Competition under time‐varying demands and dynamic lot sizing costs

We develop a competitive pricing model which combines the complexity of time‐varying demand and cost functions and that of scale economies arising from dynamic lot sizing costs. Each firm can replenish inventory in each of the T periods into which the planning horizon is partitioned. Fixed as well as variable procurement costs are incurred for each procurement order, along with inventory carrying costs. Each firm adopts, at the beginning of the planning horizon, a (single) price to be employed throughout the horizon. On the basis of each period's system of demand equations, these prices determine a time series of demands for each firm, which needs to service them with an optimal corresponding dynamic lot sizing plan. We establish the existence of a price equilibrium and associated optimal dynamic lotsizing plans, under mild conditions. We also design efficient procedures to compute the equilibrium prices and dynamic lotsizing plans.© 2008 Wiley Periodicals, Inc. Naval Research Logistics 2009     

Purchase and retrieval competition for seasonal produce

We study the competition problem of purchase and multiretrieval of perishable seasonal produce, where wholesalers purchase and stock their products in the first period, and then retrieve and sell them in subsequent periods. We first consider the duopoly case and assume that the prices are exogenous and fluctuate. In each period, after the price realization, the wholesalers retrieve some stock from their warehouses to satisfy their demands. One wholesaler's unsatisfied customers can switch to another and be satisfied by its left retrieved products. Any unsold retrieved stock has no salvage value and any unsatisfied demand is lost. The unretrieved stock is carried to the next period at a perishable rate. The wholesalers compete for the substitute demand by determining their own purchase and retrieval quantities. We show the existence and uniqueness of a pure-strategy Nash equilibrium, and that the Nash equilibrium strategy has the simple “sell-down-to” structure. We also consider the general N-person game and show the existence of the Nash equilibrium, and characterize the structure of the equilibrium strategy for the symmetric case. In addition, we consider the case with endogenous prices, and show that the problem reduces to a repeated newsvendor game with price and inventory competition. We derive the conditions under which a unique Nash equilibrium exists and characterize the equilibrium strategy. Finally, we conduct numerical studies to examine the impacts of the model parameters on the equilibrium outcomes and to generate managerial insights.     

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     

On the effectiveness of returns policies in the price‐dependent newsvendor model

We study in this paper the price‐dependent (PD) newsvendor model in which a manufacturer sells a product to an independent retailer facing uncertain demand and the retail price is endogenously determined by the retailer. We prove that for a zero salvage value and some expected demand functions, in equilibrium, the manufacturer may elect not to introduce buybacks. On the other hand, if buybacks are introduced in equilibrium, their introduction has an insignificant effect on channel efficiency improvement, but, by contrast, may significantly shift profits from the retailer to the manufacturer. We further demonstrate that the introduction of buybacks increases the wholesale price, retail price, and inventory level, as compared to the wholesale price‐only contract, and that the corresponding vertically integrated firm offers the lowest retail price and highest inventory level. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

Bayesian strategies for dynamic pricing in e‐commerce

E‐commerce platforms afford retailers unprecedented visibility into customer purchase behavior and provide an environment in which prices can be updated quickly and cheaply in response to changing market conditions. This study investigates dynamic pricing strategies for maximizing revenue in an Internet retail channel by actively learning customers' demand response to price. A general methodology is proposed for dynamically pricing information goods, as well as other nonperishable products for which inventory levels are not an essential consideration in pricing. A Bayesian model of demand uncertainty involving the Dirichlet distribution or a mixture of such distributions as a prior captures a wide range of beliefs about customer demand. We provide both analytic formulas and efficient approximation methods for updating these prior distributions after sales data have been observed. We then investigate several strategies for sequential pricing based on index functions that consider both the potential revenue and the information value of selecting prices. These strategies require a manageable amount of computation, are robust to many types of prior misspecification, and yield high revenues compared to static pricing and passive learning approaches. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Dynamic learning,pricing, and ordering by a censored newsvendor

We address the problem of determining optimal ordering and pricing policies in a finite‐horizon newsvendor model with unobservable lost sales. The demand distribution is price‐dependent and involves unknown parameters. We consider both the cases of perishable and nonperishable inventory. A very general class of demand functions is studied in this paper. We derive the optimal ordering and pricing policies as unique functions of the stocking factor (which is a linear transformation of the safety factor). An important expression is obtained for the marginal expected value of information. As a consequence, we show when lost sales are unobservable, with perishable inventory the optimal stocking factor is always at least as large as the one given by the single‐period model; however, if inventory is nonperishable, this result holds only under a strong condition. This expression also helps to explain why the optimal stocking factor of a period may not increase with the length of the problem. We compare this behavior with that of a full information model. We further examine the implications of the results to the special cases when demand uncertainty is described by additive and multiplicative models. For the additive case, we show that if demand is censored, the optimal policy is to order more as well as charge higher retail prices when compared to the policies in the single‐period model and the full information model. We also compare the optimal and myopic policies for the additive and multiplicative models. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

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     

Dynamic inventory and pricing models for competing retailers

We address infinite‐horizon models for oligopolies with competing retailers under demand uncertainty. We characterize the equilibrium behavior which arises under simple wholesale pricing schemes. More specifically, we consider a periodic review, infinite‐horizon model for a two‐echelon system with a single supplier servicing a network of competing retailers. In every period, each retailer faces a random demand volume, the distribution of which depends on his own retail price as well as those charged by possibly all competing retailers. We also derive various comparative statics results regarding the impact several exogenous system parameters (e.g., cost or distributional parameters) have on the equilibrium decisions of the retailers as well as their expected profits. We show that certain monotonicity properties, engrained in folklore as well as in known inventory models for centralized systems, may break down in decentralized chains under retailer competition. Our results can be used to optimize the aggregate profits in the supply chain (i.e., those of the supplier and all retailers) by implementing a specific wholesale pricing scheme. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

Inventory control and periodic price discounting campaigns

This paper develops an inventory model that determines replenishment strategies for buyers facing situations in which sellers offer price‐discounting campaigns at random times as a way to drive sales or clear excess inventory. Specifically, the model deals with the inventory of a single item that is maintained to meet a constant demand over time. The item can be purchased at two different prices denoted high and low. We assume that the low price goes into effect at random points in time following an exponential distribution and lasts for a random length of time following another exponential distribution. We highlight a replenishment strategy that will lead to the lowest inventory holding and ordering costs possible. This strategy is to replenish inventory only when current levels are below a certain threshold when the low price is offered and the replenishment is to a higher order‐up‐to level than the one currently in use when inventory depletes to zero and the price is high. Our analysis provides new insight into the behavior of the optimal replenishment strategy in response to changes in the ratio of purchase prices together with changes in the ratio of the duration of a low‐price period to that of a high‐price period. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007.     

Optimal pricing for a short life‐cycle product when customer price‐sensitivity varies over time

Technology products often experience a life‐cycle demand pattern that resembles a diffusion process, with weak demand in the beginning and the end of the life cycle and high demand intensity in between. The customer price‐sensitivity also changes over the life cycle of the product. We study the prespecified pricing decision for a product that exhibits such demand characteristics. In particular, we determine the optimal set of discrete prices and the times to switch from one price to another, when a limited number of price changes are allowed. Our study shows that the optimal prices and switching times show interesting patterns that depend on the product's demand pattern and the change in the customers' price sensitivity over the life cycle of the product. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

An integrated electric power supply chain and fuel market network framework: Theoretical modeling with empirical analysis for New England

In this article, we develop a novel electric power supply chain network model with fuel supply markets that captures both the economic network transactions in energy supply markets and the physical network transmission constraints in the electric power network. The theoretical derivation and analysis are done using the theory of variational inequalities. We then apply the model to a specific case, the New England electric power supply chain, consisting of six states, five fuel types, 82 power generators, with a total of 573 generating units, and 10 demand market regions. The empirical case study demonstrates that the regional electric power prices simulated by our model match the actual electricity prices in New England very well. We also compute the electric power prices and the spark spread, an important measure of the power plant profitability, under natural gas and oil price variations. The empirical examples illustrate that in New England, the market/grid‐level fuel competition has become the major factor that affects the influence of the oil price on the natural gas price. Finally, we utilize the model to quantitatively investigate how changes in the demand for electricity influence the electric power and the fuel markets from a regional perspective. The theoretical model can be applied to other regions and multiple electricity markets under deregulation to quantify the interactions in electric power/energy supply chains and their effects on flows and prices. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

A dynamic,nonstationary inventory problem for a price/quantity setting firm

A dynamic and nonstationary model is formulated for a firm which attempts to minimize total expected costs over a finite planning horizon. The control variables are price and production. The price p and the demand ζ are linked through the relationship ζ = g(p) + η, where g(p) is the riskless demand curve and η is a random variable. The general model allows for proportional ordering costs, convex holding and stockout costs, downward sloping riskless demand curve, backlogging, partial backlogging, lost sales, partial spoilage of inventory, and two modes of collecting revenue. Sufficient conditions are developed for this problem to have an optimal policy which resembles the single critical number policy known from stochastic inventory theory. It is also shown what set of parameters will satisfy these sufficiency conditions.     

Coordinated pricing and inventory control problems with capacity constraints and fixed ordering cost

This article addresses a single‐item, finite‐horizon, periodic‐review coordinated decision model on pricing and inventory control with capacity constraints and fixed ordering cost. Demands in different periods are random and independent of each other, and their distributions depend on the price in the current period. Each period's stochastic demand function is the additive demand model. Pricing and ordering decisions are made at the beginning of each period, and all shortages are backlogged. The objective is to find an optimal policy that maximizes the total expected discounted profit. We show that the profit‐to‐go function is strongly CK‐concave, and the optimal policy has an (s,S,P) ‐like structure. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

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     

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     

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.     

Solution of minisum and minimax location–allocation problems with Euclidean distances

A new method for the solution of minimax and minisum location–allocation problems with Euclidean distances is suggested. The method is based on providing differentiable approximations to the objective functions. Thus, if we would like to locate m service facilities with respect to n given demand points, we have to minimize a nonlinear unconstrained function in the 2m variables x₁,y₁, ?,x_m,y_m. This has been done very efficiently using a quasi-Newton method. Since both the original problems and their approximations are neither convex nor concave, the solutions attained may be only local minima. Quite surprisingly, for small problems of locating two or three service points, the global minimum was reached even when the initial position was far from the final result. In both the minisum and minimax cases, large problems of locating 10 service facilities among 100 demand points have been solved. The minima reached in these problems are only local, which is seen by having different solutions for different initial guesses. For practical purposes, one can take different initial positions and choose the final result with best values of the objective function. The likelihood of the best results obtained for these large problems to be close to the global minimum is discussed. We also discuss the possibility of extending the method to cases in which the costs are not necessarily proportional to the Euclidean distances but may be more general functions of the demand and service points coordinates. The method also can be extended easily to similar three-dimensional problems.     

Supply chain coordination with shelf‐space and retail price dependent demand and heterogeneous retailers

We consider supply chain coordination in which a manufacturer supplies some product to multiple heterogeneous retailers and wishes to coordinate the supply chain via wholesale price and holding cost subsidy. The retail price is either exogenous or endogenous. The market demand is described by the market share attraction model based on all retailers'shelf‐spaces and retail prices. We obtain optimal solutions for the centralized supply chain, where the optimal retail pricing is a modified version of the well‐known cost plus pricing strategy. We further get feasible contracts for the manufacturer to coordinate the hybrid and decentralized supply chains. The manufacturer can allocate the total profit free to himself and the retail market via the wholesale price when the retail price is exogenous, but otherwise he cannot. Finally, we point out that different characteristics of the retail market are due to different powers of the manufacturer, and the more power the manufacturer has, the simpler the contract to coordinate the chain will be. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Callable products with dependent demands

Capacity providers such as airlines often sell the same capacity to different market segments at different prices to improve their expected revenues. The absence of a secondary market, due to the nontransferability of airline tickets, gives rise to an opportunity for airlines to broker capacity between consumers with different willingness to pay. One way to broker capacity is by the introduction of callable products. The idea is similar to callable bonds where the issuer has the right, but not the obligation, to buy back the bonds at a certain price by a certain date. The idea of callable products was introduced before under the assumption that the fare-class demands are all independent. The independent assumption becomes untenable when there is significant demand recovery (respectively, demand cannibalization) when lower fares are closed (respectively, opened). In this case, consumer choice behavior should be modeled explicitly to make meaningful decisions. In this paper, we consider a general consumer choice model and develop the optimal strategy for callable products. Our numerical study illustrates how callable products are win-win-win, for the capacity provider and for both high and low fare consumers. Our studies also identify conditions for callable products to result in significant improvements in expected revenues.     

When to commit in a serial supply chain with forecast updating

Negotiations between an end product manufacturer and a parts supplier often revolve around two main issues: the supplier's price and the length of time the manufacturer is contractually held to its order quantity, commonly termed the “commitment time frame.” Because actual demand is unknown, the specification of the commitment time frame determines how the demand risk is shared among the members of the supply chain. Casual observation indicates that most manufacturers prefer to delay commitments as long as possible while suppliers prefer early commitments. In this paper, we investigate whether these goals are always in the firm's best interest. In particular, we find that the manufacturer may sometimes be better off with a contract that requires an early commitment to its order quantity, before the supplier commits resources and the supplier may sometimes be better off with a delayed commitment. We also find that the preferred commitment time frame depends upon which member of the supply chain has the power to set their exchange price. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2003