Inventory Models with Markovian Demands and Cost Functions of Polynomial Growth

This paper studies stochastic inventory problems with unbounded Markovian demands, ordering costs that are lower semicontinuous, and inventory/backlog (or surplus) costs that are lower semicontinuous with polynomial growth. Finite-horizon problems, stationary and nonstationary discounted-cost infinite-horizon problems, and stationary long-run average-cost problems are addressed. Existence of optimal Markov or feedback policies is established. Furthermore, optimality of (s, S)-type policies is proved when, in addition, the ordering cost consists of fixed and proportional cost components and the surplus cost is convex.     

Myopic control of stochastic inventories with intermittent updates: continuous <Emphasis Type="Italic">versus</Emphasis> periodic replenishment

A manufacturer who is responsible for supplying a retailer with a single product is considered. The retailer sells the product in response to stochastic demand and provides the manufacturer with periodic updates about his inventories. Replenishing the retailer's inventory under two myopic base-stock policies is addressed. These policies, referred to as vendor managed inventory, represent a relatively new approach to allocating responsibility in the replenishment process. Specifically, the manufacturer, who is responsible for the retailer's inventories, can replenish them either continuously at any point in time or periodically, at one point in time for each period. The myopic replenishment policies that are considered are of a base-stock type. It is shown that the selected policies become optimal as the number of review periods tends to infinity. Furthermore, the two replenishment alternatives are compared in terms of both base-stock levels and expected costs, including those for inventory holding/shortage and transportation costs. Although continuous rather than periodic replenishment is evidently more expensive in terms of transportation costs, it is shown that even when the transportation cost constitutes more than 55% of the total average cost, it may still be preferable to replenish continuously rather than periodically.     

Economic order quantity under advance payment

Though advance payment is widely used in practice, its influences on buyer’s inventory policy are rarely discussed. This paper investigates the buyer’s inventory policy under advance payment, including all payment in advance and partial-advanced–partial-delayed payment. The buyer’s ordering policy is derived by minimizing his total inventory costs including inventory holding cost, ordering cost, and interest cost caused by advance payment or delayed payment. The conclusions show that when all the payment is paid in advance, the buyer’s optimal replenishment cycle is influenced only by the price discount associated with advance payment, and the length of advance payment has no effect. For the partial-advanced–partial-delayed payment case, the buyer’s replenishment cycle is also not influenced by the length of advance period. However, in this situation, the delayed period and the price discount may have impacts on the inventory policy. We also use discounted cash flow (DCF) model to derive the buyer’s replenishment cycle and show that the replenishment cycle is negatively related to the length of advance period. Numerical examples are presented to illustrate the results.     

The optimality of myopic stocking policies for systems with decreasing purchasing prices

This paper studies the stocking/replenishment decisions for inventory systems where the purchasing price of an item decreases overtime. In a periodic review setting with stochastic demands, we model the purchasing prices of successive periods as a stochastic and decreasing sequence. To minimize the expected total discounted costs (purchasing, inventory holding and shortage penalty) for systems with backlogging and lost sales, we derive conditions, regarding the cost parameters, under which myopic stocking policies are optimal.     

Managing Dynamic Inventory Systems with Product Returns: A Markov Decision Process

This paper presents a Markov decision process for managing inventory systems with Markovian customer demand and Markovian product returns. Employing functional analysis, we prove the existence of the optimal replenishment policies for the discounted-cost and average-cost problems when demand, returns, and cost functions are of polynomial growth. Our model generalizes literature results by integrating Markovian demand, Markovian returns, and positive replenishment lead times. In particular, the optimality of the reorder point, order-up-to policies is proved when the order cost consists of fixed setup and proportional cost components and the inventory surplus cost is convex. We then make model extensions to include different cost components and to differentiate returned products from new ones. Finally, we derive managerial insights for running integrated closed-loop supply chains. At the aggregate level, returns reduce effective demand while many structural characteristics of inventory models are intact. A simple heuristic for managing systems with returns is to still utilize literature results without returns, but effective demand is lower than customer demand.     

Minimizing the Total Cost in an Integrated Vendor—Managed Inventory System

In this paper we consider a complex production-distribution system, where a facility produces (or orders from an external supplier) several items which are distributed to a set of retailers by a fleet of vehicles. We consider Vendor-Managed Inventory (VMI) policies, in which the facility knows the inventory levels of the retailers and takes care of their replenishment policies. The production (or ordering) policy, the retailers replenishment policies and the transportation policy have to be determined so as to minimize the total system cost. The cost includes the fixed and variable production costs at the facility, the inventory costs at the facility and at the retailers and the transportation costs, that is the fixed costs of the vehicles and the traveling costs. We study two different types of VMI policies: The order-up-to level policy, in which the order-up-to level quantity is shipped to each retailer whenever served (i.e. the quantity delivered to each retailer is such that the maximum level of the inventory at the retailer is reached) and the fill-fill-dump policy, in which the order-up-to level quantity is shipped to all but the last retailer on each delivery route, while the quantity delivered to the last retailer is the minimum between the order-up-to level quantity and the residual transportation capacity of the vehicle. We propose two different decompositions of the problem and optimal or heuristic procedures for the solution of the subproblems. We show that, for reasonable initial values of the variables, the order in which the subproblems are solved does not influence the final solution. We will first solve the distribution subproblem and then the production subproblem. The computational results show that the fill-fill-dump policy reduces the average cost with respect to the order-up-to level policy and that one of the decompositions is more effective. Moreover, we compare the VMI policies with the more traditional Retailer-Managed Inventory (RMI) policy and show that the VMI policies significantly reduce the average cost with respect to the RMI policy.     

Multi-item inventory control with full truckloads: A comparison of aggregate and individual order triggering

In this paper, we consider the stochastic joint replenishment problem in an environment where transportation costs are dominant and full truckloads or full container loads are required. One replenishment policy, taking into account capacity restrictions of the total order volume, is the so-called QS policy, where replenishment orders are placed to raise the individual inventory positions of all items to their order-up-to levels, whenever the aggregate inventory position drops below the reorder level. We first provide a method to compute the policy parameters of a QS policy such that item target service levels can be met, under the assumption that demand can be modeled as a compound renewal process. The approximation formulas are based on renewal theory and are tested in a simulation study which reveals good performance. Second, we compare the QS policy with a simple allocation policy where replenishment orders are triggered by the individual inventory positions of the items. At the moment when an individual inventory position drops below its item reorder level, a replenishment order is triggered and the total vehicle capacity is allocated to all items such that the expected elapsed time before the next replenishment order is maximized. In an extensive simulation study it is illustrated that the QS policy outperforms this allocation policy since it results in lower inventory levels for the same service level. Although both policies lead to similar performance if items are identical, it can differ substantially if the item characteristics vary.     

Modelling and computing (Rn, Sn) policies for inventory systems with non-stationary stochastic demand

This paper addresses the single-item, non-stationary stochastic demand inventory control problem under the non-stationary (R, S) policy. In non-stationary (R, S) policies two sets of control parameters—the review intervals, which are not necessarily equal, and the order-up-to-levels for replenishment periods—are fixed at the beginning of the planning horizon to minimize the expected total cost. It is assumed that the total cost is comprised of fixed ordering costs and proportional direct item, inventory holding and shortage costs. With the common assumption that the actual demand per period is a normally distributed random variable about some forecast value, a certainty equivalent mixed integer linear programming model is developed for computing policy parameters. The model is obtained by means of a piecewise linear approximation to the non-linear terms in the cost function. Numerical examples are provided.     

An integrated inventory–transportation system with modified periodic policy for multiple products

Efficient management of a distribution system requires an integrated approach towards various logistical functions. In particular, the fundamental areas of inventory control and transportation planning need to be closely coordinated. Our model deals with an inbound material-collection problem. An integrated inventory–transportation system is developed with a modified periodic-review inventory policy and a travelling-salesman component. This is a multi-item joint replenishment problem, in a stochastic setting, with simultaneous decisions made on inventory and transportation policies. We propose a heuristic decomposition method to solve the problem, minimizing the long-run total average costs (major- and minor-ordering, holding, backlogging, stopover and travel). The decomposition algorithm works by using separate calculations for inventory and routing decisions, and then coordinating them appropriately. A lower bound is constructed and computational experience is reported.     

A Backlog Inventory Model During Restricted Sale Periods

Optimal purchasing policies are derived for a continuous review inventory system in which backlogs are permissible and temporary price discounts are available. This paper derives such policies assuming that the sale price is not available at the regularly-scheduled replenishment time. Results indicate that, under specific conditions, maximum positive cost savings are generated when a special order is placed at the end of the sale period.     

A multi-product continuous review inventory system with stochastic demand,backorders, and a budget constraint

Common characteristics of inventory systems include uncertain demand and restrictions such as budgetary or storage space constraints. Several authors have examined budget constrained multi-item stochastic inventory systems controlled by continuous review policies without considering marginal shortage costs. Existing models assume that purchasing costs are paid at the time an order is placed, which is not always the case since in some systems purchasing costs are paid when orders arrive. In the latter case the maximum investment in inventory is random since the inventory level when an order arrives is a random variable. Hence payment of purchasing costs on delivery yields a stochastic budget constraint for inventory. This paper models a multi-item stochastic inventory system with backordered shortages when estimation of marginal backorder cost is available, and payment is due upon order arrival. The budget constraint can easily be converted into a storage constraint.     

Leadtime management in a periodic-review inventory system: A state-dependent base-stock policy

We examine a single-item, periodic-review inventory system with stochastic leadtimes, in which a replenishment order is delivered immediately or one period later, depending probabilistically on costly effort. The objective is to determine a joint inventory policy and effort-choice strategy that minimizes the expected total costs. Our analytical and computational analysis suggests that (i) a state-dependent base-stock policy is optimal, (ii) the optimal effort strategy is such that the marginal cost of effort is equal to the value of immediate delivery, and (iii) the cost impact of leadtime reduction can be very large. We also provide some counter-intuitive results, compared with the traditional multi-period newsvendor model.     

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.     

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 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.     

A stochastic program based lower bound for assemble-to-order inventory systems

In this paper we introduce a multi-stage stochastic program that provides a lower bound on the long-run average inventory cost of a general class of assemble-to-order (ATO) inventory systems. The stochastic program also motivates a replenishment policy for these systems. Our lower bound generalizes a previous result of Do?ru et al. (2010) [3] for systems with identical component replenishment lead times to those with general deterministic lead times. We provide a set of sufficient conditions under which our replenishment policy, coupled with an allocation policy, attains the lower bound (and is hence optimal). We show that these sufficient conditions hold for two examples, a single product system and a special case of the generalized W model.     

具有Weibull生存死亡特征的冷链品库存补货定价策略研究

建立了无限期内冷链品具有Weibull生存死亡特征、随机需求且受售价影响的库存补货定价模型,其中售价是连续变化的,需求率是售价的指数函数,变质率服从的三参数Weibull分布,提前期固定。系统以利润最大化为目标函数,在(r,Q)库存策略下,建立库存模型,采用直接法,对模型近似求解,得到最优补货定价策略。利用Matlab进行算例模拟和灵敏度分析发现:补货提前期和单位仓储成本对补货定价策略影响较大,二者增大会导致系统利润降低;单位处理成本的增加,在一定程度上使得系统降低最优补货量,使系统利润增加;保鲜期固定的前提下,受冷链品的流动环境因子和存储环境因子影响的变质率对补货定价策略影响较大,它的增大会使系统利润降低。这些发现能够帮助优化系统模型,对现实问题具有一定的指导意义。     

Deterministic lot-size inventory models with shortages and deterioration for fluctuating demand

We establish various inventory replenishment policies. We then analytically identify the best alternative among them based on the minimum total relevant costs. Finally, we prove that the relevant cost is convex with the number of replenishments. Consequently, the search for the optimal replenishment number is reduced to finding a local minimum.     

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.