Dynamic pricing and replenishment: Optimality,bounds, and asymptotics

In many applications, managers face the problem of replenishing and selling products during a finite time horizon. We investigate the problem of making dynamic and joint decisions on product replenishment and selling in order to improve profit. We consider a backlog scenario in which penalty cost (resulting from fulfillment delay) and accommodation cost (resulting from shortage at the end of the selling horizon) are incurred. Based on continuous‐time and discrete‐state dynamic programming, we study the optimal joint decisions and characterize their structural properties. We establish an upper bound for the optimal expected profit and develop a fluid policy by resorting to the deterministic version of the problem (ie, the fluid problem). The fluid policy is shown to be asymptotically optimal for the original stochastic problem when the problem size is sufficiently large. The static nature of the fluid policy and its lack of flexibility in matching supply with demand motivate us to develop a “target‐inventory” heuristic, which is shown, numerically, to be a significant improvement over the fluid policy. Scenarios with discrete feasible sets and lost‐sales are also discussed in this article.     

Coordination of pricing and inventory control across products

We consider the joint pricing and inventory‐control problem for a retailer who orders, stocks, and sells two products. Cross‐price effects exist between the two products, which means that the demand of each product depends on the prices of both products. We derive the optimal pricing and inventory‐control policy and show that this policy differs from the base‐stock list‐price policy, which is optimal for the one‐product problem. We find that the retailer can significantly improve profits by managing the two products jointly as opposed to independently, especially when the cross‐price demand elasticity is high. We also find that the retailer can considerably improve profits by using dynamic pricing as opposed to static pricing, especially when the demand is nonstationary. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Pricing and inventory management during new product introduction when shortage creates hype

In this study, we analyze the joint pricing and inventory management during new product introduction when product shortage creates additional demand due to hype. We develop a two‐period model in which a firm launches its product at the beginning of the first period, before it observes sales in the two periods. The product is successful with an exogenous probability, or unsuccessful with the complementary probability. The hype in the second period is observed only when the product is successful. The firm learns the actual status of the product only after observing the first‐period demand. The firm must decide the stocking level and price of the product jointly at the beginning of each of the two periods. In this article, we derive some structural properties of the optimal prices and inventory levels, and show that (i) firms do not always exploit hype, (ii) firms do not always increase the price of a successful product in the second period, (iii) firms may price out an unsuccessful product in the first period if the success probability is above a threshold, and (iv) such a threshold probability is decreasing in the first‐period market potential of the successful product. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 304–320, 2015     

Joint pricing and ordering policy for exponentially decaying inventory with known demand

This paper is concerned with the problem of simultaneously setting price and production levels for an exponentially decaying product. Such products suffer a loss in utility which is proportional to the total quantity of stock on hand. A continuous review, deterministic demand model is considered. The optimal ordering decision quantity is derived and its sensitivity to changes in perishability and product price is considered. The joint ordering pricing decision is also computed and consideration of parametric changes of these decisions indicates a non-monotonic response for optimal price to changes in product decay. Issues of market entry and extensions to a model with shortages are also analyzed.     

Evaluating the potential effects from probabilistic selling of similar products

In this article, we consider an online retailer who sells two similar products (A and B) over a finite selling period. Any stock left at the end of the period has no value (like clothes going out of fashion at the end of a season). Aside from selling the products at regular prices, he may offer an additional option that sells a probabilistic good, “A or B,” at a discounted price. Whenever a customer buys a probabilistic good, he needs to assign one of the products for the fulfillment. Considering the choice behavior of potential customers, we model the problem using continuous‐time, discrete‐state, finite‐horizon dynamic programming. We study the optimal admission decisions and devise two scenarios, whose value functions can be used as benchmarks to evaluate the demand induction effect and demand dilution effect of probabilistic selling (PS). We further investigate an extension of the base MDP (Markov Decision Process) model in which the fulfillment of probabilistic sales is uncontrollable by the retailer. A special case of the extended model can be used as a benchmark to quantify the potential inventory pooling effect of PS. Finally, numerical experiments are conducted to evaluate the overall profit improvement, and the effects from adopting the PS strategy. © 2014 Wiley Periodicals, Inc. Naval Research Logistics, 61: 604–620, 2014     

Optimal pricing and inventory control policy in periodic‐review systems with fixed ordering cost and lost sales

This paper studies a periodic‐review pricing and inventory control problem for a retailer, which faces stochastic price‐sensitive demand, under quite general modeling assumptions. Any unsatisfied demand is lost, and any leftover inventory at the end of the finite selling horizon has a salvage value. The cost component for the retailer includes holding, shortage, and both variable and fixed ordering costs. The retailer's objective is to maximize its discounted expected profit over the selling horizon by dynamically deciding on the optimal pricing and replenishment policy for each period. We show that, under a mild assumption on the additive demand function, at the beginning of each period an (s,S) policy is optimal for replenishment, and the value of the optimal price depends on the inventory level after the replenishment decision has been done. Our numerical study also suggests that for a sufficiently long selling horizon, the optimal policy is almost stationary. Furthermore, the fixed ordering cost (K) plays a significant role in our modeling framework. Specifically, any increase in K results in lower s and higher S. On the other hand, the profit impact of dynamically changing the retail price, contrasted with a single fixed price throughout the selling horizon, also increases with K. We demonstrate that using the optimal policy values from a model with backordering of unmet demands as approximations in our model might result in significant profit penalty. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

Synchronizing pricing and replenishment to serve forward-looking customers

We incorporate strategic customer waiting behavior in the classical economic order quantity (EOQ) setting. The seller determines not only the timing and quantities of the inventory replenishment, but also the selling prices over time. While similar ideas of market segmentation and intertemporal price discrimination can be carried over from the travel industries to other industries, inventory replenishment considerations common to retail outlets and supermarkets introduce additional features to the optimal pricing scheme. Specifically, our study provides concrete managerial recommendations that are against the conventional wisdom on “everyday low price” (EDLP) versus “high-low pricing” (Hi-Lo). We show that in the presence of inventory costs and strategic customers, Hi-Lo instead of EDLP is optimal when customers have homogeneous valuations. This result suggests that because of strategic customer behavior, the seller obtains a new source of flexibility—the ability to induce customers to wait—which always leads to a strictly positive increase of the seller's profit. Moreover, the optimal inventory policy may feature a dry period with zero inventory, but this period does not necessarily result in a loss of sales as customers strategically wait for the upcoming promotion. Furthermore, we derive the solution approach for the optimal policy under heterogeneous customer valuation setting. Under the optimal policy, the replenishments and price promotions are synchronized, and the seller adopts high selling prices when the inventory level is low and plans a discontinuous price discount at the replenishment point when inventory is the highest.     

Optimal inventory and dynamic admission policies for a retailer of seasonal products with affiliate programs and drop‐shipping

This article investigates the optimal inventory and admission policies for a “Clicks‐and‐Bricks” retailer of seasonal products that, in addition to selling through its own physical and online stores, also sells through third‐party websites by means of affiliate programs. Through postings on partners' webpages, an affiliate program allows a retailer to attract customers who would otherwise be missed. However, this retailer needs to pay a commission for each sale that originates from the website operators participating in the program. The retailer may also refer online orders to other sources (such as distributors and manufacturers) for fulfillment through a drop‐shipping agreement and thus earns commissions. This would be an option when, for example, the inventories at the physical stores were running low. Therefore, during the selling horizon, the retailer needs to dynamically control the opening/closing of affiliate programs and decide on the fulfillment option for online orders. On the basis of a discrete‐time dynamic programming model, the optimal admission policy of the retailer is investigated in this paper, and the structural properties of the revenue function are characterized. Numerical examples are given to show the revenue impact of optimal admission control. The optimal initial stocking decisions at the physical stores are also studied. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009     

Capacity games for partially complementary products under multivariate random demands

In this study, we consider n firms, each of which produces and sells a different product. The n firms face a common demand stream which requests all their products as a complete set. In addition to the common demand stream, each firm also faces a dedicated demand stream which requires only its own product. The common and dedicated demands are uncertain and follow a general, joint, continuous distribution. Before the demands are realized, each firm needs to determine its capacity or production quantity to maximize its own expected profit. We formulate the problem as a noncooperative game. The sales price per unit for the common demand could be higher or lower than the unit price for the dedicated demand, which affects the firm's inventory rationing policy. Hence, the outcome of the game varies. All of the prices are first assumed to be exogenous. We characterize Nash equilibrium(s) of the game. At the end of the article, we also provide some results for the endogenous pricing. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 59: 146–159, 2012     

A nonlinear continuous time optimal control model of dynamic pricing and inventory control with no backorders

In this paper, we present a continuous time optimal control model for studying a dynamic pricing and inventory control problem for a make‐to‐stock manufacturing system. We consider a multiproduct capacitated, dynamic setting. We introduce a demand‐based model where the demand is a linear function of the price, the inventory cost is linear, the production cost is an increasing strictly convex function of the production rate, and all coefficients are time‐dependent. A key part of the model is that no backorders are allowed. We introduce and study an algorithm that computes the optimal production and pricing policy as a function of the time on a finite time horizon, and discuss some insights. Our results illustrate the role of capacity and the effects of the dynamic nature of demand in the model. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Locational tying of complementary retail items

We study a selling practice that we refer to as locational tying (LT), which seems to be gaining wide popularity among retailers. Under this strategy, a retailer “locationally ties” two complementary items that we denote by “primary” and “secondary.” The retailer sells the primary item in an appropriate “department” of his or her store. To stimulate demand, the secondary item is offered in the primary item's department, where it is displayed in very close proximity to the primary item. We consider two variations of LT: In the multilocation tying strategy (LT‐M), the secondary item is offered in its appropriate department in addition to the primary item's department, whereas in the single‐location tying strategy (LT‐S), it is offered only in the primary item's location. We compare these LT strategies to the traditional independent components (IC) strategy, in which the two items are sold independently (each in its own department), but the pricing/inventory decisions can be centralized (IC‐C) or decentralized (IC‐D). Assuming ample inventory, we compare and provide a ranking of the optimal prices of the four strategies. The main insight from this comparison is that relative to IC‐D, LT decreases the price of the primary item and adjusts the price of the secondary item up or down depending on its popularity in the primary item's department. We also perform a comparative statics analysis on the effect of demand and cost parameters on the optimal prices of various strategies, and identify the conditions that favor one strategy over others in terms of profitability. Then we study inventory decisions in LT under exogenous pricing by developing a model that accounts for the effect of the primary item's stock‐outs on the secondary item's demand. We find that, relative to IC‐D, LT increases the inventory level of the primary item. We also link the profitability of different strategies to the trade‐off between the increase in demand volume of the secondary item as a result of LT and the potential increase in inventory costs due to decentralizing the inventory of the secondary item. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009     

Meeting seasonal demand for products with limited shelf lives

This article addresses a common misconception concerning production lead time and the use of inventory to meet seasonal demand for products with limited shelf lives. Two fundamental questions are answered: 1) Under what conditions will an increase in product life lead to increased ability to meet demand? 2) Under what conditions will increased levels of starting inventory be beneficial? The results of this analysis assisted a plastics manufacturing firm in making product pricing and inventory decisions. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44 : 473–483, 1997     

Spatial pricing and product allocation in online retailing

Spatial pricing means a retailer price discriminates its customers based on their geographic locations. In this article, we study how an online retailer should jointly allocate multiple products and facilitate spatial price discrimination to maximize profits. When deciding between a centralized product allocation ((i.e., different products are allocated to the same fulfillment center) and decentralized product allocation (ie, different products are allocated to different fulfillment centers), the retailer faces the tradeoff between shipment pooling (ie, shipping multiple products in one package), and demand localization (ie, stocking products to satisfy local demand) based on its understanding of customers' product valuations. In our basic model, we consider two widely used spatial pricing policies: free on board (FOB) pricing that charges each customer the exact amount of shipping cost, and uniform delivered (UD) pricing that provides free shipping. We propose a stylized model and find that centralized product allocation is preferred when demand localization effect is relatively low or shipment pooling benefit is relatively high under both spatial pricing policies. Moreover, centralized product allocation is more preferred under the FOB pricing which encourages the purchase of virtual bundles of multiple products. Furthermore, we respectively extend the UD and FOB pricing policies to flat rate shipping (ie, the firm charges a constant shipping fee for each purchase), and linear rate shipping (ie, the firm sets the shipping fee as a fixed proportion of firm's actual fulfillment costs). While similar observations from the basic model still hold, we find the firm can improve its profit by sharing the fulfillment cost with its customers via the flat rate or linear rate shipping fee structure.     

A note on a maximin disposal policy under nwue pricing

For the classical disposal model for selling an asset with unknown price distribution which is NWUE (new worse than used in expectation) with a given finite mean price, this note derives a policy which is maximin. The gain in using the maximin policy relative to the option of selling right away is convex decreasing in the continuation cost to mean price ratio. The relevant results of Derman, Lieberman and Ross also follow as a consequence of our analysis. Our theorem provides a practical justification of their main result on the cutoff bid for the disposal model subject to NWUE pricing.     

Exact algorithms for integrated facility location and production planning problems

We consider a class of facility location problems with a time dimension, which requires assigning every customer to a supply facility in each of a finite number of periods. Each facility must meet all assigned customer demand in every period at a minimum cost via its production and inventory decisions. We provide exact branch‐and‐price algorithms for this class of problems and several important variants. The corresponding pricing problem takes the form of an interesting class of production planning and order selection problems. This problem class requires selecting a set of orders that maximizes profit, defined as the revenue from selected orders minus production‐planning‐related costs incurred in fulfilling the selected orders. We provide polynomial‐time dynamic programming algorithms for this class of pricing problems, as well as for generalizations thereof. Computational testing indicates the advantage of our branch‐and‐price algorithm over various approaches that use commercial software packages. These tests also highlight the significant cost savings possible from integrating location with production and inventory decisions and demonstrate that the problem is rather insensitive to forecast errors associated with the demand streams. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Fashion products with asymmetric sales horizons

In this study, we explore an inventory model for a wholesaler who sells a fashion product through two channels with asymmetric sales horizons. The wholesaler can improve profitability by employing joint procurement and inventory reallocation as a recourse action in response to the dynamics of sales. In this research, a simple stochastic programming model is analyzed to specify the properties of the optimal inventory decisions. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

A sequential dynamic pricing model and its applications

Consider a sequential dynamic pricing model where a seller sells a given stock to a random number of customers. Arriving one at a time, each customer will purchase one item if the product price is lower than her personal reservation price. The seller's objective is to post a potentially different price for each customer in order to maximize the expected total revenue. We formulate the seller's problem as a stochastic dynamic programming model, and develop an algorithm to compute the optimal policy. We then apply the results from this sequential dynamic pricing model to the case where customers arrive according to a continuous‐time point process. In particular, we derive tight bounds for the optimal expected revenue, and develop an asymptotically optimal heuristic policy. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

Multi‐item capacitated lot‐sizing problems with setup times and pricing decisions

We study a multi‐item capacitated lot‐sizing problem with setup times and pricing (CLSTP) over a finite and discrete planning horizon. In this class of problems, the demand for each independent item in each time period is affected by pricing decisions. The corresponding demands are then satisfied through production in a single capacitated facility or from inventory, and the goal is to set prices and determine a production plan that maximizes total profit. In contrast with many traditional lot‐sizing problems with fixed demands, we cannot, without loss of generality, restrict ourselves to instances without initial inventories, which greatly complicates the analysis of the CLSTP. We develop two alternative Dantzig–Wolfe decomposition formulations of the problem, and propose to solve their relaxations using column generation and the overall problem using branch‐and‐price. The associated pricing problem is studied under both dynamic and static pricing strategies. Through a computational study, we analyze both the efficacy of our algorithms and the benefits of allowing item prices to vary over time. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Pricing and capacity allocation strategies: Implications for manufacturers with product sharing

Emerging sharing modes, like the consumer-to-consumer (C2C) sharing of Uber and the business-to-consumer (B2C) sharing of GoFun, have considerably affected the retailing markets of traditional manufacturers, who are motivated to consider product sharing when making pricing and capacity decisions, particularly electric car manufacturers with limited capacity. In this paper, we examine the equilibrium pricing for a capacity-constrained manufacturer under various sharing modes and further analyze the impact of capacity constraint on the manufacturer's sharing mode selection as well as equilibrium outcomes. We find that manufacturers with low-cost products prefer B2C sharing while those with high-cost products prefer C2C sharing except when the sharing price is moderate. However, limited capacity motivates manufacturers to enter into the B2C sharing under a relatively low sharing price, and raise the total usage level by sharing high-cost products. We also show that the equilibrium capacity allocated to the sharing market with low-cost products first increases and then decreases. Finally, we find that sharing low-cost products with a high limited capacity leads to a lower retail price under B2C sharing, which creates a win-win situation for both the manufacturer and consumers. However, sharing high-cost products with a low limited capacity creates a win-lose situation for them.     

Revenue maximization in two-station tandem queueing systems

We study optimal pricing for tandem queueing systems with finite buffers. The service provider dynamically quotes prices to incoming price sensitive customers to maximize the long-run average revenue. We present a Markov decision process model for the optimization problem. For systems with two stations, general-sized buffers, and two or more prices, we describe the structure of the optimal dynamic pricing policy and develop tailored policy iteration algorithms to find an optimal pricing policy. For systems with two stations but no intermediate buffer, we characterize conditions under which quoting either a high or a low price to all customers is optimal and provide an easy-to-implement algorithm to solve the problem. Numerical experiments are conducted to compare the developed algorithms with the regular policy iteration algorithm. The work also discusses possible extensions of the obtained results to both three-station systems and two-station systems with price and congestion sensitive customers using numerical analysis.