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.     

How Couples Manage Interracial and Intercultural Differences: Implications for Clinical Practice

This study focused on how couples managed their interracial and intercultural differences. To understand their experiences, a qualitative grounded theory analysis was used (n = 17). Analysis revealed that couples experienced most issues as cultural issues; race only occurred during their interactions with “others.” They appeared to organize their responses according to four relationship structures: Integrated, Singularly Assimilated, Coexisting, and Unresolved. Couples in each of these structures managed daily process through four sets of relationship strategies: (a) creating a “we,” (b) framing differences, (c) emotional maintenance, and (d) positioning in relationship to familial and societal context. These findings are a step toward a strength‐based and research‐informed education and clinical interventions for this population. Video Abstract     

On Khatri's characterization of the inverse-Gaussian distribution

Khatri has given a characterization of the inverse-Gaussian distribution by the independence of two statistics. His proof involves assumptions on the existence of certain moments. In this note, we offer a short proof using only the positivity of the random variable X₁.     

Differential Pricing for Information Sharing Under Competition

We consider a two‐echelon supply chain with a manufacturer supplying to multiple downstream retailers engaged in differentiated Cournot competition. Each retailer has private information about uncertain demand. The manufacturer is the Stackelberg leader who sets the contract terms with the retailers, and benefits from retailers sharing their private information. When all retailers are given the same wholesale price, truthful information sharing is not an equilibrium outcome. We propose two variants of differential pricing mechanisms that induce truthful information sharing by all retailers. The first variant rewards a retailer for providing optimistic information and achieves truthful information sharing as a unique equilibrium. The differential pricing mechanism is optimal in the class of linear‐price, incentive‐compatible, direct mechanisms. The second variant, which incorporates provision for a fixed payment in addition to wholesale prices, preserves all the equilibrium properties of the first variant and additionally “nearly coordinates” the supply chain. Our analysis of differential pricing with a fixed payment provides interesting observations regarding the relationship between product substitutability, number of retailers, information precision, and market power. As products become closer substitutes and/or number of retailers increase, the manufacturer's market power increases, enabling her to extract a larger fraction of the supply chain surplus.     

Supply Chain Design and Carbon Penalty: Monopoly vs. Monopolistic Competition

This paper studies whether imposing carbon costs changes the supply chain structure and social welfare. We explore the problem from a central policymaker's perspective who wants to maximize social welfare. We consider two stakeholders, retailers, and consumers, who optimize their own objectives (i.e., profits and net utility) and three competitive settings (i.e., monopoly, monopolistic competition with symmetric market share, and monopolistic competition with asymmetric market share). For the monopoly case, we find that when the retailer's profit is high, imposing some carbon emission charges on the retailer and the consumers does not substantially change the supply chain structure or the social welfare. However, when the retailer's profit is low, imposing carbon costs optimally can lead to a significant increase in social welfare. Moreover, the impact of imposing carbon emission charges becomes more significant when the degree of competition increases. Additionally, the quantum of benefit may depend only on factors common across industries, such as fuel and carbon costs.     

A Multi‐Supplier Sourcing Problem with a Preference Ordering of Suppliers

We study a sourcing problem faced by a firm that seeks to procure a product or a component from a pool of alternative suppliers. The firm has a preference ordering of the suppliers based on factors such as their past performance, quality, service, geographical location, and financial strength, which are commonly included in a supplier scorecard system. Thus, the firm first uses available inventory from supplier 1, if any, then supplier 2, if any, and so on. The suppliers differ in costs and prices. The buyer firm seeks to determine which suppliers to purchase from and in what quantities to maximize its total expected profit subject to the preference ordering constraint. We present the optimal solution to this problem, and show that it has a portfolio structure. It consists of a sub‐set of suppliers that are ordered by their underage and overage costs. This portfolio achieves a substantial profit gain compared to sourcing from a unique supplier. We present an efficient algorithm to compute the optimal solution. Our model applies to component sourcing problems in manufacturing, merchandizing problems in retailing, and capacity reservation problems in services.     

The Lindsay transform of natural exponential families

Let μ be an infinitely divisible positive measure on R. If the measure ρ_μ is such that x^-2[ρ_μ(dx)—ρ_μ({0})δ₀(dx)] is the Lévy measure associated with μ and is infinitely divisible, we consider for all positive reals α and β the measure T_α,β(μ) which is the convolution of μ*^α and ρ_μ*β. For example, if μ is the inverse Gaussian law, then ρ_μ is a gamma law with paramter 3/2. Then T_α,β(μ) is an extension of the Lindsay transform of the first order, restricted to the distributions which are infinitely divisible. The main aim of this paper is to point out that it is possible to apply this transformation to all natural exponential families (NEF) with strictly cubic variance functions P. We then obtain NEF with variance functions of the form □ΔP(□Δ), where A is an affine function of the mean of the NEF. Some of these latter types appear scattered in the literature.     

Multivariate distributions with generalized inverse gaussian marginals,and associated poisson mixtures

Several types of multivariate extensions of the inverse Gaussian (IG) distribution and the reciprocal inverse Gaussian (RIG) distribution are proposed. Some of these types are obtained as random-additive-effect models by means of well-known convolution properties of the IG and RIG distributions, and they have one-dimensional IG or RIG marginals. They are used to define a flexible class of multivariate Poisson mixtures.