Dynamic pricing inventory control under fixed cost and lost sales

This paper studies a periodic review pricing and inventory replenishment problem which encounters stochastic demands in multiple periods. In many inventory control problems, the unsatisfied demand is traditionally assumed to be backlogged but in this paper is assumed to be lost. In many practical problems, a consumer who could not buy what he/she wants in one store is not willing to wait until that store restocks it but tries to buy alternatives in other stores. Also, in this paper, the random variable for the demand function is assumed to be general, which means that any probability function for the random variable can be applied to our result. Cost terms consist of the holding cost by the leftover, the shortage cost by lost sales, and the strictly positive fixed ordering cost. The objective of this paper is to dynamically and simultaneously decide the optimal selling price and replenishment in each period by maximizing the expected profit over the finite selling horizon. We show that, under the general assumption on the random variable for the demand, the objective function is K$K$-concave, an (s,S)$(s\text{,}S)$ policy is optimal for the replenishment and the optimal price is determined based on the inventory level after the replenishment in each period.     

Optimal inventory replenishment policy for a queueing system with finite waiting room capacity

We treat an inventory control problem in a facility that provides a single type of service for customers. Items used in service are supplied by an outside supplier. To incorporate lost sales due to service delay into the inventory control, we model a queueing system with finite waiting room and non-instantaneous replenishment process and examine the impact of finite buffer on replenishment policies. Employing a Markov decision process theory, we characterize the optimal replenishment policy as a monotonic threshold function of reorder point under the discounted cost criterion. We present a simple procedure that jointly finds optimal buffer size and order quantity.     

Reducing lost-sales rate on the stochastic inventory model with defective goods for the mixtures of distributions

In this paper, we assume that the demands of different customers are not identical in the lead time. Thus, we investigate a continuous review inventory model involving controllable lead time and a random number of defective goods in buyer’s arriving order lot with partial lost sales for the mixtures of distributions of the lead time demand to accommodate more practical features of the real inventory systems. Moreover, we analyze the effects of increasing investment to reduce the lost sales rate when the order quantity, reorder point, lost sales rate and lead time are treated as decision variables. In our studies, we first assume that the lead time demand follows the mixture of normal distributions, and then relax the assumption about the form of the mixture of distribution functions of the lead time demand and apply the minimax distribution free procedure to solve the problem. By analyzing the total expected cost function, we develop an algorithm to obtain the optimal ordering policy and the optimal investment strategy for each case. Finally, we provide numerical examples to illustrate the results.     

Inventory management for dual sales channels with inventory-level-dependent demand

This paper studies the inventory management problem of dual channels operated by one vendor. Demands of dual channels are inventory-level-dependent. We propose a multi-period stochastic dynamic programming model which shows that under mild conditions, the myopic inventory policy is optimal for the infinite horizon problem. To investigate the importance of capturing demand dependency on inventory levels, we consider a heuristic where the vendor ignores demand dependency on inventory levels, and compare the optimal inventory levels with those recommended by the heuristic. Through numerical examples, we show that the vendor may order less for dual channels than those recommended by the heuristic, and the difference between the inventory levels in the two cases can be so large that the demand dependency on inventory levels cannot be neglected. In the end, we numerically examine the impact of different ways to treat unmet demand and obtain some managerial insights.     

The stochastic EOQ model with random sales price

The article deals with a stochastic economic order quantity (EOQ) model over a finite time horizon where uniform demand over the replenishment period is price dependent. The selling price is assumed to be a random variable that follows a probability density function. As demand is probabilistic, stock out situation may occur. Based on the partial backlogging and lost sale cases during stock out period, the author develops the criterion for the optimal solution for the replenishment size such that the integrated expected profit is maximized. Moreover, the article suggests a new function regarding price dependent demand. Finally, numerical examples and its sensitivity analysis of key parameters are given to illustrate the proposed model.     

Optimal EOQ Models for Deteriorating Items with Time-Varying Demand

In this paper, optimal inventory lot-sizing models are developed for deteriorating items with general continuous time-varying demand over a finite planning horizon and under three replenishment policies. The deterioration rate is assumed to be a constant fraction of the on-hand inventory. Shortages are permitted and are completely backordered. The proposed solution procedures are shown to generate global minimum replenishment schedules for both general increasing and decreasing demand patterns. An extensive empirical comparison using randomly generated linear and exponential demands revealed that the replenishment policy which starts with shortages in every cycle is the least cost policy and the replenishment policy which prohibits shortages in the last cycle exhibited the best service level effectiveness. An optimal procedure for the same problem with trended inventory subject to a single constraint on the minimum service level (maximum fraction of time the inventory system is out of stock during the planning horizon) is also proposed in this paper.     

On two-echelon inventory systems with Poisson demand and lost sales

We consider a two-echelon, continuous review inventory system under Poisson demand and a one-for-one replenishment policy. Demand is lost if no items are available at the local warehouse, the central depot, or in the pipeline in between. We give a simple, fast and accurate approach to approximate the service levels in this system. In contrast to other methods, we do not need an iterative analysis scheme. Our method works very well for a broad set of cases, with deviations to simulation below 0.1% on average and below 0.36% for 95% of all test instances.     

Inventory and sales effort management under unobservable lost sales

A retailer needs to make decisions regarding how much to order and how much sales effort to exert in an environment with uncertain demand. One intrinsic complexity in a typical retail environment is caused by the fact that the retailer can obtain information about demand only based on sales, as demand itself is unobservable. Taking a Bayesian approach, Lariviere and Porteus (1999) show that in such a setting a retailer should stock more to increase the probability of an exact demand observation. In this article, we extend their work by allowing the retailer to control both the stocking quantity and sales effort, which can be used to affect demand. We show that their insights with respect to information stalking carry over to this setting. In addition, our model allows gaining a better understanding of optimal sales effort strategies. We find that demand management has a dual role in supporting information gathering: while at the beginning of a product life cycle it is optimal to support learning effects by sharply reducing sales effort, at later stages of the product life cycle an aggressive strategy of increased promotional activities can be used to harvest the information gathered in earlier periods.     

An inventory control problem for deteriorating items with back-ordering and financial considerations

This paper investigates the effects of time value of money and inflation on the optimal ordering policy in an inventory control system. We proposed an economic order quantity model to manage a perishable item over the finite horizon planning under which back-ordering and delayed payment are assumed. The demand and deterioration rates are constant. The present value of total cost during the planning horizon in this inventory system is modeled first, then a three phases solution procedure is proposed to derive the optimal order and shortage quantities, and the number of replenishment during the planning horizon. Finally, the proposed model is illustrated through numerical examples and the sensitivity analysis is reported to find some managerial insights.     

Optimal pricing and inventory policies with reliable and random-yield suppliers: characterization and comparison

In this paper, we study the joint pricing and inventory replenishment problem for a periodic-review inventory system with random demand and dual suppliers, one of the suppliers is reliable but more expensive, the other supplier is less expensive but is unreliable with random yield. We characterize the firm’s optimal policies that simultaneously determine the optimal ordering and pricing decisions in each period over a finite planning horizon, and investigate the impacts of supply source diversification and supplier reliability on the firm and on its customers. We show that having source diversification or higher reliability of suppliers not only increases the firm’s expected profit, but also results in a lower optimal selling price, thus they benefit both the firm and its customers.     

Profit maximization in an inventory system with time-varying demand,partial backordering and discrete inventory cycle

In this paper, an inventory problem where the inventory cycle must be an integer multiple of a known basic period is considered. Furthermore, the demand rate in each basic period is a power time-dependent function. Shortages are allowed but, taking necessities or interests of the customers into account, only a fixed proportion of the demand during the stock-out period is satisfied with the arrival of the next replenishment. The costs related to the management of the inventory system are the ordering cost, the purchasing cost, the holding cost, the backordering cost and the lost sale cost. The problem is to determine the best inventory policy that maximizes the profit per unit time, which is the difference between the income obtained from the sales of the product and the sum of the previous costs. The modeling of the inventory problem leads to an integer nonlinear mathematical programming problem. To solve this problem, a new and efficient algorithm to calculate the optimal inventory cycle and the economic order quantity is proposed. Numerical examples are presented to illustrate how the algorithm works to determine the best inventory policies. A sensitivity analysis of the optimal policy with respect to some parameters of the inventory system is developed. Finally, conclusions and suggestions for future research lines are given.

     

Demand seasonality in retail inventory management

We investigate the value of accounting for demand seasonality in inventory control. Our problem is motivated by discussions with retailers who admitted to not taking perceived seasonality patterns into account in their replenishment systems. We consider a single-location, single-item periodic review lost sales inventory problem with seasonal demand in a retail environment. Customer demand has seasonality with a known season length, the lead time is shorter than the review period and orders are placed as multiples of a fixed batch size. The cost structure comprises of a fixed cost per order, a cost per batch, and a unit variable cost to model retail handling costs. We consider four different settings which differ in the degree of demand seasonality that is incorporated in the model: with or without within-review period variations and with or without across-review periods variations. In each case, we calculate the policy which minimizes the long-run average cost and compute the optimality gaps of the policies which ignore part or all demand seasonality. We find that not accounting for demand seasonality can lead to substantial optimality gaps, yet incorporating only some form of demand seasonality does not always lead to cost savings. We apply the problem to a real life setting, using Point-of-Sales data from a European retailer. We show that a simple distinction between weekday and weekend sales can lead to major cost reductions without greatly increasing the complexity of the retailer’s automatic store ordering system. Our analysis provides valuable insights on the tradeoff between the complexity of the automatic store ordering system and the benefits of incorporating demand seasonality.     

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     

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.     

Optimal selling price and lotsize with time varying deterioration and partial backlogging

The article deals with an EOQ (economic order quantity) model over an infinite time horizon for perishable items where demand is price dependent and partial backorder is permitted. The rate of deterioration is taken to be time proportional and it is assumed that shortage occurs at starting of the inventory cycle. Based on the partial backlogging and lost sale cases, the author develops the criterion for the optimal solution for the replenishment schedule, and proves the optimal ordering policy is unique. Moreover, the article suggests to new functions regarding price-dependent demand and time varying deterioration rate. Finally, numerical examples are illustrated to test the model in various issues.     

Production-inventory control policy under warm/cold state-dependent fixed costs and stochastic demand: partial characterization and heuristics

The motivation for our study comes from some production and inventory systems in which ordering/producing quantities that exceed certain thresholds in a given period might eliminate some setup activities in the next period. Many examples of such systems have been discussed in prior research but the analysis has been limited to production settings under deterministic demand. In this paper, we consider a periodic-review production-inventory model under stochastic demand and incorporate the following fixed-cost structure into our analysis. When the order quantity in a given period exceeds a specified threshold value, the system is assumed to be in a “warm” state and no fixed cost is incurred in the next period regardless of the order quantity; otherwise the system state is considered “cold” and a positive fixed cost is required to place an order. Assuming that the unsatisfied demand is lost, we develop a dynamic programming formulation of the problem and utilize the concepts of quasi-K-convexity and non-K-decreasing to show some structural results on the optimal cost-to-go functions. This analysis enables us to derive a partial characterization of the optimal policy under the assumption that the demands follow a Pólya or uniform distribution. The optimal policy is defined over multiple decision regions for each system state. We develop heuristic policies that are aimed to address the partially characterized decisions, simplify the ordering policy, and save computational efforts in implementation. The numerical experiments conducted on a large set of test instances including uniform, normal and Poisson demand distributions show that a heuristic policy that is inspired by the optimal policy is able to find the optimal solution in almost all instances, and that a so-called generalized base-stock policy provides quite satisfactory results under reasonable computational efforts. We use our numerical examples to generate insights on the impact of problem parameters. Finally, we extend our analysis into the infinite horizon setting and show that the structure of the optimal policy remains similar.     

The Trended Inventory Lot Sizing Problem with Shortages Under a New Replenishment Policy

In the past few years, considerable attention has been given to the inventory lot sizing problem with trended demand over a fixed horizon. The traditional replenishment policy is to avoid shortages in the last cycle. Each of the remaining cycles starts with a replenishment and inventory is held for a certain period which is followed by a period of shortages. A new replenishment policy is to start each cycle with shortages and after a period of shortages a replenishment should be made. In this paper, we show that this new type of replenishment policy is superior to the traditional one. We further propose four heuristic procedures that follow the new replenishment policy. These are the constant demand approximation method, the equal cycle length heuristic, the extended Silver approach, and the extended least cost solution procedure. We also examine the cost and computation time performances of these heuristic procedures through an empirical study. The number of test problems solved to optimality, average and maximum cost deviation from optimum were used as measures of cost performance. The results of the 10 000 test problems reveal that the extended least cost approach is most cost effective.     

需求依赖销售努力库存系统的随机比较

研究需求依赖销售努力库存系统中需求不确定性对系统最优订货量、利润和销售努力的影响.对一般需求模型给出期望利润关于订货量和努力水平为联合凹的充分条件,证明期望利润函数的超模性质.对加乘需求模型证明系统最优利润和最优努力水平都可由一类与需求分布有关的广义TTT变换来表示.通过引入定义在不同支撑分布集合上一阶、二阶和三阶随机占优,得到广义TTT变换之差与二阶和三阶随机占优之间的关系式,建立了比较库存系统最优利润或努力水平的理论基础.在一阶和二阶随机占优意义下对加乘需求模型得到比较系统最优利润和努力水平的充分条件或充分必要条件.进一步,证明存在一类需求分布当系统关键比(或市场价格)足够大时系统最优利润和努力水平随需求可变性的增加而增加.最后给出几个数值例子验证了研究结果.     

Periodic review lost-sales inventory models with compound Poisson demand and constant lead times of any length

In almost all literature on inventory models with lost sales and periodic reviews the lead time is assumed to be either an integer multiple of or less than the review period. In a lot of practical settings such restrictions are not satisfied. We develop new models allowing constant lead times of any length when demand is compound Poisson. Besides an optimal policy, we consider pure and restricted base-stock policies under new lead time and cost circumstances. Based on our numerical results we conclude that the latter policy, which imposes a restriction on the maximum order size, performs almost as well as the optimal policy. We also propose an approximation procedure to determine the base-stock levels for both policies with closed-form expressions.