Stochastic Robust Optimization Research Articles

Exploring the Impact of Random Distribution Choices on Particle Swarm Optimization: An Initial Analysis

Particle Swarm Optimization (PSO) is a robust stochastic optimization algorithm for solving complex and constrained optimization problems. This paper aims to systematically investigate the influence of diverse random number distributions on the learning parameters of PSO and evaluate the algorithm's sensitivity based on these parameters. To this end, several exhaustive numerical experiments were conducted on several test functions, including asymmetric Sphere, Easom, and Csendes. Initially, appropriate random distributions were identified for the learning parameters and proceeded to compare these distributions using all potential combinations. The sensitivity of PSO to these parameters was evaluated using the mean best score, the relative proximity of the best particle to the theoretical optimum, and the average objective function evaluations. The most significant contribution of this research lies in the insights derived from the experimental data. The discovery was that highly skewed and long-tailed distributions such as Chi, Chi-square, Lomax, and Exponential often underperform. On the other hand, distributions that are bounded proportionally to the problem domain consistently surpass the performance of the standard uniform distribution. These findings significantly enhance the understanding of PSO algorithms and provide valuable guidelines for their practical deployment.

A robust-stochastic data envelopment analysis model for supplier performance evaluation of the telecommunication industry under uncertainty

The primary activities of any organization rely on the procurement of the required goods and services at the shortest time and highest quality possible. On this basis, the problem of supplier evaluation, ranking, and selection is considered critically important. Data envelopment analysis is a well-known and successful approach in this field. In this study, we propose a robust-stochastic data envelopment analysis model to measure the efficiency of decision-making units under uncertainty. We measure efficiency through a standard and an inverted model in terms of resilience and agility. In order to demonstrate the practical potential of the proposed model, we apply the model to a case study of the Iranian telecom industry with 90 decision-making units. Numerical results reveal that human resources and cash assets are the most important input criteria. Also, the output indicators, including adaptability, reliability, visibility, and coordination, have high importance in measuring the efficiency of decision-making units. It should be noted that employing the robust-stochastic optimization approach leads to controlling the fluctuations of uncertain parameters and maintaining a desirable optimal level of efficiency for decision-making units under different scenarios. The results suggest that the model is sufficiently valid and reliable for evaluating the performance of suppliers in the telecom industry, may be employed under uncertain conditions, and can incorporate decision-makers’ varying preferences. The managerial insights derived from this research indicate that, in the short term, uncertainty throughout the evaluation process of suppliers often leads to reduced efficiency among the decision-making units. However, operating under uncertainty is associated with several advantages in the long term, such as increased decision-making consistency and improved vital ability to cope with uncertainty.

Design of a robust waste recycling network integrating social and environmental pillars of sustainability

The design of an efficient Municipal Solid Waste (MSW) management network can reduce the network costs, such as investment, operational and recycling costs, and improve its sustainability, from economic to environmental and social perspectives. In this paper, a multi-objective scenario-based robust stochastic optimization model for designing a sustainable MSW management network under uncertainty is proposed. The proposed model has four objectives to seek sustainability from two quantitative and qualitative aspects. Considering the dynamicity of the factors affecting an MSW management network as well as the multiplicity of sustainability perspectives, the proposed model allows, on one hand, reaching a robust solution considering the potential scenarios, and on the other hand, integrating sustainability indicators while creating a balance between the quantitative and qualitative evaluation of such indicators. Moreover, the waste treatment technologies, as the highest added-value echelon of the MSW management network, which also distinguishes the network from a generic waste recycling network, has been investigated in the model. Finally, fuel consumption, have been particularly emphasized as critical factor, highly contributing to transportation costs and Co2 emission that are decisive criteria from both economic and environmental points of view. The proposed model and solution approach are validated through a real case study.

Robust and resilience budget allocation for projects with a risk-averse approach: A case study in healthcare projects

One of the critical factors that force to complete projects in minimum time is Budget Allocation for Projects (BAP). This subject is vital in large-scale projects like provinces and healthcare projects that affect people's lives. The innovation of this research considers Robust, Resilience, and Risk-averse BAP (3RBAP). We utilize a hybrid robust stochastic optimization by considering worst-case in the objective function for robustness. This model minimizes the weighted expected and minimum progress functions to tackle budget fluctuation. The case study is about healthcare project construction. Finally, we assign the budget for healthcare projects with uncertain scenarios. After sensitivity analysis, we obtain that by employing 3R, we can see that the progress function is 17.07% less than without 3R. In addition, if the conservativity coefficient rises to 50%, the progress function will decrease to 7.32%, but the time computation changes a little. However, declining the resiliency coefficient makes decreases and disrupts the budget assignment. As a result, the progress function reduces. Finally, the time computation grows smoothly by increasing the scale of problems, and the progress function decreases. Because the number of projects is very much, maximizing the expected value of the progress function decreases. Eventually, by changing to exponential and sine functions, the progress function improved 41.46%.

Reliable design of a congested disaster relief network: A two-stage stochastic-robust optimization approach

The preparedness of humanitarian relief networks can be enhanced by pre-positioning resources in strategic locations and using them when disasters strike, a strategy that gives rise to a two-stage planning problem. This paper presents a novel two-stage stochastic-robust optimization approach for integrated planning of pre- and post-disaster positioning and allocation of relief resources, while taking into consideration the uncertainty about demand for relief services and disruptions in the relief facilities and the transportation network. The proposed approach enables planners to effectively use limited historical data and imperfect experts’ opinions to obtain robust solutions while avoiding the over-conservatism of classical robust optimization methods. The objective sought is to minimize the expected total time victims need to receive assistance, including both access time to facilities and waiting/service time in them. Congestion in relief facilities is accounted for by modeling them as queuing systems and penalizing waiting time. A decomposition method based on column-and-constraint generation is implemented to solve the problem, whereas the nonlinear terms corresponding to queuing in the second-stage problem are handled using a direct search procedure. Applicability of the proposed approach is demonstrated through a real case study and the numerical results are analyzed to draw managerial insights.

A robust and resilience machine learning for forecasting agri-food production

This research proposes a new framework for agri-food capacity production by considering resiliency and robustness and paying attention to disruption and risk for the first time. It is applied robust stochastic optimization by adding robustness to the constraint's objective function and resiliency situation. This research minimizes the mean absolute deviation and coefficient of standard deviation errors by linear function in the agri-food capacity production. This study suggests agri-food managers and decision-makers use this mathematical method to forecast and improve production management. The results of this research lead to better decision-making and are compared with other sine functions. The main model's Robust and Resiliency Mean Absolute Deviation (RRMAD) value is 1.28% lower than other sine-type functions. The conservativity coefficient, confidence level, weight factor, resiliency coefficient, and probability of the scenario vary. The main model's RRMAD value is 1.28% lower than other sine-type functions. Growing the weight factor will result in an increase in RRMAD and a smooth decline in R-squared. Additionally, as the resilience coefficient rises, the RRMAD function increases while the R-squared declines. By altering the probability of the scenario, the RRMAD function drops, and the R-squared goes up.

Path Optimization of Low-Carbon Container Multimodal Transport under Uncertain Conditions

The development of multimodal transport has had a significant impact on China’s transportation industry. Due to the variability of the market environment, in this study, based on the context of the official launch of the national carbon emission trading market, the uncertainty of the demand and the randomness of carbon trading prices were considered. Taking minimum total transportation cost as the objective function, a robust stochastic optimization model of container multimodal transport was constructed, and a hybrid fireworks algorithm with gravitational search operator (FAGSO) was designed to solve and verify the effectiveness of the algorithm. Using a 35-node multimodal transportation network as an example, the multimodal transportation costs and schemes under three different modes were compared and analyzed, and the influence of parameter uncertainty was determined. The results show that the randomness of carbon trading prices will lead to an increase or decrease in the total transport cost, while robust optimization with uncertain demand will be affected by the regret value constraint, resulting in an increase in the total transport cost. Multimodal carriers can reduce transportation costs, reduce carbon emissions, and improve the transportation efficiency of multimodal transportation by comprehensively weighing the randomness of carbon trading prices, the nondeterminism of demand, and the relationship between the selection of maximum regret values and transportation costs.

A multi-time scale dispatching optimal model for rural biomass waste energy conversion system-based micro-energy grid considering multi-energy demand response

Aiming at utilizing straw, garbage, domestic sewage and other biomass waste resources in rural areas, this study designed a rural biomass wastes energy conversion system (BWs)-based micro energy grid (BWs-MEG), and the mathematical modeling of BWs-MEG is carried out including multi-energy transforms system (METS) and multi-energy demand response (MEDR). Then, a two-stage optimal framework for rural BWs-MEG multi-time scale dispatching is designed. In the day-ahead stage, a multi-objective dispatching optimal model was constructed with the objectives of minimum operating costs and maximum eco-environment benefits. In the intra-day stage, the robust optimization theory was utilized to characterize the uncertainties of wind power plant (WPP) and photovoltaic power generation (PV) and a rolling dispatching optimal model was constructed with the objective of minimum deviation costs. After that, the above dispatching model was fuzzified and linearized, and then converted into a mixed integer linear programming model. Finally, a micro-energy grid in northern China was selected as an example for case study. The results showed: (1) BWs can utilize rural biomass waste resources for pyrolysis power generation (PG) and gas production to achieve energy utilization and provide power, heating and gas output. In the islanded operation mode and grid-connected operation mode, when MEG is configured with BWs, the deviation costs decrease by 25.6 % and 26.33 %, while the operating costs increases only by 7.60 % and 13.52 %, respectively. The power output of WPP and PV increase by 1.03 % and 2.19 % in the grid-connected operation mode, indicating that BWs is conducive to realizing the multi-dimension supply and demand balance of power, heating and gas loads. (2) Multi-time scale dispatching model can give play to the regulation ability of BWs, METS and MEDR and connect the day-ahead dispatching plan with the intra-day dispatching strategy to formulate the optimal dispatching strategy. When there is deviation in the day-ahead dispatching strategy, METS and MEDR could maintain the supply and demand balance of power load. AHP can change the heating period and BWs could maintain the supply and balance of heating and gas loads. Compared with the day-ahead dispatching plan, the output of PG and power-to-gas device in intra-day dispatching strategy increase by 21.22 % and 9.78 %. (3) Robust stochastic optimization method can characterize the uncertainties of WPP and PV and formulate the dispatching decision schemes with different risk attitudes. With the robust coefficient Г increasing, MEG operating costs and eco-environment benefits increase and deviation adjustment costs decreases, but the robustness of the dispatching scheme is improved. When 0.25≤Г≤0.75, the increase of the uncertainty parameter will have a direct impact on the formulation of dispatching scheme. And the decision maker belongs to risk preference type, who is willing to take certain risks to win excess benefits. Besides, if MEG operates in the grid-connected mode, the uncertainty risks will be weakened and the operation mode shall be reasonably selected according to the demand for dispatching decision. Overall, the proposed optimization model can promote the energy utilization of rural biomass waste resources, which is conducive to the realization of a clean and low-carbon transformation of the overall energy structure.

A hybrid Robust-Stochastic optimization model for planned outage based Day-Ahead scheduling of a Plug-in electric vehicles parking lot

The growth of plug-in electric vehicles (PEVs) in recent decades and their impact on the operation of distribution networks have necessitated the requirement for PEV parking lots (PEPLs). With respect to the development of technology advances, PEPLs can benefit from charging the PEVs and selling the discharge power of PEVs (with V2G ability) to the grid. Besides, the load growth, the aging of the components and their thermal limitations enforce the distribution network operator to de-energize some of the zones to lighten the peak load. This de-energization with day-ahead notification is accounted for as a planned outage (PO). Such outages can have an adverse effect on the PEPL’s daily revenue. This paper introduces a hybrid robust-stochastic model for optimal day-ahead scheduling of the PEPL with renewable generation and considering the inherent uncertainties. The proposed model of the day-ahead scheduling is subject to the PO notified by the distribution network operator and concerns profit maximization in a way that mitigates the power shortage problem in the PO period. This PO-oriented scheduling helps the owner of the PEPL to successfully provide service for the PEVs at a minimum cost.

Designing sustainable closed-loop supply chain network with considering spot-to-point inflation and carbon emission policies: A case study

This paper uses a fuzzy robust stochastic optimization (FRSO) approach and a new hybrid algorithm of genetic algorithm (GA) and multi-choice goal programming with Utility Function (MCGP-UF) to minimize environmental impacts (EI) while maximizing net present value (NPV) and social impacts (SI) in a sustainable closed-loop supply chain (SCLSC) in a case study has been proposed under hybrid uncertainty with considering spot-to-point inflation (STPI). The resulting mixed-integer linear programming (MILP) model has been applied to the electronic industry. The proposed model is evaluated in several experiments and discussed in different scenarios to prove the efficiency and performance comparison and validation of the proposed method. The results of this study double the share of the article (i) Effective improvement of the obtained solutions by reducing the solution time by 25% showed that the proposed method can be responsible for large-scale problems. (ii) Sensitivity analysis on parameters such as demand, inflation, costs, and capacity showed which scenario has the most impact on the strategic and tactical decisions of companies.

Multi-Stage Resilience Management of Smart Power Distribution Systems: A Stochastic Robust Optimization Model

Significant outages from weather and climate extremes have highlighted the critical need for resilience-centered risk management of the grid. This paper proposes a multi-stage stochastic robust optimization (SRO) model that advances the existing planning frameworks on two main fronts. First, it captures interactions of operational measures with hardening decisions. Second, it properly treats the multitude of uncertainties in planning. The SRO model coordinates hardening and system operational measures for smart power distribution systems equipped with distributed generation units and switches. To capture the uncertainty in the incurred damage by extreme events, an uncertainty set is developed by integrating probabilistic information of hurricanes with the performance of overhead structures. A novel probabilistic model for the repair time of damaged lines is derived to account for the uncertainty in the recovery process. A solution strategy based on the integration of a differential evolution algorithm and a mixed-integer solver is designed to solve the resilience maximization model. The proposed approach is applied to a modified IEEE 33-bus system with 485 utility poles and a 118-bus system with 1841 poles. The systems are mapped on the Harris County, TX, U.S. Results reveal that optimal hardening decisions can be significantly influenced by resilience operational measures.

Decision bounding problems for two-stage distributionally robust stochastic bilevel optimization

Decision bounding problems for two-stage distributionally robust stochastic bilevel optimization

An efficient energy planning model optimizing cost, emission, and social impact with different carbon tax scenarios

This study proposes a modified energy planning model that considers a broad range of future uncertainties. Modifications to hybrid stochastic robust optimization and robust optimization methodology allow for the introduction of multi-objective functions that reflect the various dimensions of energy planning including cost, emission, and social impact. A lexicographical and weighted-sum type multi-objective function are used in formulating the proposed optimization model. Changing the priorities of the objective functions generates different energy policies, which are then compared. Data envelopment analysis is applied to measure the energy efficiency of each optimal energy policy produced by the energy planning model. Energy efficiency is defined as the ability to satisfactorily address five main aspects—cost, emissions, social impact, employment, and security. An updated power development plan for Thailand is used as an illustrative case study. The empirical analysis indicates that a policy prioritizing the environment first, followed by social impact and cost, is the most efficient policy among the five alternatives considered, with an average efficiency score of 0.9988. Sensitivity analysis involving various carbon tax scenarios is used to establish the importance of weight selection in the weighted-sum method. Specifically, the empirical case study reveals that when the weighted-sum method is used, increasing the carbon tax increases the relative policy efficiency score. Results from the case study offer quantitative support for policy makers seeking to devise an efficient energy policy that meets extensive requirements while still dealing with the bounds of uncertain future projections.

Distributionally Robust Stochastic Optimization with Wasserstein Distance

Distributionally robust stochastic optimization (DRSO) is an approach to optimization under uncertainty in which, instead of assuming that there is a known true underlying probability distribution, one hedges against a chosen set of distributions. In this paper, we first point out that the set of distributions should be chosen to be appropriate for the application at hand and some of the choices that have been popular until recently are, for many applications, not good choices. We next consider sets of distributions that are within a chosen Wasserstein distance from a nominal distribution. Such a choice of sets has two advantages: (1) The resulting distributions hedged against are more reasonable than those resulting from other popular choices of sets. (2) The problem of determining the worst-case expectation over the resulting set of distributions has desirable tractability properties. We derive a strong duality reformulation of the corresponding DRSO problem and construct approximate worst-case distributions (or an exact worst-case distribution if it exists) explicitly via the first-order optimality conditions of the dual problem. Our contributions are fourfold. (i) We identify necessary and sufficient conditions for the existence of a worst-case distribution, which are naturally related to the growth rate of the objective function. (ii) We show that the worst-case distributions resulting from an appropriate Wasserstein distance have a concise structure and a clear interpretation. (iii) Using this structure, we show that data-driven DRSO problems can be approximated to any accuracy by robust optimization problems, and thereby many DRSO problems become tractable by using tools from robust optimization. (iv) Our strong duality result holds in a very general setting. As examples, we show that it can be applied to infinite dimensional process control and intensity estimation for point processes.

Bi‐objective optimization of a stochastic resilient vaccine distribution network in the context of the COVID‐19 pandemic

This paper develops an approach to optimize a vaccine distribution network design through a mixed-integer nonlinear programming model with two objectives: minimizing the total expected number of deaths among the population and minimizing the total distribution cost of the vaccination campaign. Additionally, we assume that a set of input parameters (e.g., death rate, social contacts, vaccine supply, etc.) is uncertain, and the distribution network is exposed to disruptions. We then investigate the resilience of the distribution network through a scenario-based robust-stochastic optimization approach. The proposed model is linearized and finally validated through a real case study of the COVID-19 vaccination campaign in France. We show that the current vaccination strategies are not optimal, and vaccination prioritization among the population and the equity of vaccine distribution depend on other factors than those conceived by health policymakers. Furthermore, we demonstrate that a vaccination strategy mixing the population prioritization and the quarantine restrictions leads to an 8.5% decrease in the total number of deaths.

A hybrid robust-stochastic optimization approach for day-ahead scheduling of cascaded hydroelectric system in restructured electricity market

Uncertainties arising from complicated natural and market environments pose great challenges for the efficient operation of cascaded hydroelectric systems. To overcome these challenges, this paper studies the day-ahead scheduling of cascaded hydroelectric systems in a restructured electricity market with the presence of uncertainties in electricity price and natural water inflow. To properly model the uncertainty, we consider the unique characteristics of these two types of uncertainties and capture them via the uncertainty set and stochastic scenarios, respectively. A hybrid robust-stochastic optimization model is developed to simultaneously hedge against these two types of uncertainties, which is formulated as a large-scale non-convex optimization problem with mixed integer recourse. After introducing linearization of nonlinear terms, a tailored hybrid decomposition scheme combining Lagrangian relaxation and Dantzig-Wolfe decomposition is adopted to achieve efficient computation of the proposed model. Two real-world cases are conducted to demonstrate the capability and characteristics of the proposed model and algorithms.

Strengthening Transmission System Resilience Against Extreme Weather Events by Undergrounding Selected Lines

Natural disasters, such as extreme weather events (EWEs), can cause significant damage to power systems. In fact, it is expected that the intensity and frequency of EWEs will increase the next years due to climate change, making power system resilience enhancement necessary. This paper proposes a transmission resilience planning solution by determining the lines to be placed underground in order to minimize load shedding in the most cost-efficient way taking into account historical EWEs (HEWEs). The problem is formulated as a stochastic robust optimization problem and is decomposed into a master problem determining the lines to be placed underground and a number of subproblems, where each subproblem corresponds to a HEWE and provides the worst damage scenario that leads to the maximization of load shedding. The problem is solved using a decomposition-based algorithm. The proposed method is illustrated on a modified IEEE Reliability Test System and IEEE 118-bus System and it is evaluated by applying a Sequential Monte Carlo simulation.

Hybrid Stochastic-Robust Service Restoration for Wind Power Penetrated Distribution Systems Considering Subsequent Random Contingencies

Service restoration (SR) is essential to recovering distribution systems with outages when extreme weather conditions occur. However, the occurrence of long-duration extreme weather events may cause a restored system to become susceptible to subsequent random contingencies. To address this challenge, this paper proposes a microgrid-based SR approach for recovering critical loads in wind power penetrated distribution systems incorporating the impacts from subsequent random contingencies. A hybrid stochastic-robust optimization model is developed considering both the subsequent random contingencies that may occur and the uncertainty of wind power generation. The expected load shedding is minimized by locating mobile emergency generators, dynamically adjusting microgrid (MG) topologies, and dispatching power sources. To ensure the tractability of the optimization, the original problem is reformulated and decomposed into a master problem and several robust optimization subproblems based on the column-and- constraint generation approach. In addition, the master problem, which is in the format of stochastic optimization is solved by an improved progressive hedging algorithm to accelerate optimization. The proposed method is validated on the modified IEEE distribution systems and the simulation results show the effectiveness of the proposed SR method in reducing security violation risk and reducing the computational complexity.

Coupling Two-Stage Stochastic Robust Programming with Improved Export Coefficient for Water Allocation among Industrial Sectors

Water scarcity and water pollution are essential factors limiting coordinated regional development, especially in water-deprived regions. Industrial restructuring is an effective water management solution to alleviate water scarcity and mitigate water pollution. However, due to widely existing inexact parameter information in the water resource management system, it is challenging to allocate water resources among industrial sectors. To address these problems, an export coefficient coupled with a two-stage stochastic robust programming method (EC-TSRP) was developed through integrating an export coefficient model (ECM), two-stage stochastic programming (TSP) and robust optimization. The proposed EC-TSRP model could effectively deal with the multiple uncertainties expressed as stochastic and the intervals with fluctuation ranges, and enhance the robustness of optimal plans for supporting water resource allocation among industrial sectors under complex uncertainties. It was then applied to Bayan Nur City, in arid north-west China. The optimization alternatives indicate that wheat, sheep and services would be the most sensitive sectors among all industrial sectors, when non-point source (NPS) pollution exports are restricted. In addition, comparing the EC-TSRP results with the deterministic model, the reliability of the system could be improved significantly, while the value of the objective function would be decreased slightly. The simulation results were also compared with the historical data from 2012 to 2016. Although the total revenue of Bayan Nur City would decrease by 1.52%, the pollutant loads of total nitrogen, total phosphorus and chemical oxygen demand (TN, TP and COD) would decrease by 14.5%, 7.75% and 2.07%, respectively, and total water allocation also would decrease from 4.6 billion m3 to 4.23 billion m3.

Robust Stochastic Facility Location: Sensitivity Analysis and Exact Solution

This work focuses on a broad class of facility location problems in the context of adaptive robust stochastic optimization under the state-dependent demand uncertainty. The demand is assumed to be significantly affected by related state information, such as the seasonal or socio-economic information. In particular, a state-wise ambiguity set is adopted for modeling the distributional uncertainty associated with the demand in different states. The conditional distributional characteristics in each state are described by a support, as well as by mean and dispersion measures, which are assumed to be conic representable. A robust sensitivity analysis is performed, in which, on the one hand, we analyze the impact of the change in ambiguity-set parameters (e.g., state probabilities, mean value abounds, and dispersion bounds in different states) onto the optimal worst-case expected total cost using the ambiguity dual variables. On the other hand, we analyze the impact of the change in location design onto the worst-case expected second-stage cost and show that the sensitivity bounds are fully described as the worst-case expected shadow-capacity cost. As for the solution approach, we propose a nested Benders decomposition algorithm for solving the model exactly, which leverages the subgradients of the worst-case expected second-stage cost at the location decisions formed insightfully by the associated worst-case distributions. The nested Benders decomposition approach ensures a finite-step convergence, which can also be regarded as an extension of the classic L-shaped algorithm for two-stage stochastic programming to our state-wise, robust stochastic facility location problem with conic representable ambiguity. Finally, the results of a series of numerical experiments are presented that justify the value of the state-wise distributional information incorporated in our robust stochastic facility location model, the robustness of the model, and the performance of the exact solution approach.