Distributionally robust trading strategies for renewable energy producers

Renewable energy generation is to be offered through electricity markets, quite some time in advance. This then leads to a problem of decision-making under uncertainty, which may be seen as a newsvendor problem. Contrarily to the conventional case for which underage and overage penalties are known, such penalties in the case of electricity markets are unknown, and difficult to estimate. In addition, one is actually only penalized for either overage or underage, not both. Consequently, we look at a slightly different form of a newsvendor problem, for a price-taker participant offering in electricity markets, which we refer to as Bernoulli newsvendor problem. After showing that its solution is similar to the classical newsvendor problem, we then introduce distributionally robust versions, with ambiguity possibly about both the probabilistic forecasts for power generation and the chance of success of the Bernoulli variable. All these distributionally robust Bernoulli newsvendor problems admit closed-form solutions. We finally use simulation studies, as well as a real-world case-study application, to illustrate the workings and benefits from the approach

Conic Reformulations for Kullback-Leibler Divergence Constrained Distributionally Robust Optimization and Applications

In this paper, we consider a distributionally robust optimization (DRO) model in which the ambiguity set is defined as the set of distributions whose Kullback-Leibler (KL) divergence to an empirical distribution is bounded. Utilizing the fact that KL divergence is an exponential cone representable function, we obtain the robust counterpart of the KL divergence constrained DRO problem as a dual exponential cone constrained program under mild assumptions on the underlying optimization problem. The resulting conic reformulation of the original optimization problem can be directly solved by a commercial conic programming solver. We specialize our generic formulation to two classical optimization problems, namely, the Newsvendor Problem and the Uncapacitated Facility Location Problem. Our computational study in an out-of-sample analysis shows that the solutions obtained via the DRO approach yield significantly better performance in terms of the dispersion of the cost realizations while the central tendency deteriorates only slightly compared to the solutions obtained by stochastic programming

Robust newsvendor games with ambiguity in demand distributions

In classical newsvendor games, vendors collaborate to serve their aggregate demand whose joint distribution is assumed known with certainty. We investigate a new class of newsvendor games with ambiguity in the joint demand distributions, which is represented by a Fréchet class of distributions with some, possibly overlapping, marginal information. To model this new class of games, we use ideas from distributionally robust optimization to handle distributional ambiguity and study the robust newsvendor games. We provide conditions for the existence of core solutions of these games using the structural analysis of the worst-case joint demand distributions of the corresponding distributionally robust newsvendor optimization problem

Distribution-free Inventory Risk Pooling in a Multi-location Newsvendor

With rapidly increasing e-commerce sales, firms are leveraging the virtual pooling of online demands across customer locations in deciding the amount of inventory to be placed in each node in a fulfillment network. Such solutions require knowledge of the joint distribution of demands; however, in reality, only partial information about the joint distribution may be reliably estimated. We propose a distributionally robust multi-location newsvendor model for network inventory optimization where the worst-case expected cost is minimized over the set of demand distributions satisfying the known mean and covariance information. For the special case of two homogeneous customer locations with correlated demands, we show that a six-point distribution achieves the worst-case expected cost, and derive a closed-form expression for the optimal inventory decision. The general multi-location problem can be shown to be NP-hard. We develop a computationally tractable upper bound through the solution of a semidefinite program (SDP), which also yields heuristic inventory levels, for a special class of fulfillment cost structures, namely nested fulfillment structures. We also develop an algorithm to convert any general distance-based fulfillment cost structure into a nested fulfillment structure which tightly approximates the expected total fulfillment cost.https://deepblue.lib.umich.edu/bitstream/2027.42/146785/1/1389_Govindarajan.pd

Optimized Dimensionality Reduction for Moment-based Distributionally Robust Optimization

Moment-based distributionally robust optimization (DRO) provides an optimization framework to integrate statistical information with traditional optimization approaches. Under this framework, one assumes that the underlying joint distribution of random parameters runs in a distributional ambiguity set constructed by moment information and makes decisions against the worst-case distribution within the set. Although most moment-based DRO problems can be reformulated as semidefinite programming (SDP) problems that can be solved in polynomial time, solving high-dimensional SDPs is still time-consuming. Unlike existing approximation approaches that first reduce the dimensionality of random parameters and then solve the approximated SDPs, we propose an optimized dimensionality reduction (ODR) approach. We first show that the ranks of the matrices in the SDP reformulations are small, by which we are then motivated to integrate the dimensionality reduction of random parameters with the subsequent optimization problems. Such integration enables two outer and one inner approximations of the original problem, all of which are low-dimensional SDPs that can be solved efficiently. More importantly, these approximations can theoretically achieve the optimal value of the original high-dimensional SDPs. As these approximations are nonconvex SDPs, we develop modified Alternating Direction Method of Multipliers (ADMM) algorithms to solve them efficiently. We demonstrate the effectiveness of our proposed ODR approach and algorithm in solving two practical problems. Numerical results show significant advantages of our approach on the computational time and solution quality over the three best possible benchmark approaches. Our approach can obtain an optimal or near-optimal (mostly within 0.1%) solution and reduce the computational time by up to three orders of magnitude

Inventory Sharing and Demand-Side Underweighting

Problem definition: Transshipment/inventory sharing has been used in practice because of its risk-pooling potential. However, human decision makers play a critical role in making inventory decisions in an inventory sharing system, which may affect its benefits. We investigate whether the opportunity to transship inventory influences decision makers’ inventory decisions and whether, as a result, the intended risk-pooling benefits materialize. Academic/practical relevance: Previous research in transshipment, which is focused on finding optimal stocking and sharing decisions, assumes rational decision making without any systematic bias. As one of the first to study inventory sharing from a behavioral perspective, we demonstrate a persistent stocking-decision bias relevant for inventory sharing systems. Methodology: We develop a behavioral model of a multilocation inventory system with transshipments. Using four behavioral studies, we identify, test, estimate, and mitigate a demand-side underweighting bias: although inventory sharing brings both a supply-side benefit and a demand-side benefit, players underestimate the latter. We show analytically that such bias leads to underordering. We also explore whether reframing the inventory sharing decision reduces this bias. Results: Our results show that subjects persistently reduce their order quantities when transshipments are allowed. This underordering, which persists even when a decision-support system suggests optimal quantities, causes insufficient inventory in the system, in turn reducing the risk-pooling benefits of inventory sharing. Underordering is evidently caused by an underweighting bias; although players correctly estimate the supply-side potential from transshipment, they only estimate 20% of the demand-side potential. Managerial implications: Although inventory sharing can profitably reduce inventory, too much underordering undermines its intended risk-pooling benefits. The demand-side benefits of transshipment need to be emphasized when implementing inventory sharing systems

How Big Should Your Data Really Be? Data-Driven Newsvendor and the Transient of Learning

We study the classical newsvendor problem in which the decision-maker must trade-off underage and overage costs. In contrast to the typical setting, we assume that the decision-maker does not know the underlying distribution driving uncertainty but has only access to historical data. In turn, the key questions are how to map existing data to a decision and what type of performance to expect as a function of the data size. We analyze the classical setting with access to past samples drawn from the distribution (e.g., past demand), focusing not only on asymptotic performance but also on what we call the transient of learning, i.e., performance for arbitrary data sizes. We evaluate the performance of any algorithm through its worst-case relative expected regret, compared to an oracle with knowledge of the distribution. We provide the first finite sample exact analysis of the classical Sample Average Approximation (SAA) algorithm for this class of problems across all data sizes. This allows to uncover novel fundamental insights on the value of data: it reveals that tens of samples are sufficient to perform very efficiently but also that more data can lead to worse out-of-sample performance for SAA. We then focus on the general class of mappings from data to decisions without any restriction on the set of policies and derive an optimal algorithm as well as characterize its associated performance. This leads to significant improvements for limited data sizes, and allows to exactly quantify the value of historical information