Linear Decision Rule Research Articles

Nonlinear Decision Rules Made Scalable by Nonparametric Liftings

Sequential decision making often requires dynamic policies, which are computationally not tractable in general. Decision rules provide approximate solutions by restricting decisions to simple functions of uncertainties. In this paper, we consider a nonparametric lifting framework where the uncertainty space is lifted to higher dimensions to obtain nonlinear decision rules. Current lifting-based approaches require predetermined functions and are parametric. We propose two nonparametric liftings, which derive the nonlinear functions by leveraging the uncertainty set structure and problem coefficients. Both methods integrate the benefits from lifting and nonparametric approaches, and hence provide scalable decision rules with performance bounds. More specifically, the set-driven lifting is constructed by finding polyhedrons within uncertainty sets, inducing piecewise-linear decision rules with performance bounds. The dynamics-driven lifting, on the other hand, is constructed by extracting geometric information and accounting for problem coefficients. This is achieved by using linear decision rules of the original problem, also enabling one to quantify lower bounds of objective improvements over linear decision rules. Numerical comparisons with competing methods demonstrate superior computational scalability and comparable performance in objectives. These observations are magnified in multistage problems with extended time horizons, suggesting practical applicability of the proposed nonparametric liftings in large-scale dynamic robust optimization. This paper was accepted by David Simchi-Levi, optimization. Supplemental Material: The electronic companion and data files are available at https://doi.org/10.1287/mnsc.2024.4988 .

A distributionally robust optimization approach for the potassium fertilizer product transportation considering transshipment through crossdocks

Potassium fertilizer is an essential input for agricultural productivity, and plays a critical role in various plant processes, influencing water uptake, enzyme activation, and photosynthesis. The efficient delivery of potassium fertilizer products to the end-users, typically farmers and agricultural enterprises, is of utmost importance. Due to the long traveling distance between the supplier and customers, decision makers of supplier normally set up multiple crossdocks to aid in transition and storage of potassium fertilizer products. In this situation, making an optimal transportation plan as well as the inventory level of crossdocks will directly enhance the efficiency and effectiveness of long-distance transportation. In this study, we model this problem as a dynamic transportation problem with transshipment considering the uncertain time-varying customers’ demand. To characterize the demand information, we first construct an ambiguity set of customers’ demand based in limited historical data and then develop a distributionally robust optimization-based (DRO) framework to optimize the transportation plan and related inventory level of crossdocks simultaneously. We also propose a general approach to overcome the computational challenges of DRO by transforming the original DRO into a second-order cone programming based on duality theory. Additionally, we introduce linear decision rules to adjust the optimization strategy based on the new observed demand, thus lead the model to handle the dynamic information flow in real time. In case study, all involved data are collected or derived from a real potassium fertilizer company located at Western China. The results show our model reduces the cost by 18.07% compared to stochastic model (sample average approximation), indicating a significant effectiveness of our model on the improvement of delivery efficiency and cost saving in real dynamic logistic systems. Also, we conclude the managerial insight that decision-makers should develop a comprehensive strategy, including improving communication to ensure order status updates, planning rationally to evenly distribute orders, and proactively allocating resources to meet operational demands.

A peer‐to‐peer joint energy and reserve market considering renewable generation uncertainty: A generalized Nash equilibrium approach

AbstractPeer‐to‐peer (P2P) energy trading enhances distribution network resilience by reducing energy demand from central power plants and enabling distributed energy resources to support critical loads after extreme events. However, adequate reserves from main grids are still required to ensure real‐time energy balance in distribution networks due to the uncertainty in renewable generation. This paper introduces a novel two‐stage joint energy and reserve market for prosumers, wherein local flexible resources are fully utilized to manage renewable generation uncertainty. In contrast to cooperative optimization methods, the interactions between prosumers are modelled as a generalized Nash game (GNG), considering that prosumers are self‐interested and should follow distribution network constraints. Then, linear decision rules are employed to ensure a feasible market equilibrium and develop a privacy‐preserving algorithm to guide prosumers toward the market equilibrium with a proven convergence. Finally, the numerical study on a modified IEEE 33‐power system demonstrates that the designed market effectively manages renewable generation uncertainty, and that the algorithm converges to the market equilibrium.

Bi-level distributionally robust optimization model for low-carbon planning of integrated electricity and heat systems

Interdependence among diverse energy forms within integrated electricity and heat systems (IEHSs) holds the potential to harness cooperation effects. This paper proposes a bi-level optimization model for the low-carbon planning of IEHSs, which realizes the coordination between IEHSs and their upstream networks. The precise carbon emissions of IEHSs are traced through carbon emission flows of components, power distribution networks, and district heating networks. Moreover, the distributionally robust optimization approach is employed to account for the inherent uncertainty in renewable power outputs faced by IEHS planning, by incorporating the moment information into an ambiguity set. The bi-level distributionally robust planning model of IEHSs is initially reconfigured into a second-order cone structure, employing the strong duality theory, second-order cone duality theory, and linear decision rules. Subsequently, an iterative approach is introduced to attain convergence of the reformulated bi-level model. Numerical results demonstrate the effectiveness and superiority of the proposed approach for reducing carbon emissions and economic costs of IEHSs.

Target-oriented robust satisficing models for the single machine scheduling problems with release time

In this paper, we investigate single machine scheduling problems with release times and random processing times, where the release times can be either deterministic or random. The objective is to determine a scheduling sequence that exhibits strong out-of-sample performance. To achieve this, we employ target-oriented robust satisficing models to obtain the scheduling sequence. For the scheduling problem with deterministic release times, we derive an equivalent mixed-integer linear program and an approximate reformulation to address cases involving a large number of jobs. For the scheduling problem with random release times, we first demonstrate the challenges associated with providing an equivalent reformulation. To overcome this intractability, we propose an approximate reformulation based on the linear decision rule. Numerical experiments are conducted to demonstrate the superiority of our solutions through comparisons with several benchmarks. Furthermore, the numerical results demonstrate that a relatively higher target should be chosen if the worst-case performance is valued, otherwise, the decision-maker should set a relatively lower target to obtain better average performance.

Multistage adaptive distributionally robust optimization for the medical supplies distribution problem with uncertain demand in humanitarian aid

Production, inventory, distribution, and dispensing of relief resources are critical operational functions in humanitarian aid. To design an efficient humanitarian relief network, it is beneficial to study these operational functions in an integrated way. Accounting for the demand uncertainty of medical supplies, we propose a multistage adaptive distributionally robust model for the medical supplies distribution network design that considers simultaneously the issues of production, inventory, distribution and dispensing of medical resources, as well as the life-loss due to the delays in treatment. The objective is to dynamically match the supply and demand of medical supplies so as to minimize the total cost consisting of the production cost, holding cost, dispensing cost, and life-loss cost related to the unmet demand. We also introduce a safety-stock and production capacity model to efficiently predetermine the initial supply of medical supplies and maximum available production abilities under the given demand information. To obtain tractable formulations, we approximate the developed models using an enhanced linear decision rule (ELDR) and a simplified ELDR (SELDR), respectively. Using a set of real-world COVID-19 data, we show that (i) both the ELDR and SELDR can yield feasible solutions extremely close to the optimal solution of the multistage adaptive distributionally robust model, whereas the SELDR is about one order of magnitude faster than the ELDR; and (ii) accounting for the safety-stock and production capacity model yields significant improvements of the obtained solution, which can also inform the decision-maker about at least how many initial supply of vaccines and the maximum available production abilities should be set to counter the risk of demand uncertainty. We also analyze the impact of some model parameters to gain managerial implications.

Distributionally Robust Optimization of Photovoltaic Power With Lifted Linear Decision Rule for Distribution System Voltage Regulation

Distributionally Robust Optimization of Photovoltaic Power With Lifted Linear Decision Rule for Distribution System Voltage Regulation

Joint flexibility-risk managed distributed energy trading considering network constraints and uncertainty

This paper addresses the operational challenges in distribution systems caused by the integration of fluctuating renewable energy sources and rising peak loads from widespread electrification. We propose a new distributed energy coordination framework that facilitates seamless interoperation between peer-to-peer (P2P) energy trading and the operation of distributed flexibility resources. This framework ensures network constraints are respected while promoting Energy Trading Consistency (ETC). Our approach utilizes a two-stage dynamic energy and flexibility sharing mechanism. It achieves energy efficiency through demand-management and P2P energy trading in the day-ahead (DA) stage, followed by adjustments of flexible resources and flexibility sharing through a receding-horizon feedback mechanism inspired by model predictive control (MPC) in the real-time (RT) stage. High-risk scenarios and computational overhead are managed by employing conditional value-at-risk (CVaR) measures, as well as approximation of constraints through dual robust reformulation and linear decision rules. To uphold the autonomy and privacy of prosumers, we propose distributed algorithms based on the fast consensus alternating direction method of multipliers (ADMM). Numerical results demonstrate that our framework achieves a 20% reduction in social cost while providing a coherent and interactive P2P energy trading environment, facilitating reliable network operation under high penetration of renewable energy sources without third-party intervention.

Optimizing bike rebalancing strategies in free-floating bike-sharing systems: An enhanced distributionally robust approach

Bike-sharing systems are important components of urban transportation systems that facilitate short-distance travel. In recent years, free-floating bike-sharing systems have become popular. However, a major problem in such systems is the increasing number of idle bikes due to their spatial dispersion, which leads to a loss of demand and difficulty in bike rebalancing. To overcome these drawbacks, operators reserve parking areas, and laborers are incorporated into the bike rebalancing operation. We propose a novel dynamic bike rebalancing strategy based on trucks and laborers for free-floating bike-sharing systems. This strategy combines bike collection by laborers and bike transportation by trucks for rebalancing operations. In addition, we consider broken bike detection by laborers and user dynamics in two extension models. Because of the demand uncertainty in bike-sharing systems, we employ an enhanced distributionally robust optimization (EDRO) approach to protect against uncertainty. A target-level constraint is added to the general distributionally robust optimization (DRO) model to further improve the out-of-sample performance. An adaptive robust strategy is also employed to avoid overly conservative solutions. Furthermore, a linear decision rule is adopted to reformulate the EDRO model into a mixed-integer programming model, which guarantees computational tractability. The results of numerical experiments show that the EDRO model outperforms other benchmark models on a series of measures. Moreover, we verify the value of laborers in dynamic rebalancing operations and explore the effect of different bike supply levels and several other parameters on the value of laborers. We also test how the target-level parameter affects the model performance.

A sample robust optimal bidding model for a virtual power plant

In many energy markets, the trade amount of electricity must be committed to before the actual supply. This study explores one consecutive operational challenge for a virtual power plant—the optimal bidding for highly uncertain distributed energy resources in a day-ahead electricity market. The optimal bidding problem is formulated as a scenario-based multi-stage stochastic optimization model. However, the scenario-tree approach raises two consequent issues—scenario overfitting and massive computation cost. This study addresses the issues by deploying a sample robust optimization approach with linear decision rules. A tractable robust counterpart is derived from the model where the uncertainty appears in a nonlinear objective and constraints. By applying the decision rules to the balancing policy, the original model can be reduced to a two-stage stochastic mixed-integer programming model and then efficiently solved by adopting a dual decomposition method combined with heuristics. Based on real-world business data, a numerical experiment is conducted with several benchmark models. The results verify the superior performance of our proposed approach based on increased out-of-sample profits and decreased overestimation of in-sample profits.

Robust CARA Optimization

Decision making under uncertainty involves both ambiguity and risk. In “Robust CARA Optimization,” Chen and Sim have developed innovative optimization models designed for ambiguity-averse decision makers whose risk preference is consistent with constant absolute risk aversion (CARA). The research delves into maximizing the worst-case expected exponential utility amid uncertainties from independent factors with ambiguous marginals. To enhance computational feasibility, the authors developed a series of approximations: starting with tractable concave functions in affinely perturbed cases, advancing to concave piecewise affinely perturbed scenarios, and culminating in novel multi-deflected linear decision rules for adaptive optimization. This comprehensive framework extends to a multi-period consumption model, ultimately forming an exponential conic optimization problem efficiently solvable with existing solvers. Practical applications demonstrated in project and multiperiod inventory management highlight the models’ potential to surpass existing stochastic optimization methods, especially under high risk aversion.

Two-stage chance-constrained programming based on Gaussian mixture model and piecewise linear decision rule for refinery optimization

The two-stage chance-constrained program (CCP) is studied for a refinery optimization problem. In stage-I, the refinery decision-makers determine the type and quantity of crude oil procurement under operational uncertainties to maximize the expected profit under all possibilities. In stage-II, process unit flowrates are adjusted based on the realized uncertainties and available crude oil, while introducing probabilistic constraints to manage the off-spec risk. To solve such a two-stage optimization problem, we propose a novel approach using Gaussian mixture model (GMM) to characterize uncertainties, and piecewise linear decision rule for stage-II operations. Comparing to the conventional scenario-based mixed-integer linear program (MILP), our new approach offers three advantages. First, it leverages a well-developed global optimization scheme for joint CCP to avoid scenario generation and potential bias. Second, the data-driven GMM enables CCP to handle uncertainties with general distributions. Third, the stage-II variables are parameterized via Gaussian component induced piecewise linear decision rule to strike an excellent trade-off between optimality and computational time. A simplified refinery plant, consisting of distillation, cracker, reformer, isomerization, and desulfurization units, is used as a test bed to demonstrate the superiority of the proposed optimization method in solution time, probabilistic feasibility, and optimality over the large-scale scenario-based MILP.

Reliable Frequency Regulation Through Vehicle-to-Grid: Encoding Legislation with Robust Constraints

Problem definition: Vehicle-to-grid increases the low utilization rate of privately owned electric vehicles by making their batteries available to electricity grids. We formulate a robust optimization problem that maximizes a vehicle owner’s expected profit from selling primary frequency regulation to the grid and guarantees that market commitments are met at all times for all frequency deviation trajectories in a functional uncertainty set that encodes applicable legislation. Faithfully modeling the energy conversion losses during battery charging and discharging renders this optimization problem nonconvex. Methodology/results: By exploiting a total unimodularity property of the uncertainty set and an exact linear decision rule reformulation, we prove that this nonconvex robust optimization problem with functional uncertainties is equivalent to a tractable linear program. Through extensive numerical experiments using real-world data, we quantify the economic value of vehicle-to-grid and elucidate the financial incentives of vehicle owners, aggregators, equipment manufacturers, and regulators. Managerial implications: We find that the prevailing penalties for nondelivery of promised regulation power are too low to incentivize vehicle owners to honor the delivery guarantees given to grid operators. Funding: This work was supported by the Institut Vedecom. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0154 .

Multi-Stage Decision Rules for Power Generation & Storage Investments With Performance Guarantees

We develop multi-stage linear decision rules (LDRs) for dynamic power system generation and energy storage investment planning under uncertainty and propose their chance-constrained optimization with performance guarantees. First, the optimized LDRs guarantee operational and carbon policy feasibility of the resulting dynamic investment plan even when the planning uncertainty distribution is ambiguous. Second, the optimized LDRs internalize the tolerance of the system planner towards the stochasticity (variance) of uncertain investment outcomes. They can eventually produce a quasi-deterministic investment plan, which is insensitive to uncertainty (as in deterministic planning) but robust to its realizations (as in stochastic planning). Last, we certify the performance of the optimized LDRs with the bound on their sub-optimality due to their linear functional form. Using this bound, we guarantee that the preference of LDRs over less restrictive – yet poorly scalable – scenario-based optimization does not lead to financial losses exceeding this bound. We use a testbed of the U.S. Southeast power system to reveal the trade-offs between the cost, stochasticity, and feasibility of LDR-based investments. We also conclude that the LDR sub-optimality depends on the amount of uncertainty and the tightness of chance constraints on operational, investment and policy variables.

A Decision Rule Approach for Two-Stage Data-Driven Distributionally Robust Optimization Problems with Random Recourse

We study two-stage stochastic optimization problems with random recourse, where the coefficients of the adaptive decisions involve uncertain parameters. To deal with the infinite-dimensional recourse decisions, we propose a scalable approximation scheme via piecewise linear and piecewise quadratic decision rules. We develop a data-driven distributionally robust framework with two layers of robustness to address distributional uncertainty. We also establish out-of-sample performance guarantees for the proposed scheme. Applying known ideas, the resulting optimization problem can be reformulated as an exact copositive program that admits semidefinite programming approximations. We design an iterative decomposition algorithm, which converges under some regularity conditions, to reduce the runtime needed to solve this program. Through numerical examples for various known operations management applications, we demonstrate that our method produces significantly better solutions than the traditional sample-average approximation scheme especially when the data are limited. For the problem instances for which only the recourse cost coefficients are random, our method exhibits slightly inferior out-of-sample performance but shorter runtimes compared with a competing approach. History: Accepted by Nicola Secomandi, Area Editor for Stochastic Models & Reinforcement Learning. Funding: This work was supported by the National Science Foundation [Grants 2342505 and 2343869]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2021.0306 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0306 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Multistage Adaptive Robust Binary Optimization: Uncertainty Set Lifting versus Partitioning through Breakpoints Optimization

Two methods for multistage adaptive robust binary optimization are investigated in this work. These methods referred to as binary decision rule and finite adaptability inherently share similarities in dividing the uncertainty set into subsets. In the binary decision rule method, the uncertainty is lifted using indicator functions which result in a nonconvex lifted uncertainty set. The linear decision rule is further applied to a convexified version of the lifted uncertainty set. In the finite adaptability method, the uncertainty set is divided into partitions and a constant decision is applied for each partition. In both methods, breakpoints are utilized either to define the indicator functions in the lifting method or to partition the uncertainty set in the finite adaptability method. In this work, we propose variable breakpoint location optimization for both methods. Extensive computational study on an illustrating example and a larger size case study is conducted. The performance of binary decision rule and finite adaptability methods under fixed and variable breakpoint approaches is compared.

Decision Rule Approaches for Pessimistic Bilevel Linear Programs Under Moment Ambiguity with Facility Location Applications

We study a pessimistic stochastic bilevel program in the context of sequential two-player games, where the leader makes a binary here-and-now decision, and the follower responds with a continuous wait-and-see decision after observing the leader’s action and revelation of uncertainty. We assume that only the information regarding the mean, covariance, and support is known. We formulate the problem as a distributionally robust (DR) two-stage problem. The pessimistic DR bilevel program is shown to be equivalent to a generic two-stage distributionally robust stochastic (nonlinear) program with both a random objective and random constraints under proper conditions of ambiguity sets. Under continuous distributions, using linear decision rule approaches, we construct upper bounds on the pessimistic DR bilevel program based on (1) a 0-1 semidefinite programming (SDP) approximation and (2) an exact 0-1 copositive programming reformulation. When the ambiguity set is restricted to discrete distributions, an exact 0-1 SDP reformulation is developed, and explicit construction of the worst-case distribution is derived. To further improve the computation of the proposed 0-1 SDPs, a cutting-plane framework is developed. Moreover, based on a mixed-integer linear programming approximation, another cutting-plane algorithm is proposed. Extensive numerical studies are conducted to demonstrate the effectiveness of the proposed approaches on a facility location problem. History: Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0168 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0168 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Scenario-based Distributionally Robust Optimization for the Stochastic Inventory Routing Problem

We consider a class of the inventory routing problem in a discrete and finite time horizon, where the demand for homogeneous products at retail stores is uncertain and varies across different scenarios. The supplier is required to determine the times to visit retailers, the replenishment quantities to each retailer, and the routing of a vehicle so as to minimize the sum of stockout, holding, and transportation costs. We propose a scenario-based distributionally robust optimization framework to tackle this problem. We transform the distributionally robust optimization model into a mixed-integer problem, which can be solved efficiently by our proposed algorithm. We adopt a warm-start procedure that utilizes the solution to the nominal model in our methodological framework. Then we apply a Tabu search algorithm, integrated with column generation, to solve a set-partitioning-like integer linear programming model so that a better route set can be identified. By doing so, a large-scale scenario-based distributionally robust optimization model can be solved. We conduct a case study of a fuel company and construct realistic instances to demonstrate the performance of our proposed method. Computational results suggest that the model taking into account various scenarios is more effective when random demands can be classified; the model with a linear decision rule outperforms a non-adaptive model; and the model with the route set identified by an improved algorithm can deliver a better solution than the original route set.

Winner determination problem with purchase budget for transportation procurement under uncertain shipment volume

We study a winner determination problem (WDP) to help shippers in determining the winning bids proposed by carriers in a combinatorial auction for transportation service procurement. The winning carriers, together with spot market carriers which may be optional, undertake the shipper’s annual shipment tasks that are uncertain in the auction stage; completing the tasks incurs a total cost. In particular, the shipper wants to control the total cost under a prescribed purchase budget, resulting in a new problem variant—the WDP with purchase budget (B-WDP). To mitigate both the probability and magnitude of cost overrun, we introduce the Essential Riskiness Index (Zhang et al., 2019) as the decision criterion. Considering the inaccuracy in each predicted scenario/sample of shipment volumes, we allow for some perturbation around each sample and formulate a two-stage sample robust optimization model for the B-WDP. We analyze the properties of the model and, via linear decision rule and duality theory, reformulate it as a mixed-integer linear program solvable via state-of-the-art solvers. By extensive comparative studies against the traditional stochastic and robust models for the WDP, we demonstrate that our two-stage sample robust B-WDP model effectively reduces the expectation of cost overrun and the worst-case total cost. Additional sensitivity analysis confirms that our model consistently and effectively helps shippers mitigate cost overrun.

Decentralized coordination between active distribution network and multi-microgrids through a fast decentralized adjustable robust operation framework

Due to the autonomous characteristic and heterogeneity of the individual agents in active distribution network (ADN) with multi-microgrids (MMG), this paper proposes a fully decentralized adjustable robust operation framework achieving the coordinated operation between ADN and MMG. The improved linear decision rules (LDRs) based microgrid adjustable robust operation model is proposed to reduce the solution conservatism in dealing with renewable energy uncertainty. The LDRs model is then reformulated as a computationally tractable solution such that the proposed adjustable robust extension of decentralized operation can handle renewable energy uncertainty while reducing the computation burden of decentralized optimization. Then, a tailored fast alternating direction method of multipliers algorithm with a predictor–corrector type acceleration step is developed to improve the convergence rate of decentralized optimization. The effectiveness of the proposed model is validated on a modified IEEE 69-bus distribution system with four microgrids.