Nicola Secomandi

A Rollout Policy for the Vehicle Routing Problem with Stochastic Demands

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.

Optimal Commodity Trading with a Capacitated Storage Asset

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 merchants qualification of high (selling) and low (buying) commodity prices in general depends on the merchants 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 merchants 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.

An Approximate Dynamic Programming Approach to Benchmark Practice-Based Heuristics for Natural Gas Storage Valuation

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.

Reoptimization Approaches for the Vehicle-Routing Problem with Stochastic Demands

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.

Analysis of a Rollout Approach to Sequencing Problems with Stochastic Routing Applications

The paper considers sequencing problems, the traveling salesman problem being their natural representative. It studies a rollout approach that employs a cyclic heuristic as its main base algorithm. The theoretical analysis establishes that it is guaranteed to improve (at least in a weak sense) the quality of any feasible solution to a given sequencing problem. Besides other applications, the paper shows that it is well suited for applications that are embedded in dynamic and stochastic environments. The computational performance of the approach is investigated with applications to two stochastic routing problems. The dynamic version of the heuristic appears to be the first algorithm available in the literature to approximately solve a variant of one of these problems.

On the Pricing of Natural Gas Pipeline Capacity

Pipelines play a critical role in matching the supply and demand of natural gas. The pricing of their capacity is an important problem in practice for pipeline companies and the users of this capacity, which include shippers such as natural gas merchants, producers, and local distribution companies. This paper conducts a normative analysis of how pipeline capacity should be priced by each of these players. Although the trading value of this capacity should be relevant to merchants and its substitution and congestion values to shippers and pipelines, respectively, this analysis shows that all of these are equivalent values. Thus pipeline capacity should be priced at its trading value, a prediction that can be empirically investigated. This paper also conducts an empirical analysis of this prediction based on transacted prices of transport contracts for the capacity of the Tennessee Gas Pipeline, a major interstate pipeline in the United States, and finds support for it. This analysis suggests that the uncertainty in the evolution of natural gas prices is an important driver of operational performance in the pricing of pipeline capacity. The results of this paper have potential relevance for the pricing of the capacity of other commodity conversion assets.

An Analysis of the Control-Algorithm Re-solving Issue in Inventory and Revenue Management

While inventory-and revenue-management problems can be represented as Markov decision process (MDP) models, in some cases the well-known dynamic-programming curse of dimensionality makes it computationally prohibitive to solve them exactly. An alternative solution, called here the control-algorithm approach, is to use a math program (MP) to approximately represent the MDP and use its optimal solution to heuristically instantiate the parameters of the decision rules of a given set of control policies. As new information is observed over time, the control algorithm can incorporate it by re-solving the MP and revising the parameters of the decision rules with the newly obtained solution. The re-solving issue arises when one reflects on the consequences of this revision: Does the performance of the control algorithm really improve by revising its decision-rule instantiation with the solution of the re-solved MP, or should an appropriate modification of the prior solution be used? This paper analyzes the control-algorithm re-solving issue for a class of finite-horizon inventory-and revenue-management problems. It establishes sufficient conditions under which re-solving does not deteriorate the performance of a control algorithm, and it applies these results to control algorithms for network revenue management and multiproduct make-to-order production with lost sales and positive lead time.

Valuation of Storage at a Liquefied Natural Gas Terminal

The valuation of the real option to store liquefied natural gas (LNG) at the downstream terminal of an LNG value chain is an important problem in practice. Because the exact valuation of this real option is computationally intractable, we develop a novel and tractable heuristic model for its strategic valuation that integrates models of LNG shipping, natural gas price evolution, and inventory control and sale into the wholesale natural gas market. We incorporate real and estimated data to quantify the value of this real option and its dependence on the throughput of an LNG chain, the type of price variability, the type of inventory control policy employed, and the level of stochastic variability in both the shipping model and the natural gas price model used. In addition, we develop an imperfect information dual upper bound to assess the effectiveness of our heuristic and find that our method is near optimal. Our approach also has potential relevance to value the real option to store other commodities in facilities located downstream from a commodity production or transportation stage, such as petroleum and agricultural products, chemicals, and metals, or the real option to store the input used in the production of a commodity such as electricity.

Managing Wind-Based Electricity Generation in the Presence of Storage and Transmission Capacity

We 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.

Optimal Energy Procurement in Spot and Forward Markets

Spot and forward purchases for delivery on the usage date play an important role in matching the supply and the uncertain demand of energy because storage capacity for energy, such as electricity, natural gas, and oil, is limited. Transaction costs tend to be larger in spot than forward energy markets near maturity. Partially procuring supply in the forward market, rather than entirely in the spot market, is thus a potentially valuable real option, which we call the forward procurement option . We investigate the optimal value and management of this real option as well as their sensitivities to parameters of interest. Our research quantifies the value of the forward procurement option on realistic natural gas instances, also suggesting that procuring the demand forecast in the forward market is nearly optimal. This policy greatly simplifies the management of this real option without an appreciable loss of value. We provide some theoretical support for this numerical finding. Beyond energy, our research has potential relevance for the procurement of other commodities, such as metals and agricultural products.