Risk-averse Stochastic Programming Research Articles

Multi-period emergency facility location-routing problems under uncertainty and risk aversion

This paper addresses a multi-period emergency facility location-routing problem, in which the uncertainties in material demands and transportation time, as well as dynamic inventory replenishment and carryover are incorporated in the design of multi-level emergency logistics networks. To measure the risks stemming from uncertain transportation time, mean-CVaR method is used. Then, a risk-averse stochastic programming model for the presented problem is formulated to minimize the total rescue time of the network. Moreover, a genetic algorithm is developed to solve the proposed model. Extensive numerical experiments including the randomly generated instances and a case study on the Wenchuan earthquake in China are conducted to verify the effectiveness of the presented model and algorithm. Experimental results show that the genetic algorithm significantly performs better than the Gurobi solver in terms of both solution quality and solution time.

First order asymptotics of the sample average approximation method to solve risk averse stochastic programs

We investigate statistical properties of the optimal value of the Sample Average Approximation of stochastic programs, continuing the study (Krätschmer in Nonasymptotic upper estimates for errors of the sample average approximation method to solve risk averse stochastic programs, 2023. Forthcoming in SIAM J. Optim.). Central Limit Theorem type results are derived for the optimal value. As a crucial point the investigations are based on a new type of conditions from the theory of empirical processes which do not rely on pathwise analytical properties of the goal functions. In particular, continuity or convexity in the parameter is not imposed in advance as usual in the literature on the Sample Average Approximation method. It is also shown that the new condition is satisfied if the paths of the goal functions are Hölder continuous so that the main results carry over in this case. Moreover, the main results are applied to goal functions whose paths are piecewise Hölder continuous as e.g. in two stage mixed-integer programs. The main results are shown for classical risk neutral stochastic programs, but we also demonstrate how to apply them to the Sample Average Approximation of risk averse stochastic programs. In this respect we consider stochastic programs expressed in terms of absolute semideviations and divergence risk measures.

Risk-averse stochastic capacity planning and P2P trading collaborative optimization for multi-energy microgrids considering carbon emission limitations: An asymmetric Nash bargaining approach

The increasing penetration of renewable energy with stochastic characteristics and continuous refinement of carbon emission policies put forward higher requirements for the construction of new power systems. This paper proposes a risk-averse stochastic capacity planning and peer-to-peer (P2P) trading collaborative optimization method for multi-energy microgrids (MEMGs) considering carbon emission limitations. First, a cooperative operation model for MEMGs considering the capacity planning of distributed generation units, uncertainty from renewable energy generations, carbon emission limitations and P2P electricity trading among MEMGs is formulated. Second, a risk-averse stochastic programming method is applied to avoid the potential risk losses caused by randomness and intermittence of renewable energy. Third, an asymmetric Nash bargaining approach is adopted to ensure the fair allocation of benefits and maintain the willingness of individual MEMGs to participate in cooperation. Then, to protect the privacy of individual MEMGs belonging to different stakeholders, the alternating direction method of multipliers is used to solve the two subproblems in a distributed manner. Meanwhile, to further alleviate the computation burdens, a diagonal quadratic approximation method is applied to linearize the quadratic penalty term in the augmented Lagrangian function and realize the parallel solution of all optimization subproblems. Moreover, the influence of different carbon emission targets on the optimal resource combination strategy of the system is investigated by introducing carbon emission factors. Simulations on different models, strategies, and distributed algorithms are conducted to verify the effectiveness and superiority of the proposed method.

Risk-averse restoration of coupled power and water systems with small pumped-hydro storage and stochastic rooftop renewables

Modern coupled power and water (CPW) systems exhibit increasing integration and interdependence, which challenges system performance to disasters and makes service restoration complex during post-disruption. Meanwhile, new technologies, such as small pumped-hydro storage (PHS) and rooftop renewables, are being widely installed and further deepen the interdependencies. To capture these features and improve overall performance, this paper proposes a coordinated restoration framework for a CPW system to respond to disruptions. The proposed CPW model comprises physical networks and mechanisms, considering available units, such as water desalination/treatment plants, pump stations and small PHS, in the water system, and rooftop renewables, distributed generators, in power system. The interdependencies are modeled through component-wise connections and consumer behavior, then grouped into three phases: production, distribution, and consumption. Aggregate service loss with respect to different consumer loads and time periods, is chosen as performance metric and to be minimized using network reconfiguration, energy/water dispatching, load curtailment, and operation management of components. A two-stage risk-averse stochastic programming is applied for reliable restoration and manage risks, to tackle the uncertainties in renewable power generations and water/power demands that affect method effectiveness. Finally, the method is implemented on a modified 33-bus/25-node CPW system, and the results demonstrate the effectiveness of the proposed restoration framework.

Risk-Averse Stochastic Programming for Planning Hybrid Electrical Energy Systems: A Brazilian Case

This work presents a risk-averse stochastic programming model for the optimal planning of hybrid electrical energy systems (HEES), considering the regulatory policy applied to distribution systems in Brazil. Uncertainties associated with variables related to photovoltaic (PV) generation, load demand, fuel price for diesel generation and electricity tariff are considered, through the definition of scenarios. The conditional value-at-risk (CVaR) metric is used in the optimization problem to consider the consumer’s risk propensity. The model determines the number and type of PV panels, diesel generation, and battery storage capacities, in which the objective is to minimize investment and operating costs over the planning horizon. Case studies involving a large commercial consumer are carried out to evaluate the proposed model. Results showed that under normal conditions only the PV system is viable. The PV/diesel system tends to be viable in adverse hydrological conditions for risk-averse consumers. Under this condition, the PV/battery system is viable for a reduction of 87% in the battery investment cost. An important conclusion is that the risk analysis tool is essential to assist consumers in the decision-making process of investing in HEES.

Data-Driven Transportation Network Company Vehicle Scheduling With Users’ Location Differential Privacy Preservation

With the popularity of mobile devices with global positioning system (GPS), transportation network company (TNC) service has become an indispensable option of people's daily commute. However, it also provides opportunities for malicious parties to compromise TNC users’ location privacy. There are great challenges to preserve TNC users’ location privacy while improving the revenue of TNC and its quality of service (QoS). To address this issue, we propose a novel scheme to schedule the TNC vehicles while preserving the TNC users’ location differential privacy. Briefly, we add high dimensional Laplace noises to guarantee the TNC users’ geo-indistinguishability. Due to the differential private obfuscation, the demand for TNC vehicles in an area becomes uncertain. Thus, we employ the data-driven approach to characterize users’ demand uncertainty, formulate the TNC's revenue maximization problem into risk-averse stochastic programming, and provide corresponding feasible solutions. Using the released public data of Didi Chuxing, we conduct extensive simulations to evaluate the performance of the proposed scheduling scheme and compare the results under different <inline-formula><tex-math notation="LaTeX">$\zeta$</tex-math></inline-formula> -structure metrics. The results show that the proposed scheme can efficiently schedule the TNC vehicles, maximize the TNC's revenue and provide a better service for TNC users while protecting the TNC users’ location privacy.

Distributed tri-layer risk-averse stochastic game approach for energy trading among multi-energy microgrids

This paper discusses a tri-layer non-cooperative energy trading approach among multiple grid-tied multi-energy microgrids (MEMGs) in the restructured integrated energy market. The heterogeneous uncertainties from renewable energy, market prices, and electric energy loads are also considered via the risk-averse stochastic programming (SP) approach. First, comprehensive operation models of individual MEMGs are presented with the consideration of practical electric energy and thermal network flows as well as battery degradation. Second, to guarantee fair multi-energy trading among MEMGs and deal with adverse effects from all uncertainty sources, a tri-layer Cournot Nash game-based energy bidding method is developed and solved by the SP approach. In the first layer, i.e., day-ahead multi-energy market, optimal energy bids, dispatches of energy storage assets, and thermal flows against uncertainty scenarios are acquired in a risk-averse manner; In the second layer, i.e., intra-day multi-energy market, optimal intra-day energy bids and dispatches of all resources against uncertainty realizations are sequentially calculated; In the third layer, i.e., the real-time multi-energy market, transactions between each MEMG and the wholesale multi-energy market are finalized. Third, for protecting the privacy of individual MEMGs and alleviating the computation burdens, the distributed alternating search procedure is employed to compute the Nash equilibriums in the day-ahead and intra-day markets. In the end, numerical case studies are conducted to verify the effectiveness of our method. From the simulation results, it can be inferred that compared with the traditional cooperative, deterministic and risk-natural methods in the literature, our proposed method is more practical and economical for real-world applications since it comprehensively considers the market competition, uncertainty handling, and energy trading risk.

Two-stage risk-averse stochastic programming approach for multi-item single source ordering problem: CVaR minimisation with transportation cost

Integrating inventory and transportation decisions is vital in supply chain management and can enable decision-makers to achieve competitive advantages. This study considers a multi-item replenishment problem (MIRP) with a piece-wise linear transportation cost under demand uncertainty, which usually occurs both in retail and production environment when several items must be ordered from a single supplier. Conventionally, two-stage stochastic programming formulation is risk-neutral, and it lacks robustness in the presence of high data variability. Hence, we introduce the Conditional Value at Risk (CVaR) approach for MIRP. Additionally, we deploy both single and multi-cut L-shaped and the sample average approximation method to circumvent the computational complexity to solve large-scale instances. The data-driven simulation study is used to benchmark the results from deterministic, risk-neutral, and risk-averse stochastic models. The results indicate that under higher data variations, the risk-averse model provides better perspectives for a decision-maker. The results show a 40–50% reduction in lost sales with marginal growth in total cost while considering CVaR instead of a risk-neutral approach.

Sequential Resource Planning Decisions in an Epidemic Based on an Innovative Spread Model

Making rapid decisions in intervention resource planning is crucial for mitigating morbidity, mortality, and costs to the societies during epidemic outbreaks. This study presents a data-driven optimization approach for multiperiod resources planning, based on a sequential decision framework, considering up-to-date information and uncertainty in the spread of epidemics. In this method, a new <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$SEI^{3}H^{2}RD$ </tex-math></inline-formula> spread model is constructed to generate the most potential scenarios of an epidemic, based on all the historical information, and risk-averse stochastic programming was proposed to arrive at an optimal resource planning solution. The data-based numerical experiments demonstrate that our approach could control the epidemic by reducing the infected cases and deaths with similar or fewer resources than in the reality. In addition, we also find that the risk-averse design of the objective was able to take a steadier approach to resource planning helped avoid large fluctuations in resource allocations compared to a risk-neutral design. The other insight obtained from these experiments was that a moderate decision interval along with a planning horizon, which is slightly larger than the decision interval, would be a good choice for the sequential planning problem. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —In the event of an outbreak of a new infectious disease, the uncertainty of the epidemic parameters limits the ability of the model to provide accurate predictions. However, decision-makers cannot wait for more information to alleviate this uncertainty and must immediately make decisions to control the epidemic. This article proposes a sequential decision-making approach with an innovative <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$SEI^{3}H^{2}RD$ </tex-math></inline-formula> spread model and a risk-averse scenario-based stochastic programming to help decision-makers arrive at real-time decisions. We illustrate the performance of this approach using the COVID-19 outbreak in Wuhan. The results show that our model predictions closely fit the real outbreak data and prove that our decision approach could reduce the cost of using the available resources and achieve the goal of controlling the epidemic. The proposed modeling framework can be adopted to study other infectious diseases and provide tangible policy recommendations for controlling outbreaks of such diseases.

Risk-Averse Stochastic Programming vs. Adaptive Robust Optimization: A Virtual Power Plant Application

This paper compares risk-averse optimization methods to address the self-scheduling and market involvement of a virtual power plant (VPP). The decision-making problem of the VPP involves uncertainty in the wind speed and electricity price forecast. We focus on two methods: risk-averse two-stage stochastic programming (SP) and two-stage adaptive robust optimization (ARO). We investigate both methods concerning formulations, uncertainty and risk, decomposition algorithms, and their computational performance. To quantify the risk in SP, we use the conditional value at risk (CVaR) because it can resemble a worst-case measure, which naturally links to ARO. We use two efficient implementations of the decomposition algorithms for SP and ARO; we assess (1) the operational results regarding first-stage decision variables, estimate of expected profit, and estimate of the CVaR of the profit and (2) their performance taking into consideration different sample sizes and risk management parameters. The results show that similar first-stage solutions are obtained depending on the risk parameterizations used in each formulation. Computationally, we identified three cases: (1) SP with a sample of 500 elements is competitive with ARO; (2) SP performance degrades comparing to the first case and ARO fails to converge in four out of five risk parameters; (3) SP fails to converge, whereas ARO converges in three out of five risk parameters. Overall, these performance cases depend on the combined effect of deterministic and uncertain data and risk parameters. Summary of Contribution: The work presented in this manuscript is at the intersection of operations research and computer science, which are intrinsically related with the scope and mission of IJOC. From the operations research perspective, two methodologies for optimization under uncertainty are studied: risk-averse stochastic programming and adaptive robust optimization. These methodologies are illustrated using an energy scheduling problem. The study includes a comparison from the point of view of uncertainty modeling, formulations, decomposition methods, and analysis of solutions. From the computer science perspective, a careful implementation of decomposition methods using parallelization techniques and a sample average approximation methodology was done . A detailed comparison of the computational performance of both methods is performed. Finally, the conclusions allow establishing links between two alternative methodologies in operations research: stochastic programming and robust optimization.

Resilient NdFeB magnet recycling under the impacts of COVID-19 pandemic: Stochastic programming and Benders decomposition

Neodymium-iron-boron (NdFeB) magnets are the most powerful magnets per unit volume sold in the commercial market. Despite the increasing demand for clean energy applications such as electric vehicles and wind turbines, disruptive events including the COVID-19 pandemic have caused significant uncertainties in the supply and demand for NdFeB magnets. Therefore, this study aims to alleviate the risk of supply shortage for NdFeB magnets and the containing critical materials, rare-earth elements (REEs), through the development of a resilient reverse supply chain and logistics network design. We develop scenarios to model the unique impact of the COVID-19 pandemic on the proposed business, incorporating both disruption intensity and recovery rate. We formulate a chance-constrained two-stage stochastic programming model to maximize the profit while guaranteeing the network resiliency against disruption risks. To solve the problem in large-scale instances, we develop an efficient Benders decomposition algorithm that reduces the computational time by 98.5% on average compared to the default CPLEX algorithm. When applied to the United States, the model suggests the optimal facility locations, processing capacities, inventory levels, and material flows for NdFeB magnet recyclers that could meet 99.7% of the demand. To the best of our knowledge, this study is the first to incorporate the impacts of the COVID-19 pandemic to design a resilient NdFeB magnet recycling supply chain and logistics network, leveraging risk-averse stochastic programming.

Risk-Averse Stochastic Programming and Distributionally Robust Optimization Via Operator Splitting

Risk-Averse Stochastic Programming and Distributionally Robust Optimization Via Operator Splitting

Risk-Averse Stochastic Programming: Time Consistency and Optimal Stopping

Decision making under uncertainty includes reassessing and reevaluating risk after initial decisions. To this end, it is essential to consider a governing value process and to track its evolution over time. The paper, “Risk-Averse Stochastic Programming: Time Consistency and Optimal Stopping,” by Pichler, Liu, and Shapiro, develops a consistent framework by consolidating risk and optimal stopping. The paper develops the notion of time consistency for the stochastic multistage optimization problem. Supermartingales and envelopes characterizing optimal decisions are given explicitly. With that, dynamic equations are derived, which gradually reveal the optimal policy. Taking risk into account requires updating optimal policies, as an explicit example on American options demonstrates.

A Risk-Averse Adaptively Stochastic Optimization Method for Multi-Energy Ship Operation Under Diverse Uncertainties

In this paper, an optimal coordination method for energy dispatch and voyage scheduling is proposed for a renewable-energy-integrated hybrid AC/DC multi-energy ship (MES) microgrid under the continuous ship swinging. In the MES microgrid, all the onboard units are dispatched coordinately with higher flexibility for providing multiple energies. To guarantee the reliable ship operation, diverse uncertainties from solar irradiation, ship swinging angle, and onboard multi-energy demands are managed by an adaptive risk-averse stochastic programming approach to minimize the voyage cost and conditional value-at-risk. Besides, chance constraints are introduced to leverage the quality of thermal service given the thermal inertia. To speed up the solution process, the original nonlinear/nonconvex operation constraints are reformulated to a mixed-integer quadratically constrained programming form by linearization/convexification and scenario generation/reduction methods. Then the problem can be efficiently solved by commercial solvers. Finally, case studies are conducted on a test MES microgrid. The simulation results verify that the proposed method is effective in coordinating multi-energy dispatch and voyage scheduling, minimizing operating cost/risk, and immunizing against diverse uncertainties.

A Risk-Averse Shelter Location and Evacuation Routing Assignment Problem in an Uncertain Environment.

Disasters such as hurricanes, earthquakes and floods continue to have devastating socioeconomic impacts and endanger millions of lives. Shelters are safe zones that protect victims from possible damage, and evacuation routes are the paths from disaster zones toward shelter areas. To enable the timely evacuation of disaster zones, decisions regarding shelter location and routing assignment (i.e., traffic assignment) should be considered simultaneously. In this work, we propose a risk-averse stochastic programming model with a chance constraint that takes into account the uncertainty in the demand of disaster sites while minimizing the total evacuation time. The total evacuation time reflects the efficacy of emergency management from a system optimal (SO) perspective. A conditional value-at-risk (CVaR) is incorporated into the objective function to account for risk measures in the presence of uncertain post-disaster demand. We resolve the non-linear travel time function of traffic flow by employing a second-order cone programming (SOCP) approach and linearizing the non-linear chance constraints into a new mixed-integer linear programming (MILP) reformulation so that the problem can be directly solved by state-of-the-art optimization solvers. We illustrate the application of our model using two case studies. The first case study is used to demonstrate the difference between a risk-neutral model and our proposed model. An extensive computational study provides practical insight into the proposed modeling approach using another case study concerning the Black Saturday bushfire in Australia.

Epi-Regularization of Risk Measures

Uncertainty pervades virtually every branch of science and engineering, and in many disciplines, the underlying phenomena can be modeled by partial differential equations (PDEs) with uncertain or random inputs. This work is motivated by risk-averse stochastic programming problems constrained by PDEs. These problems are posed in infinite dimensions, which leads to a significant increase in the scale of the (discretized) problem. In order to handle the inherent nonsmoothness of, for example, coherent risk measures and to exploit existing solution techniques for smooth, PDE-constrained optimization problems, we propose a variational smoothing technique called epigraphical (epi-)regularization. We investigate the effects of epi-regularization on the axioms of coherency and prove differentiability of the smoothed risk measures. In addition, we demonstrate variational convergence of the epi-regularized risk measures and prove the consistency of minimizers and first-order stationary points for the approximate risk-averse optimization problem. We conclude with numerical experiments confirming our theoretical results.

A multistage risk-averse stochastic programming model for personal savings accrual: the evidence from Lithuania

In this paper we consider the problem of choosing the optimal pension fund in the second pillar of Lithuanian pension system by providing some guidelines to individuals with defined contribution pension plans. A multistage risk-averse stochastic optimization model is proposed that can be used to plan a long-term pension accrual under two different cases: minimum and maximum accumulation plans as possible options in the system. The investment strategy of personal savings is based on the optimal solutions over possible scenario realizations generated for a particular participant. The concept of the risk-averse decision-maker is implemented by choosing the conditional value at risk as the risk measure defined by a nested formulation that guarantees the time consistency in the multistage model. The paper focuses on three important decision-making moments corresponding to the duration of periods to be modelled. The first period is a short-term accumulation, while the second period is a long-term accumulation with possibly high deviation of objective function value. The third period is designed to implement the concept of target date fund in the second pillar pension scheme as the subsequent need to protect against potential losses at risky pension funds. The experimental findings of this research provide insights for individuals as decision-makers to select pension funds, as well as for policy-makers by revealing the vulnerability of pension system.

Strong convexity in risk-averse stochastic programs with complete recourse

We give sufficient conditions for the expected excess and the mean-upper-semideviation of recourse functions to be strongly convex. This is done in the setting of two-stage stochastic programs with complete linear recourse and random right-hand side. This work extends results on strong convexity of risk-neutral models.

A Central Limit Theorem and Hypotheses Testing for Risk-averse Stochastic Programs

We study statistical properties of the optimal value and optimal solutions of the sample average approximation of risk-averse stochastic problems. Central limit theorem-type results are derived for the optimal value when the stochastic program is expressed in terms of a law invariant coherent risk measure having a discrete Kusuoka representation. The obtained results are applied to hypotheses testing problems aiming at comparing the optimal values of several risk-averse convex stochastic programs on the basis of samples of the underlying random vectors. We also consider nonasymptotic tests based on confidence intervals on the optimal values of the stochastic programs obtained using the stochastic mirror descent algorithm. Numerical simulations show how to use our developments to choose among different distributions and on the considered class of risk-averse stochastic programs the asymptotic tests show better results.

Supply chain network design under uncertainty: A comprehensive review and future research directions

Supply chain network design (SCND) is one of the most crucial planning problems in supply chain management (SCM). Nowadays, design decisions should be viable enough to function well under complex and uncertain business environments for many years or decades. Therefore, it is essential to make these decisions in the presence of uncertainty, as over the last two decades, a large number of relevant publications have emphasized its importance. The aim of this paper is to provide a comprehensive review of studies in the fields of SCND and reverse logistics network design under uncertainty. The paper is organized in two main parts to investigate the basic features of these studies. In the first part, planning decisions, network structure, paradigms and aspects related to SCM are discussed. In the second part, existing optimization techniques for dealing with uncertainty such as recourse-based stochastic programming, risk-averse stochastic programming, robust optimization, and fuzzy mathematical programming are explored in terms of mathematical modeling and solution approaches. Finally, the drawbacks and missing aspects of the related literature are highlighted and a list of potential issues for future research directions is recommended.