We formulate the merchant trading of energy in a network of storage and transport assets as a Markov decision process with uncertain energy prices, generalizing known models. Because of the intractability of our model, we develop heuristics and both lower and dual (upper) bounds on the optimal policy value estimated within Monte Carlo simulation. We achieve tractability using linear optimization, extending near optimal approximate dynamic programming techniques for the case of a single storage asset, versions of two of which are commercially available. We propose (i) a generalization of a deterministic reoptimization heuristic, (ii) an iterative version of the least squares Monte Carlo approach, and (iii) a perfect information dual bound. We apply our methods to a set of realistic natural gas instances. The combination of our reoptimization heuristic and dual bound emerges as a practical approach to nearly optimally solve our model. Our iterative least squares Monte Carlo heuristic is also close to optimal. Compared to our other heuristic, it exhibits slightly larger optimality gaps and requires some tuning, but is faster to execute in some cases. Our methods could enhance single energy storage asset software and have potential relevance beyond our specific application.

Least squares Monte Carlo (LSM) is commonly used to manage and value intractable early and multiple exercise financial and 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 LSM is itself an ALP relaxation that potentially corrects this ALP shortcoming. Our analysis consolidates two separate streams of research and suggests using LSM rather than ALP for the considered class of problems.

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 hydropower-reservoir 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.

The real option management of commodity conversion assets gives rise to intractable Markov decision processes (MDPs), in part due to 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, which we address by developing a novel approach to derive relaxations of approximate linear programs (ALPs). 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 corresponding ALPs. Applied to existing natural gas storage instances, our ALP relaxations significantly outperform their corresponding ALPs. 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.

Commodity conversion assets play important economic roles. It is well known that the market value of these assets can be maximized by managing them as real options on the prices of their inputs and/or outputs. In particular, when futures on these inputs and outputs are traded, managing such real options, that is, valuing, hedging, and exercising them, is analogous to managing options on such futures, using risk neutral valuation and delta hedging methods. This statement holds because dynamically trading portfolios of these futures and a risk less bond can replicate the cash flows of these assets. This basic principle is not always appreciated by managers of commodity conversion assets. Moreover, determining the optimal operational cash flows of such an asset requires optimizing the asset operating policy. This issue complicates the real option management of commodity conversion assets. This chapter illustrates the application of this approach to manage a hydrocarbon cracker, a specific commodity conversion asset, using linear programming and Monte Carlo simulation. The discussion is based on a simplified representation of the operations of this asset. However, the material presented here has potential applicability to the real option management of more realistic models of hydrocarbon cracking assets, as well as other energy and commodity conversion assets.

Less-Than-Truckload (LTL) carriers generally serve geographical regions that are more localized than the inter-city line-hauls served by truckload carriers. That localization can lead to urban freight transportation routes that overlap. If trucks are traveling with less than full loads, there typically exist opportunities for carriers to collaborate over such routes. We introduce a two stage framework for LTL carrier collaboration. Our first stage involves collaboration between multiple carriers at the entrance to the city and can be formulated as a vehicle routing problem with time windows (VRPTW). We employ guided local search for solving this VRPTW. The second stage involves collaboration between carriers at transshipment facilities while executing their routes identified in phase one. For solving the second stage problem, we develop novel local search heuristics, one of which leverages integer programming to efficiently explore the union of neighborhoods defined by new problem-specific move operators. Our computational results indicate that integrating integer programming with local search results in at least an order of magnitude speed up in the second stage problem. We also perform sensitivity analysis to assess the benefits from collaboration. Our results indicate that distance savings of 7–15 % can be achieved by collaborating at the entrance to the city. Carriers involved in intra-city collaboration can further save 3–15 % in total distance traveled, and also reduce their overall route times.

Merchant commodity and energy production assets operate in markets with volatile prices and exchange rates. Producers can choose in each period between production, production suspension, and permanent shutdown. Plant closures, however, 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 plant's operating flexibility. Our social commerce perspective towards managing shutdown decisions deviates from the commonly used asset value maximization objective in merchant operations. We formulate a high-dimensional constrained Markov decision process to manage shutdown decisions. We approximate this intractable model using unconstrained stochastic dynamic programs and define operating policies that account for preferences to delay and reduce shutdowns. Our first policy leverages anticipated regret theory in behavioral psychology 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. We also propose a reoptimization heuristic to simplify the anticipated-regret policy. We show that anticipated-regret policies possess desirable asymptotic properties absent in classification- and reoptimization-based policies. On instances created using real data, anticipated-regret and classification-based policies outperform practice-based production margin strategies and, to a lesser extent, reoptimization. Specifically, the former policies decrease the shutdown probability by 25% and, in addition, delay shutdown decisions by an average of 4 years for a 4% asset value loss. Our operating policies show that unaccounted social costs amounting to a few percent of the maximum asset value can justify delaying or avoiding the use of a plant's shutdown option by adapting its operating flexibility in our application. Thus, taking a social commerce perspective towards managing a plant's operating flexibility appears financially viable.

Approximate linear programs (ALPs) are well-known models for computing value function approximations (VFAs) of intractable Markov decision processes (MDPs) arising in applications. VFAs from ALPs have desirable theoretical properties, define an operating policy, and provide a lower bound on the optimal policy cost, which can be used to assess the suboptimality of heuristic policies. However, solving ALPs near-optimally remains challenging, for example, when approximating MDPs with nonlinear cost functions and transition dynamics or when rich basis functions are required to obtain a good VFA. We address this tension between theory and solvability by proposing a convex saddle-point reformulation of an ALP that includes as primal and dual variables, respectively, a vector of basis function weights and a constraint violation density function over the state-action space. To solve this reformulation, we develop a proximal stochastic mirror descent (PSMD) method. We establish that PSMD returns a near-optimal ALP solution and a lower bound on the optimal policy cost in a finite number of iterations with high probability. We numerically compare PSMD with several benchmarks on inventory control and energy storage applications. We find that the PSMD lower bound is tighter than a perfect information bound. In contrast, the constraint sampling approach to solve ALPs may not provide a lower bound and applying row generation to tackle ALPs is not computationally viable. PSMD policies outperform problem-specific heuristics and are comparable or better than the policies obtained using constraint sampling. Overall, our ALP reformulation and solution approach broadens the applicability of approximate linear programming.

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 aggregations 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.

Large-scale constrained convex optimization problems arise in several application domains. First-order methods are good candidates to tackle such problems due to their low iteration complexity and memory requirement. The level-set framework extends the applicability of first-order methods to tackle problems with complicated convex objectives and constraint sets. Current methods based on this framework either rely on the solution of challenging subproblems or do not guarantee a feasible solution, especially if the procedure is terminated before convergence. We develop a level-set method that finds an $\epsilon$-relative optimal and feasible solution to a constrained convex optimization problem with a fairly general objective function and set of constraints, maintains a feasible solution at each iteration, and only relies on calls to first-order oracles. We establish the iteration complexity of our approach, also accounting for the smoothness and strong convexity of the objective function and constraints when these properties hold. The dependence of our complexity on $\epsilon$ is similar to the analogous dependence in the unconstrained setting, which is not known to be true for level-set methods in the literature. Nevertheless, ensuring feasibility is not free. The iteration complexity of our method depends on a condition number, while existing level-set methods that do not guarantee feasibility can avoid such dependence. We numerically validate the usefulness of ensuring a feasible solution path by comparing our approach with an existing level set method on a Neyman-Pearson classification problem.

Some prominent hotel chains quote a booking price of a particular type of rooms on each day and dynamically update these prices over time. We present a novel Markov decision process formulation that determines the optimal booking price for a single type of rooms under this strategy, while considering the availability of rooms throughout the multiple-day stays requested by customers. We unveil the importance of modeling multiple-day stays by comparing the optimal policy with a single-day decomposition approach (SDD). Our analysis of the optimal policy around peak-demand events (such as public holidays or conferences) suggests that hotels should substantially raise the booking prices for some high-demand days, and simultaneously, significantly lower the booking prices for the low-demand days that are immediately adjacent to these high-demand days. This finding may potentially simplify the implementation of pricing policies. Since computing an optimal policy is intractable, we develop heuristics based on a fluid approximation and approximate linear programming (ALP). We numerically benchmark them against an SDD and an adaptation of a fixed-price heuristic. The ALP-based heuristic (i) outperforms the other methods; (ii) generates up to 7% and 6% more revenue than the SDD and fixed-price heuristic respectively; and (iii) incurs a revenue loss of only less than 1% when using our pricing structure around peak-demand events, which supports the use of this simple pricing profile. Our findings are potentially relevant beyond the hotel domain for applications involving the dynamic pricing of capacitated resources used by multiple products.

The generalized intersection cut (GIC) paradigm is a recent framework for generating cutting planes in mixed integer programming with attractive theoretical properties. We investigate this computationally unexplored paradigm and observe that a key hyperplane activation procedure embedded in it is not computationally viable. To overcome this issue, we develop a novel replacement to this procedure called partial hyperplane activation (PHA), introduce a variant of PHA based on a notion of hyperplane tilting, and prove the validity of both algorithms. We propose several implementation strategies and parameter choices for our PHA algorithms and provide supporting theoretical results. We computationally evaluate these ideas in the COIN-OR framework on MIPLIB instances. Our findings shed light on the the strengths of the PHA approach as well as identify properties related to strong cuts that can be targeted in the future.