Georgia Perakis

Regret in the Newsvendor Model with Partial Information

Traditional stochastic inventory models assume full knowledge of the demand probability distribution. However, in practice, it is often difficult to completely characterize the demand distribution, especially in fast-changing markets. In this paper, we study the newsvendor problem with partial information about the demand distribution (e.g., mean, variance, symmetry, unimodality). In particular, we derive the order quantities that minimize the newsvendors maximum regret of not acting optimally. Most of our solutions are tractable, which makes them attractive for practical application. Our analysis also generates insights into the choice of the demand distribution as an input to the newsvendor model. In particular, the distributions that maximize the entropy perform well under the regret criterion. Our approach can be extended to a variety of problems that require a robust but not conservative solution.

The Price of Anarchy in Supply Chains: Quantifying the Efficiency of Price-Only Contracts

In this paper, we quantify the efficiency of decentralized supply chains that use price-only contracts. With a price-only contract, a buyer and a seller agree only on a constant transaction price, without specifying the amount that will be transferred. It is well known that these contracts do not provide incentives to the parties to coordinate their inventory/capacity decisions. We measure efficiency with the price of anarchy (PoA), defined as the largest ratio of profits between the integrated supply chain (that is, fully coordinated) and the decentralized supply chain. We characterize the efficiency of various supply chain configurations: push or pull inventory positioning, two or more stages, serial or assembly systems, single or multiple competing suppliers, and single or multiple competing retailers.

A Robust Optimization Approach to Dynamic Pricing and Inventory Control with no Backorders

In this paper, we present a robust optimization formulation for dealing with demand uncertainty in a dynamic pricing and inventory control problem for a make-to-stock manufacturing system. We consider a multi-product capacitated, dynamic setting. We introduce a demand-based fluid 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 show that the robust formulation is of the same order of complexity as the nominal problem and demonstrate how to adapt the nominal (deterministic) solution algorithm to the robust problem.

Competitive Multi-period Pricing for Perishable Products: A Robust Optimization Approach

We study a multi-period oligopolistic market for a single perishable product with fixed inventory. Our goal is to address the competitive aspect of the problem together with demand uncertainty using ideas from robust optimization and variational inequalities. The demand function for each seller has some associated uncertainty and we assume that the sellers would like to adopt a policy that is robust to adverse uncertain circumstances. We believe this is the first paper that uses robust optimization for dynamic pricing under competition. In particular, starting with a given fixed inventory, each seller competes over a multi-period time horizon in the market by setting prices and protection levels for each period at the beginning of the time horizon. Any unsold inventory at the end of the horizon is worthless. The sellers do not have the option of periodically reviewing and replenishing their inventory. We study non-cooperative Nash equilibrium policies for sellers under such a model. This kind of a setup can be used to model pricing of air fares, hotel reservations, bandwidth in communication networks, etc. In this paper we demonstrate our results through some numerical examples.

Dynamic Pricing: A learning Approach

We present an optimization approach for jointly learning the demand as a function of price, and dynamically setting prices of products in order to maximize expected revenue. The models we consider do not assume that the demand as a function of price is known in advance, but rather assume parametric families of demand functions that are learned over time. In the first part of the paper, we consider the noncompetitive case and present dynamic programming algorithms of increasing computational intensity with incomplete state information for jointly estimating the demand and setting prices as time evolves. Our computational results suggest that dynamic programming based methods outperform myopic policies often significantly. In the second part of the paper, we consider a competitive oligopolistic environment. We introduce a more sophisticated model of demand learning, in which the price elasticities are slowly varying functions of time, and allows for increased flexibility in the modeling of the demand. We propose methods based on optimization for jointly estimating the Firm’s own demand, its competitors’ demands, and setting prices. In preliminary computational work, we found that optimization based pricing methods offer increased expected revenue for a firm independently of the policy the competitor firm is following.

The “Price of Anarchy” Under Nonlinear and Asymmetric Costs

In this paper we characterize the “price of anarchy,” i.e., the inefficiency between user and system optimal solutions, when costs are nonseparable, asymmetric and nonlinear, generalizing earlier work that has addressed “the price of anarchy” under separable costs. The results in this paper apply primarily to nonatomic games such as the traffic equilibrium problem, but also in competitive multiperiod pricing and competitive supply chain settings. The bounds established in this paper are tight and explicitly account for the degree of asymmetry and nonlinearity of the cost function. We first provide a proof method for problems with a positive definite Jacobian matrix. Subsequently, we use ideas from semidefinite optimization in order to account for problems with a positive semidefinite Jacobian matrix (where the first approach does not apply). This latter connection also provides a different application of semidefinite optimization.

Robust Controls for Network Revenue Management

Revenue management models traditionally assume that future demand is unknown but can be described by a stochastic process or a probability distribution. Demand is, however, often difficult to characterize, especially in new or nonstationary markets. In this paper, we develop robust formulations for the capacity allocation problem in revenue management using the maximin and the minimax regret criteria under general polyhedral uncertainty sets. Our approach encompasses the following open-loop controls: partitioned booking limits, nested booking limits, displacement-adjusted virtual nesting, and fixed bid prices. In specific problem instances, we show that a booking policy of the type of displacement-adjusted virtual nesting is robust, both from maximin and minimax regret perspectives. Our numerical analysis reveals that the minimax regret controls perform very well on average, despite their worst-case focus, and outperform the traditional controls when demand is correlated or censored. In particular, on real large-scale problem sets, the minimax regret approach outperforms by up to 2% the traditional heuristics. The maximin controls are more conservative but have the merit of being associated with a minimum revenue guarantee. Our models are scalable to solve practical problems because they combine efficient (exact or heuristic) solution methods with very modest data requirements.

The Price of Anarchy when Costs Are Non-separable and Asymmetric

In this paper we characterize the “price of anarchy”, i.e., the inefficiency between user and system optimal solutions, when costs are non-separable, asymmetric and nonlinear, generalizing earlier work that has addressed “the price of anarchy” under separable costs. This generalization models traffic equilibria, competitive multi-period pricing and competitive supply chains. The bounds established in this paper are tight and explicitly account for the degree of asymmetry and nonlinearity of the cost function. We introduce an alternate proof method for providing bounds that uses ideas from semidefinite optimization. Finally, in the context of multi-period pricing our analysis establishes that user and system optimal solutions coincide.

Dynamic Pricing and Inventory Control: Uncertainty and Competition

In this paper, we study a make-to-stock manufacturing system where two firms compete through dynamic pricing and inventory control. Our goal is to address competition (in particular a duopoly setting) together with the presence of demand uncertainty. We consider a dynamic setting where multiple products share production capacity. We introduce a demand-based fluid model where the demand is a linear function of the price of the supplier and of her competitor, the inventory and production costs are quadratic, and all coefficients are time dependent. A key part of the model is that no backorders are allowed and the strategy of a supplier depends on her competitors strategy. First, we reformulate the robust problem as a fluid model of similar form to the deterministic one and show existence of a Nash equilibrium in continuous time. We then discuss issues of uniqueness and address how to compute a particular Nash equilibrium, i.e., the normalized Nash equilibrium.

Consumer Choice Model for Forecasting Demand and Designing Incentives for Solar Technology

In this paper, we develop a model for the adoption of solar photovoltaic technology by residential consumers. In particular, we assume consumers purchase these solar panels according to a discrete choice model. The technology adoption process is reinforced by network externalities such as imitating customer behavior and cost improvements through learning-by-doing. Using this model, we develop a framework for policy makers to find optimal subsidies in order to achieve a desired adoption target with minimum cost for the system. We discuss the structure of the optimal subsidy policy and how the overall system cost changes with the adoption target. Furthermore, we validate the model through an empirical study of the German solar market, where we estimate the model parameters, generate adoption forecasts and demonstrate how to solve the policy design problem. We use this framework to show that the current policies in Germany are not efficient. In particular, our study suggests that their subsidies should be higher in the near future and the gradual phase-out of the subsidies should occur faster.