Dynamic Pricing and Inventory Management with Regular and Expedited Supplies

We consider a periodic‐review inventory system with regular and expedited supply modes. The expedited supply is faster than the regular supply but incurs a higher cost. Demand for the product in each period is random and sensitive to its selling price. The firm determines its order quantity from each supply in each period as well as its selling price to maximize the expected total discounted profit over a finite or an infinite planning horizon. We show that, in each period if it is optimal to order from both supplies, the optimal inventory policy is determined by two state‐independent thresholds, one for each supply mode, and a list price is set for the product; if only the regular supply is used, the optimal policy is a state‐dependent base‐stock policy, that is, the optimal base‐stock level depends on the starting inventory level, and the optimal selling price is a markdown price that decreases with the starting inventory level. We further study the operational impact of such supply diversification and show that it increases the firm's expected profit, reduces the optimal safety‐stock levels, and lowers the optimal selling price. Thus that diversification is beneficial to both the firm and its customers. Building upon these results, we conduct a numerical study to assess and compare the respective benefit of dynamic pricing and supply diversification.     

消费者锚定效应下的动态定价与库存控制研究

在零售商-消费者两级供应链中, 零售商垄断销售市场, 研究基于消费者锚定效应的动态定价与库存问题。将消费者的锚定效应纳入决策模型中, 探讨消费者锚定的参考价格与产品的市场需求、价格的动态变化关系;利用动态规划理论建立了消费者锚定效应的动态定价与库存模型;在此基础上考虑利润折现和库存积压问题, 并分别构建了相应模型。数值分析表明, 消费者会根据市场价格的变化决定最佳的购买时机, 零售商的期望收益随着销售时间的延长而增加。     

Inventory Management for Customers with Alternative Lead Times

It is common for suppliers operating in batch‐production mode to deal with patient and impatient customers. This paper considers inventory models in which a supplier provides alternative lead times to its customers: a short or a long lead time. Orders from patient customers can be taken by the supplier and included in the next production cycle, while orders from impatient customers have to be satisfied from the on‐hand inventory. We denote the action to commit one unit of on‐hand inventory to patient or impatient customers as the inventory‐commitment decision, and the initial inventory stocking as the inventory‐replenishment decision. We first characterize the optimal inventory‐commitment policy as a threshold type, and then prove that the optimal inventory‐replenishment policy is a base‐stock type. Then, we extend our analysis to models to consider cases of a multi‐cycle setting, a supply‐capacity constraint, and the on‐line charged inventory‐holding cost. We also evaluate and compare the performances of the optimal inventory‐commitment policy and the inventory‐rationing policy. Finally, to further investigate the benefits and pitfalls of introducing an alternative lead‐time choice, we use the customer‐choice model to study the demand gains and losses, known as demand‐induction and demand‐cannibalization effects, respectively.     

Inventory‐Based Dynamic Pricing with Costly Price Adjustment

We study an average‐cost stochastic inventory control problem in which the firm can replenish inventory and adjust the price at anytime. We establish the optimality to change the price from low to high in each replenishment cycle as inventory is depleted. With costly price adjustment, scale economies of inventory replenishment are reflected in the cycle time instead of lot size—An increased fixed ordering cost leads to an extended replenishment cycle but does not necessarily increase the order quantity. A reduced marginal cost of ordering calls for an increased order quantity, as well as speeding up product selling within a cycle. We derive useful properties of the profit function that allows for reducing computational complexity of the problem. For systems requiring short replenishment cycles, the optimal solution can be easily computed by applying these properties. For systems requiring long replenishment cycles, we further consider a relaxed problem that is computational tractable. Under this relaxation, the sum of fixed ordering cost and price adjustment cost is equal to (greater than, less than) the total inventory holding cost within a replenishment cycle when the inventory holding cost is linear (convex, concave) in the stock level. Moreover, under the optimal solution, the time‐average profit is the same across all price segments when the inventory holding cost is accounted properly. Through a numerical study, we demonstrate that inventory‐based dynamic pricing can lead to significant profit improvement compared with static pricing and limited price adjustment can yield a benefit that is close to unlimited price adjustment. To be able to enjoy the benefit of dynamic pricing, however, it is important to appropriately choose inventory levels at which the price is revised.     

Joint Inventory and Pricing Coordination with Incomplete Demand Information

In retailing operations, retailers face the challenge of incomplete demand information. We develop a new concept named K‐approximate convexity, which is shown to be a generalization of K‐convexity, to address this challenge. This idea is applied to obtain a base‐stock list‐price policy for the joint inventory and pricing control problem with incomplete demand information and even non‐concave revenue function. A worst‐case performance bound of the policy is established. In a numerical study where demand is driven from real sales data, we find that the average gap between the profits of our proposed policy and the optimal policy is 0.27%, and the maximum gap is 4.6%.     

考虑成本和收益净现值的确定性存储决策模型研究

净现值是企业决策中需要考虑的重要宏观经济因素.本文建立了考虑成本和收益净现值的连续时间有限时段确定性库存系统的最优存储和定价决策模型,证明了给定价格下最优策略中任意两个相邻订货周期之间的递推关系,分析了该关系的解析性质,并得出订货周期长度的上下限.在此基础上提出求解最优存储策略和最优价格的两步优化算法.最后通过数值算例对本文模型及结果做出说明.     

随机需求下考虑不同竞争情形的网约车平台动态定价

针对乘车需求波动下网约车平台间存在乘车需求竞争和乘运供应竞争的最优定价问题,以平台期望收益最大化为目标,运用最优控制论方法,构建不同竞争情形下的网约车平台动态定价模型,并利用哈密尔顿函数及模型推导,求得最优动态竞争价格解以及乘运供应率与需求率的变化轨迹。结果表明：平台最优动态竞争价格随市场需求的波动而动态变化,且最优价格可以有效调控平台供应能力,促使平台供需匹配,优化平台期望收益。此外,乘车需求市场竞争越激烈,平台最优价格越低,而乘运供应市场竞争越激烈,最优价格越高。平台间竞争的加剧将降低平台的期望收益,且平台期望收益随着固定佣金报酬率的提高先增大后减小。     

Optimal Inventory Control with Dual‐Sourcing,Heterogeneous Ordering Costs and Order Size Constraints

We consider a dual‐sourcing inventory system, where procuring from one supplier involves a high variable cost but negligible fixed cost whereas procuring from the other supplier involves a low variable cost but high fixed cost, as well as an order size constraint. We show that the problem can be reduced to an equivalent single‐sourcing problem. However, the corresponding ordering cost is neither concave nor convex. Using the notion of quasi‐convexity, we partially characterize the structure of the optimal policy and show that it can be specified by multiple thresholds which determine when to order from each supplier and how much. In contrast to previous research, which does not consider order size constraints, we show that it is optimal to simultaneously source from both suppliers when the beginning inventory level is sufficiently low. We also show that the decision to source from the low‐cost supplier is not monotonic in the inventory level. Our results require that the variable costs satisfy a certain condition which guarantees quasi‐convexity. However, extensive numerical results suggest that our policy is almost always optimal when the condition is not satisfied. We also show how the results can be extended to systems with multiple capacitated suppliers.     

Dual Sourcing Inventory Systems: On Optimal Policies and the Value of Costless Returns

We study dual sourcing inventory systems with backordering and with stationary, stochastic demands. The two supply sources differ in their unit prices and lead times. We focus on the option of making costless returns to the cheaper, longer leadtime supplier. We show that the value of this option is zero. Our analysis leading to this result includes the derivation of several structural properties of the optimal policies for dual sourcing systems with and without the return option.     

Static and Dynamic Pricing of Excess Capacity in a Make‐to‐Order Environment

We consider a make‐to‐order manufacturer that serves two customer classes: core customers who pay a fixed negotiated price, and “fill‐in” customers who make submittal decisions based on the current price set by the firm. Using a Markovian queueing model, we determine how much the firm can gain by explicitly accounting for the status of its production facility in making pricing decisions. Specifically, we examine three pricing policies: (1) static, state‐independent pricing, (2) constant pricing up to a cutoff state, and (3) general state‐dependent pricing. We determine properties of each policy, and illustrate numerically the financial gains that the firm can achieve by following each policy as compared with simpler policies. Our main result is that constant pricing up to a cutoff state can dramatically outperform a state‐independent policy, while at the same time achieving most of the increase in revenue achievable from general state‐dependent pricing. Thus, we find that constant pricing up to a cutoff state presents an attractive tradeoff between ease of implementation and revenue gain. When the costs of policy design and implementation are taken into account, this simple heuristic may actually out‐perform general state‐dependent pricing in some settings.