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.     

Sourcing from suppliers with random yield for price-dependent demand

We study a multi-period inventory planning problem. In each period, the firm under consideration can source from two possibly unreliable suppliers for a price-dependent demand. Our analysis suggests that the optimal procurement policy is neither a simple reorder-point policy nor a complex one without any structure, as previous studies suggest. Instead, we prove the existence of a reorder point for each supplier. No order is placed to that supplier for any inventory level above the reorder point and a positive order is issued to that supplier for almost every inventory level below the reorder point. We characterize conditions under which the optimal policy reveals monotone response to changes in the inventory level. Furthermore, two special cases of our model are examined in detail to demonstrate how our analysis generalizes a number of well-known results in the literature.     

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.     

An economic order quantity with imperfect quality and quantity discounts

Previous studies in the issue of inventory models with imperfect quality assumed the defectives could be sold in a batch by the end of the inspection process and the manufacturing systems were push systems. However, the above assumptions may not be true in the pull system in which buyer is powerful. Therefore, in this paper, we develop a new inventory model for items with imperfect quality and quantity discounts where buyer has exerted power over its supplier. Based on the concept of powerful buyer, there are three considerations included in this new model: (1) the order quantity is manufactured at one setup and is shipped over multiple deliveries, (2) the defectives are screened out by a 100% inspection for each shipment but sold in a batch by the end of inspection at the last shipment of each cycle, and (3) the supplier offers quantity discounts to response the request of the powerful buyer. Further, an algorithm is developed to help the powerful buyer to determine the optimal order policy accurately and quickly. Two numerical examples are available in this paper to illustrate the proposed model and algorithm. Besides, based on the numerical examples, a sensitivity analysis is made to investigate the effects of four important parameters (the inspection rate, the defective rate, the receiving cost, and the ordering cost) on the optimal solution.     

Monotonicity properties of a class of stochastic inventory systems

We consider inventory systems which are governed by an (r,q) or (r,nq) policy. We derive general conditions for monotonicity of the three optimal policy parameters, i.e., the optimal reorder level, order quantity and order-up-to level, as well as the optimal cost value, as a function of the various model primitives, be it cost parameters or complete cost rate functions or characteristics of the demand and leadtime processes. These results are obtained as corollaries from a few general theorems, with separate treatment given to the case where the policy parameters are continuous variables and that where they need to be restricted to integer values. The results are applied both to standard inventory models and to those with general shelf age and delay dependent inventory costs.     

Optimal ordering policies in inventory systems with random demand and random deal offerings

In this paper, we determine the optimal order policies for a firm facing random demand and random deal offerings. In a periodic review setting, a firm may first place an order at the regular price. Later in the period, if a price promotion is offered by the supplier (with a certain probability), the firm may decide to place another order. We consider two models in the paper. In the first model, the firm does not share the cost savings (due to the promotion offered by the supplier) with its own customers, i.e. its demand distribution remains fixed. In the second model, the cost savings are shared with the final customers. As a result, the demand distribution shifts to the right. For both the models, in a dynamic finite-horizon problem, the order policy structure is divided into three regions and is as follows. If the initial inventory level for the firm exceeds a certain threshold level, it is optimal not to order anything. If it is in the medium range, it is optimal to wait for the promotion and order only if it is offered. The order quantity when the promotion is offered has an ‘order up to’ policy structure. Finally, if the inventory level is below another threshold, it is optimal to place an order at the regular price, and to place a second order if the promotion is offered. The low initial inventory level makes it risky to just wait for the promotion to be offered. The sum of the order quantities in this case has an ‘order up to’ structure. Finally, we model the supplier's problem as a Stackelberg game and discuss the motivation for the supplier to offer a promotion for the case of uniform demand distribution for the firm. In the first model (when the firm does not share the cost savings with its customers), we show that it is rarely optimal for the supplier to offer a promotion. In the second model, the supplier may offer a promotion depending on the price elasticity of the product.     

Modeling and analysis for determining optimal suppliers under stochastic lead times

The policy of simultaneously splitting replenishment orders among several suppliers has received considerable attention in the last few years and continues to attract the attention of researchers. In this paper, we develop a mathematical model which considers multiple-supplier single-item inventory systems. The item acquisition lead times of suppliers are random variables. Backorder is allowed and shortage cost is charged based on not only per unit in shortage but also per time unit. Continuous review (s,Q)$(s\text{,}Q)$ policy has been assumed. When the inventory level depletes to a reorder level, the total order is split among n suppliers. Since the suppliers have different characteristics, the quantity ordered to different suppliers may be different. The problem is to determine the reorder level and quantity ordered to each supplier so that the expected total cost per time unit, including ordering cost, procurement cost, inventory holding cost, and shortage cost, is minimized. We also conduct extensive numerical experiments to show the advantages of our model compared with the models in the literature. According to our extensive experiments, the model developed in this paper is the best model in the literature which considers order splitting for n-supplier inventory systems since it is the nearest model to the real inventory system.     

The scheduling of deliveries in a production-distribution system with multiple buyers

In a real production and distribution business environment with one supplier and multiple heterogeneous buyers, the differences in buyers’ ordering cycles have influence on production arrangements. Consequently, the average inventory level (AIL) at the supplier’s end is affected by both the production policy and the ordering policy, typically by the scheduling of deliveries. Consequently, the average inventory holding cost is most deeply affected. In this paper, it is proposed that the scheduling of deliveries be formulated as a decision problem to determine the time point at which deliveries are made to buyers in order to minimize the supplier’s average inventory. A formulation of the average inventory level (AIL) in a production cycle at the supplier’s end using a lot-for-lot policy is developed. Under the lot-for-lot policy, the scheduling of deliveries (SP) is formulated as a nonlinear programming model used to determine the first delivery point for each buyer with an objective to minimize the sum of the product of the individual demand quantity and the first delivery time for each buyer. Thus, the SP model determines not only the sequence of the first deliveries to individual buyers, but also the time when the deliveries are made. An iterative heuristic procedure (IHP) is developed to solve the SP model assuming a given sequence of buyers. Six sequence rules are considered and evaluated via simulation.     

在中心仓库有损失销售的二级分布库存管理模型的研究

通过对一个中心仓库和N个零售商的二级分布库存系统进行分析,采用基本(S-1,S)库存策略,综合运用了排队法和M ETR IC近似法,提出了一种在中心仓库有损失销售的二级库存管理模型,该模型描述在中心仓库缺货情况下,多数零售商不等待延期付货,而直接与供应商订货,导致中心仓库就会因损失销售而产生机会成本.该模型可达到二级分布库存系统的总成本最小.     

Optimal policy for brownian inventory models with general convex inventory cost

We study an inventory system in which products are ordered from outside to meet demands, and the cumulative demand is governed by a Brownian motion. Excessive demand is backlogged. We suppose that the shortage and holding costs associated with the inventory are given by a general convex function. The product ordering from outside incurs a linear ordering cost and a setup fee. There is a constant leadtime when placing an order. The optimal policy is established so as to minimize the discounted cost including the inventory cost and ordering cost.     

Joint pricing and inventory replenishment decisions with returns and expediting

We study a single-item periodic-review model for the joint pricing and inventory replenishment problem with returns and expediting. Demand in consecutive periods are independent random variables and their distributions are price sensitive. At the end of each period, after the demand is realized, a buyer can return excess stocks to a supplier. Or, if there are stockouts, the buyer can place an expediting order at the supplier to reduce the amount of shortage. Unfilled demands are fully backlogged. We characterize the optimal dynamic policy that determines the pricing, inventory replenishment, and adjustment decisions in each period so that the total expected discounted profit is maximized. For a very general stochastic demand function, we can show that the optimal replenishment policy is a modified base-stock policy, the optimal pricing policy is a modified base-stock-list-price policy, and the optimal policy for inventory adjustment follows a dual-threshold policy. We further study the operational effect of returns and expediting. Analytical and numerical results demonstrate that returns and expediting lead to a significant profit increase in a number of situations, including limited supply capacity, sufficient flexibility of the expediting order, high demand uncertainty, and a price-sensitive market.     

A dynamic inventory model with supplier selection in a serial supply chain structure

Considering the inherent connection between supplier selection and inventory management in supply chain networks, this article presents a multi-period inventory lot-sizing model for a single product in a serial supply chain, where raw materials are purchased from multiple suppliers at the first stage and external demand occurs at the last stage. The demand is known and may change from period to period. The stages of this production–distribution serial structure correspond to inventory locations. The first two stages stand for storage areas for raw materials and finished products in a manufacturing facility, and the remaining stages symbolize distribution centers or warehouses that take the product closer to customers. The problem is modeled as a time-expanded transshipment network, which is defined by the nodes and arcs that can be reached by feasible material flows. A mixed integer nonlinear programming model is developed to determine an optimal inventory policy that coordinates the transfer of materials between consecutive stages of the supply chain from period to period while properly placing purchasing orders to selected suppliers and satisfying customer demand on time. The proposed model minimizes the total variable cost, including purchasing, production, inventory, and transportation costs. The model can be linearized for certain types of cost structures. In addition, two continuous and concave approximations of the transportation cost function are provided to simplify the model and reduce its computational time.     

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.     

Evaluating shortfall distributions in periodic inventory systems with stochastic endogenous demands and lead-times

This paper addresses a multi-period production/inventory problem with two suppliers, where demand sizes and supplier lead time are stochastic and correlated. A discrete time, single item inventory system is considered, where inventory levels are reviewed periodically and managed using a base-stock policy. At the end of each period, a replenishment order is placed, which enters a queue at the buffer stage and is consequently forwarded to the first available supplier. We present a mathematical model of this inventory system and determine optimal safety stock levels for it, in closed form, using matrix analytic techniques and the properties of phase type distributions. To account for the effect of order crossovers, which occur whenever replenishment orders do not arrive in the sequence in which they were placed, the inventory shortfall distribution is analyzed. Finally, a set of numerical experiments with a system with two suppliers is presented, where the proposed model is compared to other existing models.     

Analysis of a fuzzy economic order quantity model for deteriorating items under retailer partial trade credit financing in a supply chain

This paper investigates the economic order quantity (EOQ) — based inventory model for a retailer under two levels of trade credit to reflect the supply chain management situation in the fuzzy sense. It is assumed that the retailer maintains a powerful position and can obtain the full trade credit offered by the supplier yet the retailer just offers a partial trade credit to customers. The demand rate, holding cost, ordering cost, purchasing cost and selling price are taken as fuzzy numbers. Under these conditions, the retailer can obtain the most benefits. Study also investigates the retailer’s inventory policy for deteriorating items in a supply chain management situation as a cost minimization problem in the fuzzy sense. The annual total variable cost for the retailer in fuzzy sense is defuzzified using Graded Mean Integration Representation method. Then the present study shows that the defuzzified annual total variable cost for the retailer is convex, that is, a unique solution exists. Mathematical theorems and algorithms are developed to efficiently determine the optimal inventory policy for the retailer. Numerical examples are given to illustrate the theorems and the algorithms. Finally, the results in this paper generalize some already published results in the crisp sense.     

Optimal solutions for the economic lot-sizing problem with multiple suppliers and cost structures

This paper considers a multi-supplier economic lot-sizing problem in which the retailer replenishes his inventory from several suppliers. Each supplier is characterized by one of three types of order cost structures: incremental quantity discount cost structure, multiple set-ups cost structure and all-unit quantity discount cost structure. The problem is challenging due to the mix of different cost structures. For all cases of the problem where each supplier is characterized by one of the first two cost structures, some optimality properties are proposed and optimal algorithms based on dynamic programming are designed. For the case where all suppliers are characterized by all-unit quantity discount cost structures, it is hard to design a polynomial time algorithm by the analyzed optimal properties. However, it is proved that one of its special cases can be solved in polynomial time.     

A Production Lot-Size Model for Perishable Items Under Two Level Trade Credit Policy for a Retailer with a Powerful Position in a Supply Chain System

This paper investigates a production lot-size inventory model for perishable items under two levels of trade credit for a retailer to reflect the supply chain management situation. We assume that the retailer maintains a powerful position and can obtain full trade credit offered by supplier yet retailer just offers the partial trade credit to customers. Under these conditions, retailer can obtain the most benefits. Then, we investigate the retailer’s inventory policy as a cost minimization problem to determine the retailer’s inventory policy. A rigorous mathematical analysis is used to prove that the annual total variable cost for the retailer is convex, that is, unique and global-optimal solution exists. Mathematical theorems are developed to efficiently determine the optimal ordering policies for the retailer. The results in this paper generalize some already published results. Finally, numerical examples are given to illustrate the theorems and obtain a lot of managerial phenomena.     

Inventory problems with perishable items: Fixed lifetimes and backlogging

Our model deals with a single-product and a single-stock location with Poisson demand. The replenishment leadtime from the external supplier is fixed. The lifetime of the product is also fixed, and aging is assumed to begin when the order is placed. When the age of a unit has reached its lifetime, the unit is useless and thus discarded from the system. The replenishment policy is assumed to be an order-up-to S-policy. Demand that cannot be met immediately is backordered. We consider three different cases where the service requirements are represented by: (1) backorder costs per unit, (2) a service level constraint, (3) backorder costs per unit and time unit. Cases 1 and 2 are solved exactly, while an approximation is developed for case 3. We show how the results from an earlier paper assuming lost sales can be used to solve the considered problems. Our results are compared to the results in a related paper considering (Q, r)-policies.     

Inventory control with an order-time constraint: optimality,uniqueness and significance

This paper analyzes a stochastic inventory problem with an order-time constraint that restricts the times at which a manufacturer places new orders to a supplier. This constraint stems from the limited upstream capacity in a supply chain, such as production capacity at a supplier or transportation capacity between a supplier and a manufacturer. Consideration of limited upstream capacity extends the classical inventory literature that unrealistically assumes infinite supplier/transporter capacity. But this consideration increases the complexity of the problem. We study the constraint under a Poisson demand process and allow for a fixed ordering cost. In presence of the constraint, we establish the optimality of an (s,S) policy under both the discounted and average cost objectives. Under the average cost objective, we show the uniqueness of the order-up-to level S. We numerically compare our model with the classical unconstrained model. We report significant savings in costs that can be achieved by using our model when the order time is constrained.     

Determining optimal order splitting and reorder level for N-supplier inventory systems

This paper considers multiple-supplier single-item inventory systems, where the item acquisition lead times of suppliers and demand arrival are random, and backorder is allowed. The acquisition takes place when the inventory level depletes to a reorder level, and the order is split among multiple suppliers. The acquisition lead times may have different distributions, the unit purchasing prices from suppliers may be different, and thus the order quantities for different suppliers may be different. The problem is to determine the reorder level and order quantity for each supplier so that the expected total cost per unit time, consisting of the fixed ordering cost, procurement cost, inventory holding cost and shortage cost, is minimized. We develop a mathematical model describing the system in detail. We also conduct extensive numerical experiments to analyze the advantages and distinct characteristics of multiple-supplier systems.