Base-stock List-price Policy Research Articles

Dynamic pricing and inventory management with demand learning: A bayesian approach

We consider a retail firm selling a durable product in a volatile market where the demand is price-sensitive and random but its distribution is unknown. The firm dynamically replenishes inventory and adjusts prices over time and learns about the demand distribution. Assuming that the demand model is of the multiplicative form and unmet demand is partially backlogged, we take the empirical Bayesian approach to formulate the problem as a stochastic dynamic program. We first identify a set of regularity conditions on demand models and show that the state-dependent base-stock list-price policy is optimal. We next employ the dimensionality reduction approach to separate the scale factor that captures observed demand information from the optimal profit function, which yields a normalized dynamic program that is more tractable. We also analyze the effect of demand learning on the optimal policy using the system without Bayesian update as a benchmark. We further extend our analysis to the case with unobserved lost sales and the case with additive demand.

Coordinating Inventory and Pricing Decisions Under Total Minimum Commitment Contracts

A total minimum commitment contract is a supply contract under which a firm commits to buying a minimum quantity of a product from its supplier during the contract duration (e.g., one year). Such contracts are widely used in industries, because they provide the buyer with flexibility in terms of the timing and size of each individual order and the supplier with a guaranteed total order volume. Previous studies on such contracts are primarily concerned with the firm's inventory decisions, but none of them has considered the coordination of inventory and pricing decisions. In this paper, we fill this gap by studying dynamic inventory and pricing problems under a commitment contract. Under a general demand model with backlogging and zero lead time, we prove that the optimal policy is a committed-inventory-position-dependent base-stock list-price policy and characterize its structural properties. We also conduct an extensive numerical study to derive further managerial insights. In particular, we find that dynamic pricing can substantially improve the firm's profitability and enables it to select contracts with large committed quantities and price discount rates, and ignoring the commitment in joint pricing and ordering decisions can result in substantial profit loss. Finally, we partly extend our results to a lost-sales model and a backlogging model with positive lead times.

Joint pricing and inventory management under servitisation

We analyse a problem involving the joint pricing of a product and associated service, together with the management of inventory under servitisation. For a finite planning horizon, the firm sells both a product and a product-centric service whose market size depends on the past sales of the product. At the beginning of each period, the firm simultaneously decides the price of the product, the price of the service, and the replenishment quantity. We prove that a modified base-stock list-price policy is optimal to this problem. In addition, we find that there is a trade-off between realising current profit from the product and stimulating demand to increase future profit, and that overlooking this trade-off results in overall loss of profit for the firm. Moreover, the optimal policy that takes the trade-off into account leads to a lower optimal product price compared with a myopic policy, and the optimal price increases as a function of the past sales of the product.

Dynamic Stochastic Inventory Management with Reference Price Effects

We analyze the joint inventory and pricing decisions of a firm when demand depends on not only the current selling price but also a memory-based reference price and customers are loss averse. The presence of reference price effect leads to a nonconcave one-period expected revenue in price and reference price. We introduce a transformation technique that allows us to prove under some mild assumptions the optimality of a reference-price-dependent base-stock list-price policy, which is characterized by a base-stock level and a target reference price. In addition, we show that the target reference price is increasing in the reference price, but except in the loss-neutral case, the base-stock level is not monotone in the reference price. We also show that in the steady state of the model with the reference price effect, the optimal price is lower while the optimal base-stock level is higher than their counterparts in the model without the reference price effect.

Base stock list price policy in continuous time

We study the problem of inventory control, with simultaneous pricing optimization in continuous time. For the classical inventory control problem in continuous time, see [ 5 ], as a recent reference. We incorporate pricing decisions together with inventory decisions. We consider the situation without fixed cost for an infinite horizon. Without pricing, under very natural assumptions, the optimal ordering policy is given by a stock, which we review briefly. With pricing, the natural generalization is the so called Base Stock list price (BSLP) term coined by E. Porteus, see [ 36 ], and was shown in discrete time by A. Federgruen and A. Herching to be the optimal strategy, see [ 14 ]. We extend the concept to continuous time which not only complicates the dynamics of the problem, which has never been considered before.

Integrating Dynamic Pricing with Inventory Decisions Under Lost Sales

Inventory-based pricing under lost sales is an important yet notoriously challenging problem in the operations management literature. The vast existing literature on this problem focuses on identifying optimality conditions for a simple management policy while restricting to special classes of demand functions and to the special case of single-period or long-term stationary settings. In view of the existing developments, it seems unlikely to find general, easy-to-verify conditions for a tractable optimal policy in a possibly nonstationary environment. Instead, we take a different approach to tackle this problem. Specifically, we refine our analysis to a class of intuitively appealing policies, under which the price is decreasing in the postorder inventory level. Using properties of stochastic functions, we show that, under very general conditions on the stochastic demand function, the objective function is concave along such price paths, leading to a simple base stock list price policy. We identify the upper and lower boundaries for a candidate set of decreasing price paths and show that any decreasing path outside of this set is always dominated by some inside the set in terms of profit performance. The boundary policies can be computed efficiently through a single-dimensional search. An extensive numerical analysis suggests that choosing boundary policies yields close-to-optimal profit—in most instances, one of the boundary policies indeed generates the optimal profit; even when they are not, the profit loss is very marginal. This paper was accepted by David Simchi-Levi, operations management.

Technical Note—Joint Inventory and Pricing Control with General Additive Demand

This paper analyzes a periodic-review, joint inventory and pricing control problem for a firm that faces stochastic, price-sensitive demand under a nonstationary environment with fixed ordering costs. Any unsatisfied demand is backlogged. The objective is to maximize expected profit over a finite selling horizon by coordinating the inventory and pricing decisions in each period. We show that for an additive demand model, an (s, S, p) policy is optimal when the expected revenue is quasi-concave in price, the inventory cost (of holding and/or backlogging) is quasi-convex, and the nonnegative random demand has a Pólya or uniform density function. For the special case with no fixed ordering cost, the optimality of a base stock list price policy is demonstrated for more general demand distributions and convex inventory cost. These sets of sufficient conditions generalize the existing conditions in the literature that require, for example, the demand and/or revenue functions to be concave or the model parameters to be stationary in time. Our generalization makes the structural results applicable to models broadly supported by economic theory and empirical data. In addition, our proof uses a distinct sequential optimization technique for iteratively establishing the quasi-K-concavity of dynamic optimal value functions.

Optimal pricing and inventory policy with order cancelations under the cash-on-delivery payment scheme

Considering a periodic review system where the online seller allows the customers to pay when the products are delivered to them (referred as cash-on-delivery payment scheme in this paper), the authors investigate the seller’s joint pricing and inventory control policy with a finite planning horizon. In particular, the authors incorporate the customers’ possible order cancellation behavior with the cash-on-delivery scheme. It can be proven that the base-stock list price policy is optimal under mild conditions. The authors also analyze the impact of the customers’ forward looking behavior on the optimal policy.

Joint Pricing and Production Decisions in an Assemble-to-Order System

This paper studies coordinated pricing and production decisions in an assemble-to-order system. We first show that unlike in make-to-stock systems, a state-dependent base-stock list-price policy is optimal. The optimal state-dependent base-stock levels and list prices may increase or decrease as demand backlogs increase, whereas demand backlogs always improve the optimal expected profit. Because the problem easily becomes intractable under general system settings, we next develop a simple heuristic policy. The heuristic policy decouples inventory replenishment, pricing, and component allocation decisions in a coordinated way. We provide a sufficient condition that ensures the optimality of the heuristic policy, and present a numerical study to demonstrate its performance when the condition is not met. The numerical study also shows how the performance of the heuristic policy is affected by various market and operational conditions, and by the structure of the assemble-to-order system. By focusing on the simple W-model, we show how the heuristic pricing decisions are made in response to changes in inventory levels and various cost parameters.

Optimal replenishment and pricing decisions under the collect-on-delivery payment scheme

In this paper, we study how the timing of payments affects joint replenishment and pricing decisions. Specifically, we consider a periodic review system in which consumers pay upon receiving their goods, referred to as the collect- on-delivery scheme, as opposed to paying when placing their orders, referred to as the instant payment scheme commonly assumed in the literature. We show that a base-stock list price policy is optimal without a fixed ordering cost, and provide a sufficient condition for an $$(s, S, p)$$ policy to be optimal with a fixed ordering cost. We also perform both analytical and numerical studies to compare the two schemes.

Sourcing from Multiple Suppliers for Price‐Dependent Demands

We analyze a model that integrates demand shaping via dynamic pricing and risk mitigation via supply diversification. The firm under consideration replenishes a certain product from a set of capacitated suppliers for a price‐dependent demand in each period. Under deterministic capacities, we derive a multilevel base stock list price policy and establish the optimality of cost‐based supplier selection, that is, ordering from a cheaper source before more expensive ones. With general random capacities, however, neither result holds. While it is optimal to price low for a high inventory level, the optimal order quantities are not monotone with respect to the inventory level. In general, a near reorder‐point policy should be followed. Specifically, there is a reorder point for each supplier such that no order is issued to him when the inventory level is above this point and a positive order is placed almost everywhere when the inventory level is below this point. Under this policy, it may be profitable to order exclusively from the most expensive source. We characterize conditions under which a strict reorder‐point policy and a cost‐based supplier‐selection criterion become optimal. Moreover, we quantify the benefit from dynamic pricing, as opposed to static pricing, and the benefit from multiple sourcing, as opposed to single sourcing. We show that these two strategies exhibit a substitutable relationship. Dynamic pricing is less effective under multiple sourcing than under single sourcing, and supplier diversification is less valuable with price adjustments than without. Under limited supply, dynamic pricing yields a robust, long‐term profit improvement. The value of supply diversification, in contrast, mainly comes from added capacities and is most significant in the short run.

Integrating inventory control and a price change in the presence of reference price effects: a two-period model

Demand and procurement planning for consumer electronics products must cope with short life cycles, limited replenishment opportunities and a willingness to pay that is influenced by past prices and decreases over time. We therefore propose the use of an integrated pricing and inventory control model with a two-period linear demand model, in which demand also depends on the difference between a price-history-based reference price and the current price. For this model we prove that the optimal joint pricing/inventory policy for the replenishment opportunity after the first period is a base-stock list-price policy. That is, stock is either replenished up to a base-stock level and a list-price is charged, or it is not replenished and a discount is given that increases with the stock-level. Furthermore, we use real-world cell phone data to study the differences between an integrated policy and traditional sequential optimization, where prices are initially optimized based on the expected demand and ordering cost, and the resulting demand distribution is used to determine an optimal inventory policy. Finally, we discuss possible extensions of the model.

Integrating Dynamic Pricing and Replenishment Decisions Under Supply Capacity Uncertainty

This paper examines an integrated decision-making process regarding pricing for uncertain demand and sourcing from uncertain supply, which are often studied separately in the literature. Our analysis of the integrated system suggests that the base stock list price policy fails to achieve optimality even under deterministic demand. Instead, the optimal policy is characterized by two critical values: a reorder point and a target safety stock. Under this policy, a positive order is issued if and only if the inventory level is below the reorder point. When this happens, the optimal order and price are coordinated to achieve a constant target safety stock, which aims at hedging the demand uncertainty. We further investigate the profit improvement obtained from deploying dynamic pricing, as opposed to static pricing. Our results indicate that either supply limit or supply uncertainty may induce a significant benefit from dynamic pricing, and the compound effect of supply limit and uncertainty can be much more pronounced than the individual effects. Whether or not the supply capacity is limited has a major implication on the value of dynamic pricing. Under unlimited supply, dynamic pricing is more valuable when procurement cost is high or when demand is more sensitive to price. With limited supply, however, the capacity restriction tends to be relaxed, reducing the value of dynamic pricing.