order waiting time

In this paper an analytical model is presented which permits the determination of the minimal inventory reorder point subject to (a) a maximum specified expected customer order waiting time or (b) a maximum specified probability of a customer order waiting more than a predetermined time span. The probability distribution of the customer order waiting time is determined for an arbitrary demand distribution (e.g. normal, gamma or exponential) and under stochastic replenishment lead time conditions by use of arguments from renewal theory. The results established can be used for the customer-oriented control of inventory policy as well as for the analysis of multi-echelon inventory systems.     

A comparison of outbound dispatch policies for integrated inventory and transportation decisions

This paper investigates the impact of alternative outbound dispatch policies on integrated stock replenishment and transportation decisions. The logistics literature reports that two different types of such policies are popular in current practice. These are time-based and quantity-based dispatch policies. Considering the case of stochastic demand, the paper presents analytical and numerical results showing that the cost savings obtained through quantity-based policies can be substantial. However, under a quantity-based policy, a specific delivery time cannot be quoted when the customer places an order. Hence, the paper also investigates the cost and customer waiting time implications of hybrid policies and demonstrates that hybrid policies are superior to time-based policies in terms of the resulting costs. Furthermore, although hybrid policies are not superior to quantity-based policies in terms of the resulting costs, they are superior in terms of a service measure which is quantified by the long-run average cumulative waiting time.     

The bullwhip effect in supply chain networks

This paper analyzes the propagation and amplification of order fluctuations (i.e., the bullwhip effect) in supply chain networks operated with linear and time-invariant inventory management policies. The supply chain network is allowed to include multiple customers (e.g., markets), any network structure, with or without sharing information. The paper characterizes the stream of orders placed by any supplier for any stationary customer demand processes, and gives exact formulas for the variance of the orders placed and the amplification of order fluctuations. The paper also derives robust analytical conditions, based only on inventory management policies, to predict the presence of the bullwhip effect for any network structure, any inventory replenishment policies, and arbitrary customer demand processes. Numerical examples show that the analytical results accurately quantify the bullwhip effect; managerial insights are drawn from the analysis. The methodology presented in this paper generalizes those in previous studies for serial supply chains.     

Control policies for single-stage production systems with perishable inventory and customer impatience

We consider problems of inventory and admission control for make-to-stock production systems with perishable inventory and impatient customers. Customers may balk upon arrival (refuse to place orders) and renege while waiting (withdraw delayed orders) during stockouts. Item lifetimes and customer patience times are random variables with general distributions. Processing, setup, and customer inter-arrival times are however assumed to be exponential random variables. In particular, the paper studies two models. In the first model, the system suspends its production when its stock reaches a safety level and can resume later without incurring any setup delay or cost. In the second model, the system incurs setup delays and setup costs; during stockouts, all arriving customers are informed about anticipated delays and either balk or place their orders but cannot withdraw them later. Using results from the queueing literature, we derive expressions for the system steady-state probabilities and performance measures, such as profit from sales and costs of inventory, setups, and delays in filling customer orders. We use these expressions to find optimal inventory and admission policies, and investigate the impact of product lifetimes and customer patience times on system performance.