This paper considers the so-called warehouse problem with both space and injection/withdrawal capacity limits. This is a foundational problem in the merchant management of assets for the storage of commodities, such as energy sources and natural resources. When the commodity spot price evolves according to an exogenous Markov process, this work shows that the optimal inventory-trading policy of a risk-neutral merchant is characterized by two stage and spot-price dependent basestock targets. Under some assumptions, these targets are monotone in the spot price and partition the available inventory and spot-price space in each stage into three regions, where it is, respectively, optimal to buy and inject, do nothing, and withdraw and sell. In some cases of practical importance, one can easily compute the optimal basestock targets. The structure of the optimal policy is nontrivial because in each stage the merchant's qualification of high (selling) and low (buying) commodity prices in general depends on the merchant's inventory availability. This is a consequence of the interplay between the capacity and space limits of the storage asset and brings to light the nontrivial nature of the interface between trading and operations. A computational analysis based on natural gas data shows that mismanaging this interface can yield significant value losses. Moreover, adapting the merchant's optimal trading policy to the spot-price stochastic evolution has substantial value. This value can be almost entirely generated by reacting to the unfolding of price uncertainty, that is, by sequentially reoptimizing a model that ignores this source of uncertainty.inventory, production, policies, dynamic programming, Markov, finance, asset pricing, real options, industries, petroleum, natural gas

The valuation of the real option to store natural gas is a practically important problem that entails dynamic optimization of inventory trading decisions with capacity constraints in the face of uncertain natural gas price dynamics. Stochastic dynamic programming is a natural approach to this valuation problem, but it does not seem to be widely used in practice because it is at odds with the high-dimensional natural gas price evolution models that are widespread among traders. According to the practice-based literature, practitioners typically value natural gas storage heuristically. The effectiveness of the heuristics discussed in this literature is currently unknown because good upper bounds on the value of storage are not available. We develop a novel and tractable approximate dynamic programming method that, coupled with Monte Carlo simulation, computes lower and upper bounds on the value of storage, which we use to benchmark these heuristics on a set of realistic instances. We find that these heuristics are extremely fast to execute but significantly suboptimal compared to our upper bound, which appears to be fairly tight and much tighter than a simpler perfect information upper bound; computing our lower bound takes more time than using these heuristics, but our lower bound substantially outperforms them in terms of valuation. Moreover, with periodic reoptimizations embedded in Monte Carlo simulation, the practice-based heuristics become nearly optimal, with one exception, at the expense of higher computational effort. Our lower bound with reoptimization is also nearly optimal, but exhibits a higher computational requirement than these heuristics. Besides natural gas storage, our results are potentially relevant for the valuation of the real option to store other commodities, such as metals, oil, and petroleum products.

The paper considers the single vehicle routing problem with stochastic demands. While most of the literature has studied the a priori solution approach, this work focuses on computing a reoptimization-type routing policy. This is obtained by sequentially improving a given a priori solution by means of a rollout algorithm. The resulting rollout policy appears to be the first computationally tractable algorithm for approximately solving the problem under the reoptimization approach. After describing the solution strategy and providing properties of the rollout policy, the policy behavior is analyzed by conducting a computational investigation. Depending on the quality of the initial solution, the rollout policy obtains 1% to 4% average improvements on the a priori approach with a reasonable computational effort.

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We consider the vehicle routing problem with stochastic demands (VRPSD) under reoptimization. We develop and analyze a finite-horizon Markov decision process (MDP) formulation for the single vehicle case, and establish a partial characterization of the optimal policy. We also propose a heuristic solution methodology for our MDP, named partial reoptimization, based on the idea of restricting attention to a subset of all the possible states and computing an optimal policy on this restricted set of states. We discuss two families of computationally efficient partial reoptimization heuristics and illustrate their performance on a set of instances with up to and including 100 customers. Comparisons with an existing heuristic from the literature and a lower bound computed with complete knowledge of customer demands show that our best partial reoptimization heuristics outperform this heuristic and are on average no more than 10-13% away from this lower bound, depending on the type of instances.

W e investigate the management of a merchant wind energy farm co-located with a grid-level storage facility and connected to a market through a transmission line. We formulate this problem as a Markov decision process (MDP) with stochastic wind speed and electricity prices. Consistent with most deregulated electricity markets, our model allows these prices to be negative. As this feature makes it difficult to characterize any optimal policy of our MDP, we show the optimality of a stage-and partial-state-dependent-threshold policy when prices can only be positive. We extend this structure when prices can also be negative to develop heuristic one (H1) that approximately solves a stochastic dynamic program. We then simplify H1 to obtain heuristic two (H2) that relies on a price-dependent-threshold policy and derivative-free deterministic optimization embedded within a Monte Carlo simulation of the random processes of our MDP. We conduct an extensive and data-calibrated numerical study to assess the performance of these heuristics and variants of known ones against the optimal policy, as well as to quantify the effect of negative prices on the value added by and environmental benefit of storage. We find that (i) H1 computes an optimal policy and on average is about 17 times faster to execute than directly obtaining an optimal policy; (ii) H2 has a near optimal policy (with a 2.86% average optimality gap), exhibits a two orders of magnitude average speed advantage over H1, and outperforms the remaining considered heuristics; (iii) storage brings in more value but its environmental benefit falls as negative electricity prices occur more frequently in our model.Note: All experiments are run on a computer with Intel(R) Core(TM) i7-3770K 3.40 GHz CPU and 8 GB RAM.Zhou, Scheller-Wolf, Secomandi, and Smith: Managing Wind-Based Electricity Generation Production and Operations Management 28(4), pp. 970-989,

Electricity cannot yet be stored on a large scale, but technological advances leading to cheaper and more efficient industrial batteries make grid-level storage of electricity surpluses a natural choice. Because electricity prices can be negative, it is unclear how the presence of negative prices might affect the storage policy structure known to be optimal when prices are only nonnegative, or even how important it is to consider negative prices when managing an industrial battery. For fast storage (a storage facility that can both be fully emptied and filled up in one decision period), we show analytically that negative prices can substantially alter the optimal storage policy structure, e.g., all else being equal, it can be optimal to empty an almost empty storage facility and fill up an almost full one. For more typical slow grid-level electricity storage, we numerically establish that ignoring negative prices could result in a considerable loss of value when negative prices occur more than 5% of the time. Negative prices raise another possibility: rather than storing surpluses, a merchant might buy negatively priced electricity surpluses and dispose of them, e.g., using load banks. We find that the value of such a disposal strategy is substantial, e.g., about $118 per kilowatt-year when negative prices occur 10% of the time, but smaller than that of the storage strategy, e.g., about $391 per kilowatt-year using a typical battery. However, devices for disposal are much cheaper than those for storage. Our results thus have ramifications for merchants as well as policy makers. This paper was accepted by Serguei Netessine, operations management.

Least squares Monte Carlo (LSM) is a state-of-the-art approximate dynamic programming approach used in financial engineering and real options to value and manage options with early or multiple exercise opportunities. It is also applicable to capacity investment and inventory/production management problems with demand/supply forecast updates arising in operations and hydropowerreservoir management. LSM has two variants, referred to as regress-now/later (LSMN/L), which compute continuation/value function approximations (C/VFAs). We provide novel numerical evidence for the relative performance of these methods applied to energy swing and storage options, two typical real options, using a common price evolution model. LSMN/L estimate C/VFAs that yield equally accurate (near optimal) and precise lower and dual (upper) bounds on the value of these real options. Estimating the LSMN/L C/VFAs and their associated lower bounds takes similar computational effort. In contrast, the estimation of a dual bound using the LSML VFA instead of the LSMN CFA takes seconds rather than minutes or hours. This finding suggests the use of LSML in lieu of LSMN when estimating dual bounds on the value of early or multiple exercise options, as well as of related capacity investment and inventory/production policies.