Distributionally Robust Optimization Model Research Articles

An integrated distributionally robust model for two-echelon patient appointment scheduling

We develop a distributionally robust optimization (DRO) model for the outpatient appointment scheduling problem of a set of patients served in two serial stages, consultation and examination. The arrival sequence of patients is known, and the problem of scheduling is to assign appointment time for each patient to minimize total cost with random service time for two serial stages. A max–min problem is formulated for the two-stage appointment scheduling as a whole, in which the waiting time exhibits a high degree of coupling due to the continuous two-stage process. To address this, we devise a two-stage network maximum flow model that provides an equivalent linear expression for the waiting time. For the inner maximum problem, we employ a conic programming approach for equivalent representation, incorporate the scheduling decision of the outer minimum problem, and convert the model to its equivalent copositive programming by taking the conic duality. We conduct numerical experiments and sensitivity analysis using real and simulated data, and the results verify the effectiveness of our proposed DRO model.

Two-stage multi-objective distributionally robust operation optimization and benefits equalization of an off-grid type electric-hydrogen-ammonia-methanol coupling system

As the proportion of renewable energy has increased significantly, electric-hydrogen coupling technology has emerged as a pivotal strategy for optimizing the utilization of renewable energy resources. In particular, within the power and chemical sectors, the preparation of ammonia and methanol using electric-hydrogen coupling technology has emerged as the two principal avenues for the utilization of hydrogen energy. However, as a nascent technology, the preparation of ammonia and methanol by electric-hydrogen coupling is confronted with a multitude of challenges, including the considerable uncertainty surrounding the generation of renewable energy, the dearth of economic benefits, and the need to ensure a fair allocation of benefits among the various stakeholders. To enhance the risk resistance of the electric-hydrogen-ammonia-methanol coupling system (EHAMCS), improve the economic benefits, ensure the environmentality and stability of the system, and realize the fair distribution of benefits among the energy participants, this paper constructs a multi-objective distributionally robust optimization (DRO) and multi-dimensional benefits equilibrium model for the EHAMCS and performs a numerical analysis with an off-grid type EHAMCS as an example. The results demonstrate that: 1) The cooperative operation of the EHAMCS offers notable economic advantages. When compared with only pursuing economic benefits, under multi-objective optimization, the economic benefits of the coupling system can be reduced by 1.49 %, but the carbon emission can be reduced by 10.06 %, and the instability is reduced by 0.02, which can create derivative benefits. 2) The two-stage DRO model is an effective means of addressing the impact of system operation uncertainty and the affine strategy enables the rapid adjustment of the day-ahead dispatching plan, thereby balancing the power deviation. 3) The multi-dimensional factor benefit balancing scheme is more conducive to the participation of clean energy and downstream ammonia and methanol energy subsystems in cooperative operation and promotes the consumption of clean energy while improving the energy conversion value of the EHAMCS to increase economic benefits. 4) EHAMCS operators should adjust their operational strategies in time with market and policy signals, and select historical samples reasonably, so as to realize the balance of the DRO model in terms of economy, robustness, and practicality. The effectiveness of the model and the superiority of the EHAMCS are thus verified.

A distributionally robust approach for the parallel machine scheduling problem with optional machines and job tardiness

This paper investigates a parallel machine scheduling problem with uncertain job processing time, where the job tardiness and optional machines are considered. To address the factor of energy saving, only a subset of all available machines are turned on, which is referred to as not-all-machine (NAM). To depict the uncertain processing time, a mean–mean absolute deviation (MAD) ambiguity set is utilized, and the cost of job tardiness is minimized under the worst-case distribution scenario over the ambiguity set. After building a distributionally robust optimization (DRO) model, theoretical bounds of the optimal number of machines are obtained. Since the model is not computationally scalable, an upper bound on its inner minimization problem is employed, and a mixed integer linear programming (MILP) approximation is obtained based on McCormick inequalities. For the DRO model, tailored speedup techniques are employed, significantly enhancing the computational performance. To evaluate the validity of the proposed DRO model, we compare it with its stochastic programming (SP) counterpart under various parameter settings. Numerical experiments demonstrate that the DRO model exhibits strong performance in the worst-case scenarios. As the problem size increases, the DRO model casts clear advantages over the SP model in terms of computational efficiency and reliability. It is observed that the performance of the DRO model is more stable than that of the nominal sequence, especially with loose due dates. Furthermore, the out-of-sample performance under various decision making preferences shed new lights into the trade-off between energy saving and production efficiency.

Distributionally robust optimization configuration method for island microgrid considering extreme scenarios

The marine climate conditions are intricate and variable. In scenarios characterized by high proportions of wind and solar energy access, the uncertainty regarding the energy sources for island microgrid is significantly exacerbated, presenting challenges to both the economic viability and reliability of the capacity configuration for island microgrids. To address this issue, this paper proposes a distributionally robust optimization (DRO) method for island microgrids, considering extreme scenarios of wind and solar conditions. Firstly, to address the challenge of determining the probability distribution functions of wind and solar in complex island climates, a conditional generative adversarial network (CGAN) is employed to generate a scenario set for wind and solar conditions. Then, by combining k-means clustering with an extreme scenario selection method, typical scenarios and extreme scenarios are selected from the generated scenario set, forming the scenario set for the DRO model of island microgrids. On this basis, a DRO model based on multiple discrete scenarios is constructed with the objective of minimizing the sum of investment costs, operation and maintenance costs, fuel purchase costs, penalty costs of wind and solar curtailment, and penalty costs of load loss. The model is subjected to equipment operation and power balance constraints, and solved using the columns and constraints generation (CCG) algorithm. Finally, through typical examples, the effectiveness of this paper’s method in balancing the economic viability and robustness of the configuration scheme for the island microgrid, as well as reducing wind and solar curtailment and load loss, is verified.

Distributionally robust optimal dispatching method for integrated energy system with concentrating solar power plant

Concentrating solar power (CSP) plants have significant potential to complement the growing wind energy in power scheduling. This study examines an integrated energy system (IES) that incorporates a wind turbine (WT), CSP, and combined heat and power (CHP) to promote the utilization of renewable energy (RE), reduce fluctuations caused by uncertainty, and enhance the economic viability of the system. We propose a distributionally robust optimization (DRO) model for IES scheduling that considers the uncertainty of wind power by using an ambiguity set defined by the Wasserstein metric. Before the occurrence of uncertainties, the system determines the initial dispatching scheme based on the forecast data. In the second stage, the system aims to minimize the adjustment cost expectation under the worst distribution of the ambiguity set and adjusts the flexible resources in real time to offset the fluctuations caused by forecasting errors. The proposed DRO model is transformed into a conventional two-stage robust problem using strong dual theory and KKT conditions, and then solved with a modified column-and-constraint generation (C&CG) algorithm. The results of case studies show that the CSP plant enhances the system's flexibility and controllability through thermal energy storage (TES).

Novel integer L-shaped method for parallel machine scheduling problem under uncertain sequence-dependent setups

We study scheduling problems on unrelated parallel machines with uncertainty in job processing and sequence-dependent setup times. We first formulate this problem as a two-stage stochastic program using a scenario-based approach, in which the first-stage decisions involve job-machine assignments, and the second-stage deals with job sequencing. We then propose an innovative modified integer L-shaped method, uniquely designed to decompose the primary problem into independent subproblems at both the scenario and machine levels. We have augmented this method with a specialized integer optimality cut, which is further strengthened using a sequential lifting procedure. To broaden the applicability of our model and solution, we have adapted it to a distributionally robust optimization (DRO) framework under Wasserstein ambiguity. This process involves reformulating the DRO model into a mixed-integer linear program (MILP) using conic duality theory. Additionally, we integrate a distribution separation program to define the worst-case probability distribution during each iteration of the proposed integer L-shaped method, enhancing the computational speed for solving the DRO model. We conducted a series of numerical experiments to illustrate the effectiveness and scalability of the proposed solution methodology, which demonstrated highly promising results. Our innovative decomposition algorithms showed superior performance, achieving solution times that are significantly faster (up to two orders of magnitude) compared to the traditional integer L-shaped method for the stochastic scheduling model and the direct MILP reformulation of the DRO model. These results were obtained for problem instances solved to the exactness within the time limit. Moreover, the solution derived from the DRO model exhibited greater robustness in out-of-sample analysis with simulated data compared to the solution obtained from the stochastic model.

EXPRESS: Fairness as a Robust Utilitarianism

This article focuses on the social choice problem in which decisions are based on the utility of multiple stakeholder types. The sum of these utilities – Utilitarian welfare - is one of the most important objectives in solving the social choice problem. While it is the most efficient solution, maximizing Utilitarian welfare may lead to unfair outcomes. However, Encouraging a Utilitarian decision-maker to adopt a fair decision is challenging due to the associated efficiency loss. This article takes a novel perspective by motivating a Utilitarian decision-maker to make fair decisions from an uncertainty-averse standpoint. We study the problem where the proportions of stakeholder types are uncertain and propose a distributionally robust optimization (DRO) model that maximizes the worst-case Utilitarian welfare over an ϕ-divergence-based uncertainty set. We provide three aspects of the relationship between fairness and the uncertain-averse Utilitarian welfare maximization. First, we establish that the worst-case Utilitarian welfare adheres to all five axioms of unfairness-averse cardinal welfare functions with two stakeholder types and satisfies four of these when this number exceeds two. Second, we demonstrate that with the maximal extent of uncertainty aversion, the DRO model identifies the Egalitarian welfare maximizer, which prioritizes fairness. Further, given serveral conventional assumptions, the proposed model selects the Nash welfare maximizer, an objective trade-off between efficiency and fairness, with moderate levels of uncertainty aversion. Lastly, we present numerical studies of two specific instances of the social choice problem – resource allocation and facility location – to show that, as uncertainty aversion increases, our model increasingly emphasizes fairness.

A distributionally robust optimization approach for optimal load dispatch of energy hub considering multiple energy storage units and demand response programs

The coupling relationship among various energy sources has been consistently reinforced by the advancement of the energy Internet. In the context of modeling multi-energy systems, the energy hub (EH) represents a significant and valuable approach. To attain optimal operation of the EH, this study proposes an optimal load dispatch model for a system that incorporates a combined heat and power (CHP) unit, gas boiler (GB), electric chiller (EC), absorption chiller (AC), heat energy storage (HES) unit, and electrical energy storage (EES) unit. A two-stage distributionally robust optimization (DRO) method that is driven by data is employed to address the uncertainties of electricity prices. Additionally, the proposed two-stage DRO model is effectively solved through the utilization of the column and constraint generation (C&CG) algorithm. Three cases are discussed in the paper with various energy storage units and demand response (DR) settings. The simulation results show that by deploying energy storage units and participating in DR projects, the EH system can reduce the total costs by 2.56 % and 10 %, respectively. Meanwhile, simulation results reveal that the proposed data-driven DRO model can achieve a compromise between the economy and robustness of the scheduling model compared with stochastic optimization (SO) and robust optimization (RO) methods. The simulation results illustrate the effectiveness of the proposed model for ensuring the economical and efficient operation of the system in the face of electricity price uncertainties.

Distributionally robust optimization model considering deep peak shaving and uncertainty of renewable energy

In order to achieve the goal of carbon neutrality, the capacity of renewable power generation is continuously expanding while thermal power units are transitioning from main power source to auxiliary power source. To alleviate the peak shaving burden of thermal power units under the uncertainty of renewable energy and improve the absorption level of renewable energy, a two-stage distributionally robust optimization (DRO) model considering deep peak shaving and the uncertainty of renewable energy is proposed. The day-ahead unit commitment solutions are determined in the first stage, and the detailed scheduling strategies are obtained in the second stage. Column-and-constraint generation (C&CG) algorithm is applied to solve the model, and the master problem and subproblem are reformulated as duality-free mixed integer linear programming problems. The results show that the scheduling strategy obtained based on the model can alleviate the peak shaving burden brought by the uncertainty of renewable energy and reduce the abandonment rate of wind resources and solar resources, and the proposed DRO model provides a good trade-off between economy and robustness compared to stochastic optimization (SO) and robust optimization (RO).

Ensuring the robustness of link flow observation systems in sensor failure events

Network link flow data are an intuitive information for monitoring the traffic condition of the entire network, and can be used to enhance traffic management and control. Link flow observation systems are typically designed using flow conservation equations to obtain the information of flow on unobserved links by inference. The occurrence of sensor failures in such systems may lead to flow information loss on both observed and inferred links. Most studies on this issue have considered sensor deployment and failure evaluation as separate processes. In contrast, in this study, both processes are integrated to establish a link flow observation system that withstands sensor failures. First, we propose a novel model to solve the sensor location problem for full link flow observability. The proposed model is then modified to evaluate the link flow information loss in sensor failure event, and incorporated into a distributionally robust optimization (DRO) model for the sensor location problem concerned. The DRO model minimizes the worst-case expected information loss of the system during the planning horizon with different types of sensors. Moreover, we extend the DRO model to a target-based version, into which a convex risk measure named Observation fulfillment risk index is introduced to evaluate the risk of failing to meet the predetermined observation target for any sensor installation schemes. The devised models can be directly solved by commercial solvers for networks like Nguyen–Dupuis, and a matheuristic genetic algorithm is designed for large-scale example networks. Numerical experiments are performed for networks with different sizes. The DRO model generates robust sensor location schemes with worst-case performances that are superior to those achieved using benchmark methods, such as stochastic programming. The use of the Observation fulfillment risk index enhances the system stability and target fulfillment level and decreases the standard deviation of the link flow information loss. We also make use of numerical experiments to derive some insightful conclusions on installation budget, coverage ratio, failure risks, etc.

An Auto-Tuned Robust Dispatch Strategy for Virtual Power Plants to Provide Multi-Stage Real-Time Balancing Service

To fully exploit the flexible potential of distributed energy resources (DERs) in providing balancing service to the power system, Virtual Power Plants (VPPs) act as control centers to conduct the optimal real-time dispatch of their managed DERs. This study investigates a VPP’s auto-tuned robust policy based on a multi-stage distributionally robust optimization model (DRO) in response to the uncertainties from both the setpoint of the top-level system operator (SO) and the outputs of renewable DERs. We propose a concise paradigm to reduce the complexity of the original large-scale optimization task. Specifically, we first cast the multi-stage DRO problem into a dynamic programming (DP) formulation and further simplify it to derive a single-stage convex optimization control policy (COCP) at each time stage. Further, an automatic update method based on implicit differentiation is employed to tune the parameters of COCP. Case studies show that this method ensures higher solution quality and faster convergence during training than conventional tuning methods. The proposed COCP outperforms other stochastic optimization techniques in terms of robustness, efficiency, and computational speed.

A multistage distributionally robust optimization approach for generation dispatch with demand response under endogenous and exogenous uncertainties

AbstractDecision‐dependent (endogenous) uncertainties (DDUs), as a new type of uncertainties revealed recently, couple dispatch decisions with uncertainty parameters and thus render power system dispatch more challenging. However, most previous works handled various DDUs via stochastic programming (SP) or robust optimization (RO) in a two‐stage framework, which undoubtedly introduces the drawbacks of SP and RO, and cannot meet the nonanticipativity requirements in power scheduling. In this paper, a multistage distributionally robust optimization (DRO) method for generation dispatch with demand response (DR) is proposed considering the DDUs of deferrable loads and the decision‐independent (exogenous) uncertainties (DIUs) of wind power and regular loads. By analyzing the structure of decision‐dependency parameters, a novel data‐driven decision‐dependent ambiguity set is proposed, which provides a generic framework for formulating DDUs and DIUs simultaneously. Then a multistage DRO model with nested max‐min structure is developed to integrate the merits of DRO and nonanticipativity into generation dispatch. The proposed model is solved by tailored reformulation method and improved stochastic dual dynamic integer programming (SDDiP). Case studies illustrate the effectiveness of the proposed approach by comparing with the multistage SP, RO, and decision‐independent DRO methods.

Designing a resilient supply chain network under ambiguous information and disruption risk

In case of supply chain disruption following severe disasters, many supply chains tend to collapse and take a long time to recover. Resilient supply chain network design (RSCND) is an important research problem in supply chain management, which means that the supply chain can maintain continuous supply and quickly restore the supply capability in part destruction. Based on the limited distribution information of uncertain demand, a two-stage distributionally robust optimization (DRO) model with ambiguous chance constraint (ACC) is proposed to solve the RSCND problem under demand uncertainty and disruption scenario to provide decision support for planning the supply chain network. Finally, to verify the effectiveness and practicability of the proposed DRO model, we apply the method to a real case study in Wuhan, China, about designing a resilient RSC network to withstand disruption. By comparison and sensitivity analysis in numerical experiments, some management insights of industry decision-makers are obtained.

Distributionally robust location-allocation with demand and facility disruption uncertainties in emergency logistics

Emergency logistics is vital to disaster relief management. In this paper we develop a distributionally robust optimization model (DROM) for optimizing the locations of distribution centres and backup warehouses, and the distribution of disaster relief supplies in emergency logistic networks by minimizing the expected total cost and the total delivery time. Based on limited historical distribution information, the model considers uncertain demand and uncertain facility disruptions, and describes their distributions through ambiguity sets. Following the adaptability and tractability of the ambiguity sets, we show that the model can be equivalently re-formulated as a mixed-integer linear program. To solve the model, we propose an exact algorithm based on Benders decomposition (BD). We also introduce an in-out Benders cut generation strategy to improve the efficiency of the BD algorithm. Finally, we perform extensive numerical studies to test the performance of the BD algorithm, ascertain the benefits of the proposed DROM over the corresponding deterministic and stochastic models, and examine the impacts of the key model parameters to gain managerial insights.

Distributionally robust facility location with uncertain facility capacity and customer demand

This study explores a capacitated facility location problem where facility capacity and customer demand are subject to uncertainties simultaneously. This problem decides the subset of facilities to open at the system design phase to serve customers during the operational phase. The objective is to minimize the total cost, including the first-stage location cost and the second-stage recourse cost, and guarantee the system’s reliability, i.e., meeting demand as much as possible when uncertainties arise. A distributionally robust optimization (DRO) framework is utilized to model the problem. A scenario-wise ambiguity set with partial distributional information of random variables is constructed, which can capture uncertainties caused by different random events or different magnitudes of the same event type and explicitly represent the correlation between facilities’ uncertain capacity and customers’ uncertain demand. We apply an adaptation policy to the DRO model and reformulate it to a mixed-integer linear programming model, which is solvable by off-the-shelf solvers. Numerical results show that the scenario-wise DRO framework can provide a better trade-off between cost and service level than the stochastic programming model and the DRO model with a marginal moment-based ambiguity set, demonstrating that the proposed scenario-wise DRO model offers a practical decision-making tool for enhancing supply chain robustness via facility location.

Distributionally Robust Programming of Berth-Allocation-with-Crane-Allocation Problem with Uncertain Quay-Crane-Handling Efficiency

In order to promote the efficient and intelligent construction of container ports, we focus on the optimization of berth-and-quay-crane (QC) allocation in tidal terminal operations. This paper investigates the quay-crane-profile-(QC-profile)-based assignment problem, and considers the uncertainty in QC profiles regarding QC efficiency for the first time. A mixed-integer programming (MIP) model is established for a discrete berth allocation with a crane-assignment problem (BACAP), considering the tide time window. We aim to minimize the total time loss caused by anchorage and the delay of vessels. Leveraging the theory of uncertainty optimization, the proposed deterministic model is extended into a stochastic programming (SP) model and a distributionally robust optimization (DRO) model, via the consideration of the random QC efficiency. To solve the proposed models, a column generation (CG) algorithm is employed, utilizing the mathematical method and subproblem-solving approach. The numerical experiments with different instances demonstrate that the DRO model yields a smaller variation in the objective function values, and the effectiveness of the CG method. The experimental results verify the robustness of the constructed models, and the efficiency of the proposed algorithm.

Distributionally robust hierarchical multi-objective voltage risk management for unbalanced distribution systems

The large-scale integration of renewable energy generators (REGs) is key to reducing carbon emissions. However, the intermittency of REG outputs poses operational risk challenges on distribution systems (DSs). This paper proposes a novel Wasserstein-based distributionally robust assessment for the voltage risk index that represents both likelihood and non-linearly increasing severity of voltage violations. We propose a hierarchical DRO model for risk management that integrates a multi-objective distributionally robust optimization (DRO) at the distribution system operator (DSO) level and a basic DRO at the user level. Then, a systematic solution method is developed, where the proposed model is equivalently reformulated into tractable bilevel mixed-integer convex programs. The effectiveness, advantages, and scalability of the proposed method are validated on the IEEE 33-bus system and unbalanced IEEE-123-bus system.

Humanitarian transportation network design via two-stage distributionally robust optimization

Natural disasters are highly unpredictable, with varying degrees of magnitude, and thus require a reliable and robust humanitarian relief network. Faced with the adverse effects of disasters, we advocate taking pre-disaster preventive actions, e.g., road link strengthening, to mitigate post-disaster disruptions to road networks. In this paper, we study a highly integrated humanitarian relief network design problem, in which the network strengthening plan and inventory pre-positioning scheme are optimized cooperatively. The proposed problem is formulated as a two-stage distributionally robust optimization (DRO) model, in which the first and second stages correspond to pre- and post-disaster relief operations, respectively. By leveraging prior data and the Wasserstein metric, a tailored ambiguity set is constructed to capture both node- and link-wise uncertainties. We demonstrate that the two-stage DRO model over the Wasserstein ambiguity set has an equivalent reformulation that can be solved via the L-shaped method with bilinear subproblems. Based on observations of the problem structure and the spirit of separation optimization, both exact and heuristic approaches are proposed to address the bilinear subproblems. Via the case study of the Yushu earthquake, we highlight the value of jointly optimizing network strengthening and inventory pre-positioning decisions. Specifically, with the same investment budget, integrating network strengthening into inventory pre-positioning can easily achieve an approximately 60% reduction in shortage penalties. Furthermore, our method can produce reliable solutions, based on which we explore several managerial implications.

TimeGAN based distributionally robust optimization for biomass-photovoltaic-hydrogen scheduling under source-load-market uncertainties

The Biomass-Photovoltaic-Hydrogen Integrated Energy System (BPH-IES) is receiving increasing attentions due to the utilization of hydrogen storage for renewables accommodation. However, owing to its internal components and capability for participating in electricity and carbon markets, source-load-market uncertainties such as the random photovoltaic output, loads, as well as electricity and carbon prices are non-negligible, which seriously hinders the stable operation and realization of economic and environmental benefits for BPH-IES. In this paper, in view of the insufficient samples and the time-series attribute for the uncertainties, a novel Time-series Generative Adversarial Network (TimeGAN) based Distributionally Robust Optimization (DRO) model is proposed for BPH-IES scheduling. Based on TimeGAN learning the dynamics of the limited samples of uncertainties, sufficient generated samples are obtained with approximating the real distributions. Then, the temporal covariance conditions are introduced in the ambiguity set construction, aiming at getting rid of those distributions unmatching the temporal correlations of the enlarged samples. Finally, DRO model for scheduling is established, which jointly optimizes multi-energy operation and electricity-carbon trading strategy for BPH-IES considering the “worst-case” distributions within the narrowed ambiguity set. The solution procedure is developed according to the duality and semi-definite programming theories. Simulations have verified that for BPH-IES, the proposed model: 1) improves the economic and environmental benefits by 10.04% and 7.79% respectively, due to the TimeGAN and temporal covariance based modifications; 2) maintains operation stability and 100% renewables accommodation under the source-load uncertainties; 3) reduces the fluctuation of benefits via considering the market uncertainties.

Distributionally robust optimization for collaborative emergency response network design

The post-disaster emergency response capacity of a single country is limited, and a collaborative approach that pools emergency resources can improve disaster resilience. In this paper, we study a multi-country collaborative emergency response network design problem with uncertain demand and transportation time. Considering the benefits of inter-regional cooperation in sharing emergency facilities and resources, we construct a collaborative emergency response network (CERN) design framework. The cost of CERN is allocated among the partner countries according to their expect standalone response cost and the level of economic development. In practice, the distribution information of random parameters is not perfectly known, so we propose a distributionally robust optimization (DRO) model to design the CERN. A scenario-wise ambiguity set is constructed to characterize the uncertain parameters based on disaster-level-related events. To solve the proposed DRO model, we propose a decomposition-based algorithm with a valid inequality. In the numerical study, we first verify the advantage of the CERN design approach. The proposed scenario-wise DRO method is subsequently compared with alternative modeling approaches to assess its out-of-sample performance. The findings confirm the efficacy of the constructed ambiguity set in capturing the uncertainty stemming from varying magnitudes of catastrophic events. The computational study demonstrates that the proposed algorithm exhibits superior computational efficiency compared to the commercial solver CPLEX for large-scale problems. Additionally, we conduct sensitivity analyses on various parameter configurations and provide managerial insights for the CERN design problem.