skip to main content

Title: Joint Inventory‐Pricing Optimization with General Demands: An Alternative Approach for Concavity Preservation

In this study, we provide an alternative approach for proving the preservation of concavity together with submodularity, and apply it to finite‐horizon non‐stationary joint inventory‐pricing models with general demands. The approach characterizes the optimal price as a function of the inventory level. Further, it employs the Cauchy–Schwarz and arithmetic‐geometric mean inequalities to establish a relation between the one‐period profit and the profit‐to‐go function in a dynamic programming setting. With this relation, we demonstrate that the one‐dimensional concavity of the price‐optimized profit function is preserved as a whole, instead of separately determining the (two‐dimensional) joint concavities in price (or mean demand/risk level) and inventory level for the one‐period profit and the profit‐to‐go function in conventional approaches. As a result, we derive the optimality condition for a base‐stock, list‐price (BSLP) policy for joint inventory‐pricing optimization models with general form demand and profit functions. With examples, we extend the optimality of a BSLP policy to cases with non‐concave revenue functions in mean demand. We also propose the notion of price elasticity of the slope (PES) and articulate the condition as that in response to a price change of the commodity, the percentage change in the slope of the expected sales is greater than the percentage change in the slope of the expected one‐period profit. The concavity preservation conditions for the additive, generalized additive, and location‐scale demand models in the literature are unified under this framework. We also obtain the conditions under which a BSLP policy is optimal for the logarithmic and exponential form demand models.

more » « less
Author(s) / Creator(s):
 ;  ;  
Publisher / Repository:
SAGE Publications
Date Published:
Journal Name:
Production and Operations Management
Medium: X Size: p. 2390-2404
["p. 2390-2404"]
Sponsoring Org:
National Science Foundation
More Like this
  1. We consider the periodic review dynamic pricing and inventory control problem with fixed ordering cost. Demand is random and price dependent, and unsatisfied demand is backlogged. With complete demand information, the celebrated [Formula: see text] policy is proved to be optimal, where s and S are the reorder point and order-up-to level for ordering strategy, and [Formula: see text], a function of on-hand inventory level, characterizes the pricing strategy. In this paper, we consider incomplete demand information and develop online learning algorithms whose average profit approaches that of the optimal [Formula: see text] with a tight [Formula: see text] regret rate. A number of salient features differentiate our work from the existing online learning researches in the operations management (OM) literature. First, computing the optimal [Formula: see text] policy requires solving a dynamic programming (DP) over multiple periods involving unknown quantities, which is different from the majority of learning problems in OM that only require solving single-period optimization questions. It is hence challenging to establish stability results through DP recursions, which we accomplish by proving uniform convergence of the profit-to-go function. The necessity of analyzing action-dependent state transition over multiple periods resembles the reinforcement learning question, considerably more difficult than existing bandit learning algorithms. Second, the pricing function [Formula: see text] is of infinite dimension, and approaching it is much more challenging than approaching a finite number of parameters as seen in existing researches. The demand-price relationship is estimated based on upper confidence bound, but the confidence interval cannot be explicitly calculated due to the complexity of the DP recursion. Finally, because of the multiperiod nature of [Formula: see text] policies the actual distribution of the randomness in demand plays an important role in determining the optimal pricing strategy [Formula: see text], which is unknown to the learner a priori. In this paper, the demand randomness is approximated by an empirical distribution constructed using dependent samples, and a novel Wasserstein metric-based argument is employed to prove convergence of the empirical distribution. This paper was accepted by J. George Shanthikumar, big data analytics. 
    more » « less

    When supply disruptions occur, firms want to employ an effective pricing strategy to reduce losses. However, firms typically do not know precisely how customers will react to price changes in the short term, during a disruption. In this article, we investigate the customer's order variability and the firm's profit under several representative heuristic pricing strategies, including no change at all (fixed pricing strategy), changing the price only (naive pricing strategy), and adjusting the belief and price simultaneously (one‐period correction [1PC] and regression pricing strategies). We show that the fixed pricing strategy creates the most stable customer order process, but it brings lower profit than the naive pricing strategy in most cases. The 1PC pricing strategy produces a more volatile customer order process and smaller profit than the naive one does. Although the regression pricing strategy is a more advanced approach, it leads to lower profit and greater customer order variability than the naive pricing strategy (but the opposite when compared to the 1PC strategy). We conclude that (i) completely eliminating the customer order variability by employing a fixed pricing strategy is not advisable and adjusting the price to match supply with demand is necessary to improve the profit; (ii) frequently adjusting the belief about customer behaviors under imperfect information may increase the customer's order variability and reduce the firm's profit. The conclusions are robust to the inventory assumption (i.e., without or with inventory carryover) and the firm's objective (i.e., market clearance or profit maximization).

    more » « less
  3. This paper studies a dynamic pricing problem under model misspecification. To characterize model misspecification, we adopt the ε-contamination model—the most fundamental model in robust statistics and machine learning. In particular, for a selling horizon of length T, the online ε-contamination model assumes that demands are realized according to a typical unknown demand function only for [Formula: see text] periods. For the rest of [Formula: see text] periods, an outlier purchase can happen with arbitrary demand functions. The challenges brought by the presence of outlier customers are mainly due to the fact that arrivals of outliers and their exhibited demand behaviors are completely arbitrary, therefore calling for robust estimation and exploration strategies that can handle any outlier arrival and demand patterns. We first consider unconstrained dynamic pricing without any inventory constraint. In this case, we adopt the Follow-the-Regularized-Leader algorithm to hedge against outlier purchase behavior. Then, we introduce inventory constraints. When the inventory is insufficient, we study a robust bisection-search algorithm to identify the clearance price—that is, the price at which the initial inventory is expected to clear at the end of T periods. Finally, we study the general dynamic pricing case, where a retailer has no clue whether the inventory is sufficient or not. In this case, we design a meta-algorithm that combines the previous two policies. All algorithms are fully adaptive, without requiring prior knowledge of the outlier proportion parameter ε. Simulation study shows that our policy outperforms existing policies in the literature. 
    more » « less
  4. Motivated by applications from gig economy and online marketplaces, we study a two-sided queueing system under joint pricing and matching controls. The queueing system is modeled by a bipartite graph, where the vertices represent customer or server types and the edges represent compatible customer-server pairs. We propose a threshold-based two-price policy and queue length-based maximum-weight matching policy and show that it achieves a near-optimal profit. We study the system under the large-scale regime, wherein the arrival rates are scaled up, and under the large-market regime, wherein both the arrival rates and numbers of customer and server types increase. We show that two-price policy is a primary driver for optimality in the large-scale regime. We demonstrate the advantage of maximum-weight matching with respect to the number of customer and server types. Concurrently, we show that the interplay of pricing and matching is crucial for optimality in the large-market regime. 
    more » « less
  5. Sustainability and long‐term prosperity are chronic challenges in the agriculture sector of many countries. To address such challenges, farmer cooperatives are formed as an innovative approach to improve the livelihoods of millions of farmers around the world. Inspired by real‐life practice in the Kenya coffee industry, we study a class of stochastic and dynamic inventory models for storable agricultural products with random exogenous supply and price. For a variety of cost functions relevant in practice, we characterize the optimal selling policies to maximize the farmer cooperatives’ expected profit. We show that for concave inventory holding cost, thesell‐all‐or‐retain‐all(rR) (orsell‐all‐or‐retain‐all R) policies are optimal with (without) the fixed selling cost; for convex holding cost, thesell‐down‐to(Ss) (orsell‐down‐to s) policies are optimal with (without) the fixed selling cost. For the special case of linear holding cost, the optimal policy is acut‐off pricepolicy and we derive closed‐form expressions for the optimal policy and the optimal total discounted profit. We discuss the model extensions to include general stochastic harvest and price processes, selling/storage capacity limits, price‐dependent random demand with a spot market, and the flexibility of procurement from other producers, and then perform a numerical study to quantify the impact of the optimal solutions. Reconciling the theory with practice, useful insights and guidelines are provided to help farmer cooperatives make strategic selling decisions.

    more » « less