Selvaprabu Nadarajah

Relaxations of approximate linear programs for the real option management of commodity storage

The real option management of commodity conversion assets gives rise to intractable Markov decision processes (MDPs), in part because of the use of high-dimensional models of commodity forward curve evolution, as commonly done in practice. Focusing on commodity storage, we identify a deficiency of approximate linear programming (ALP), which we address by developing a novel approach to derive relaxations of approximate linear programs. We apply our approach to obtain a class of tractable ALP relaxations, also subsuming an existing method. We provide theoretical support for the use of these ALP relaxations rather than their associated approximate linear programs. Applied to existing natural gas storage instances, our ALP relaxations significantly outperform their corresponding approximate linear programs. Our best ALP relaxation is both near optimal and competitive with, albeit slower than, state-of-the-art methods for computing heuristic policies and lower bounds on the value of commodity storage, but is more directly applicable for dual (upper) bound estimation than these methods. Our approach is potentially relevant for the approximate solution of MDPs that arise in the real option management of other commodity conversion assets, as well as the valuation of real and financial options that depend on forward curve dynamics. This paper was accepted by Dimitris Bertsimas, optimization.

Managing Shutdown Decisions in Merchant Commodity and Energy Production: A Social Commerce Perspective

Problem definition: Merchant commodity and energy production assets operate in markets with volatile prices and exchange rates. Plant closures adversely affect societal entities beyond the specific plant being shutdown such as the parent company and the local community. Motivated by an aluminum producer, we study if mitigating these hard-to-assess broader impacts of a shutdown is financially viable using the plants operating flexibility. Academic/Practical relevance: Our social commerce perspective towards managing shutdown decisions deviates from the commonly used asset value maximization objective in merchant operations. Identifying operating policies that delay or decrease the likelihood of a shutdown without incurring a significant asset value loss supports socially-responsible plant shutdown decisions. Methodology: We formulate a constrained Markov decision process to manage shutdown decisions and limit the probability of future plant closures. We provide theoretical support for approximating this intractable model using unconstrained stochastic dynamic programs with modified shutdown costs and explore two classes of operating policies. Our first policy leverages anticipated regret theory while the second policy generalizes production-margin heuristics used in practice using machine learning. We compute the former and latter policies using a least squares Monte Carlo method and combining this method with binary classification, respectively. Results: Anticipated-regret policies possess desirable asymptotic properties absent in classification-based policies. On instances created using real data, anticipated-regret and classification-based policies outperform practice based production-margin strategies. Significant reductions in shutdown probability and delays in plant closures are possible while incurring small asset value losses. Managerial implications: A plants operating flexibility provides an effective lever to balance the social objective to reduce closures and the financial goal to maximize asset value. Adhering to both objectives requires combining short-term commitments with external stakeholders to avoid shutdown with longer-term internal efforts to reduce the probability of plant closures.

Relationship between Least Squares Monte Carlo and Approximate Linear Programming

Least squares Monte Carlo (LSM) is commonly used to manage and value early or multiple exercise financial or real options. Recent research in this area has started applying approximate linear programming (ALP) and its relaxations, which aim at addressing a possible ALP drawback. We show that regress-later LSM is itself an ALP relaxation that potentially corrects this ALP shortcoming. Our analysis consolidates two streams of research and supports using this LSM version rather than ALP on the considered models.

Dynamic Pricing for Hotel Rooms When Customers Request Multiple-Day Stays

We study the dynamic pricing problem faced by a hotel that maximizes expected revenue from a single type of rooms. Demand for the rooms is stochastic and non-stationary. Our Markov decision process formulation of this problem determines the optimal booking price of rooms (resources) for each individual day, while considering the availability of room capacity throughout the multiple-day stays (products) requested by customers. To offer attractive average daily prices for multiple-day stays, the hotel should not only substantially raise the booking prices for some high-demand days, but also significantly lower the booking prices for the low-demand days that are immediately adjacent to these high-demand days. This policy-structure insight is based on analysis and exact numerical solutions of small problem instances. For larger problem instances, we develop an approximate linear programming based pricing policy and numerically benchmark it against both a fixed-price heuristic and a single-day decomposition approach. Our pricing policy outperforms these benchmarks and generates up to 7% more revenue than the single-day decomposition approach. Enforcing our policy-structure insight, which may simplify the implementation, results in revenue losses of less than 1%. Our findings are potentially relevant beyond the hotel domain for applications involving the dynamic pricing of capacitated resources used by multiple products.

Merchant Energy Trading in a Network

The operations of merchant energy trading in wholesale markets across different locations and current and future dates can be represented as a network where storage and transport trades compete for the capacity of storage and transport assets. We study the tradeoff between storage and transport trading for a network with a single storage asset and multiple transport assets, a realistic situation that we model as a Markov decision problem (MDP). Due to the intractability of computing an optimal policy of this MDP, we leverage our structural analysis of this model to modify a least squares Monte Carlo method to obtain a heuristic policy, also computing both lower and upper bounds on the market value of an optimal policy. On a realistic natural gas application, we document a substantial tradeoff between storage and transport trading. This tradeoff is difficult to manage, as sequential storage and transport trading is considerably suboptimal, especially when prioritizing transport over storage. In contrast, our joint policy is near optimal. A practicebased method based on sequentially reoptimizing a deterministic model is also near optimal, but, even after simplification, is computationally more intensive than our approach. Moreover, we highlight the operational differences between managing storage jointly with transport assets versus as a single asset. Beyond natural gas, our research has relevance for managing the merchant trading operations of other energy sources, natural resources, and other storable commodities.

Network-Based Approximate Linear Programming for Discrete Optimization

We develop a new class of approximate linear programs (ALPs) that project the high-dimensional value function of dynamic programs onto a class of basis functions, each defined as a network that represents aggregrations over the state space. The resulting ALP is a minimum-cost flow problem over an extended variable space that synchronizes flows across multiple networks. Its solution provides a value function approximation that can be used to obtain feasible solutions and optimistic bounds. Such bounds from multiple networks are weakly stronger than their counterparts from a single network. We present a scheme for iteratively constructing a finite sequence of network ALPs with improving optimistic bounds that converge to the optimal solution value of the original problem. In addition, we provide a tractable approximation of a network ALP based on the chordalization of an auxiliary graph. We assess the performance of the ALP bounds and feasible solutions using a branch-and-bound scheme to obtain optimal solutions. We apply this scheme to challenging bilinear and routing problems arising in marketing analytics and preemptive maintenance, respectively. Numerical results show that the network ALP significantly outperforms a state-of-the-art mathematical programming solver both in solution quality and time.