Stochastic Multi-objective Optimization Problem Research Articles

Optimal sizing of an isolated microgrid with electric vehicles using stochastic programming

An isolated microgrid system with renewable sources and energy storage systems ensures sustainable access to electricity and is especially suitable to handle inadequate electrical infrastructure in rural areas. The capacity configuration of the system depends on the uncertainties of renewable energy sources and home user behaviour, and the stochastic behaviour of electric vehicle (EV) users. This study determines the optimal configuration of the isolated microgrid that comprises wind and photovoltaic generation systems as well as batteries and EVs by proposing a two-stage stochastic programming-based multi-objective optimization problem where the objective functions are minimization of the life cycle cost and reliability. The problem is solved using the deterministic equivalent of the stochastic programming model to find the exact solution. The scenarios in the model are generated for one year period to consider the effects of uncertainties throughout seasons. Cases for different roles and configurations of electric vehicles are simulated and analyzed to compare their impacts on the sizing of the microgrid. Also, the effects of charger size on the microgrid configuration are investigated through simulations. The simulation results show that a slight increase in LPSP values causes a drop in cost values of around 10%–20% in each EV scenario. The minimum cost is obtained when more efficient charge & discharge plugs for EVs are used. Furthermore, the robustness of the model results is observed through repeated experiments with different random parameters. • An isolated PV-wind-battery-EV-based microgrid system is designed by considering the uncertainties due to random demand, renewable generation and EV behaviour. • A two-stage stochastic multi-objective optimization problem is solved by minimizing the LCC and LPSP. • The impacts of EVs on the system design are investigated using different roles and configurations of EVs. • As bigger-sized chargers replace smaller ones, the cost of microgrid decreases.

Nonlinear Stochastic Multiobjective Optimization Problem in Multivariate Stratified Sampling Design

Decision-making in survey sampling planning is a tricky situation; sometimes it involves multiple objectives, with various decision variables emanating from heterogeneous and homogeneous populations. Dealing with the entire population under study and its uncertain nature becomes a challenging issue for researchers and policymakers. Hence, an appropriate sampling design and optimization methodology are imperative. The study presents a useful discussion on stochastic multiobjective multivariate stratified sampling (MSS) models theoretically, and the concepts are illustrated with numerical examples. Also, it has been found that the linearization of sampling variance in survey sampling does not help determine the optimal sampling allocation problem with minimum variability. Optimal allocation problems under the weighted goal programming, stochastic goal programming, and Chebyshev goal programming methods are also discussed with numerical examples. Finally, the study discussed the linear approximation of the MSS problem with examples. The study is a conceptual and theoretical framework for MSS under a stochastic environment. The numerical data is simulated using the stratifyR package.

Accuracy and fairness trade-offs in machine learning: a stochastic multi-objective approach

In the application of machine learning to real-life decision-making systems, e.g., credit scoring and criminal justice, the prediction outcomes might discriminate against people with sensitive attributes, leading to unfairness. The commonly used strategy in fair machine learning is to include fairness as a constraint or a penalization term in the minimization of the prediction loss, which ultimately limits the information given to decision-makers. In this paper, we introduce a new approach to handle fairness by formulating a stochastic multi-objective optimization problem for which the corresponding Pareto fronts uniquely and comprehensively define the accuracy-fairness trade-offs. We have then applied a stochastic approximation-type method to efficiently obtain well-spread and accurate Pareto fronts, and by doing so we can handle training data arriving in a streaming way.

Stochastic multi-objective scheduling of a wind farm integrated with high-temperature heat and power storage in energy market

In spite of significant opportunities created by wind energy, the high uncertainty of wind farms is problematic for their operators, making their participation in the energy market quite challenging. The combination of Air-based High-Temperature Heat and Power Storage (HTHPS), which is known as a novel energy storage technology, with a wind farm as an integrated system can make new opportunities for a Generation Company (GenCo) to overcome the wind generation challenges. This paper proposes a novel Stochastic Multi-objective Optimization (SMOO) problem to determine the optimal charging and discharging scheduling and the best bidding strategy in the day-ahead energy market for an integrated energy system including a wind farm and HTHPS unit. The proposed model presents a comprehensive and coherent approach from historical data analysis as input to final decision making as output, that it can maximize Genco’s profit by avoiding incorrect commitment and energy offering. In the phase of uncertainty handling, via a scenario-based approach, the wind uncertainty is modeled by Monte-Carlo Simulation (MCS) method based on wind speed forecast errors. In the optimization phase, the Pareto optimal set is obtained by Non-dominated Sorting Genetic Algorithm-II (NSGA-II), and then, a new method is also provided to choose the best possible solution based on the entropy Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) method and the minimax regret criterion. The historical data related to wind generation and energy prices provided by the Danish energy market in West Denmark is used in simulation analysis, and the results demonstrate pleasantly the effectiveness of the proposed model.

A Survey on Metaheuristics for Solving Stochastic Multi-Objective Optimization Problems

A Survey on Metaheuristics for Solving Stochastic Multi-Objective Optimization Problems

Equivalent Characterization of McRow Optimal Solutions to Multiobjective Optimization Problems

Based on the McRow model for multiobjective optimization problems, the relationships between the W-robust efficient solution (also known as the McRow optimal solution) and the weakly efficient solution, the efficient solution and the properly efficient solution were established. Firstly, the relationship between the W-robust efficient solution and various exact solutions to multiobjective optimization problems was studied. Then, the concept of the McRow optimal solution to stochastic multiobjective optimization problems was introduced, and the relationship between the McRow optimal solution and other solutions was given. The examples show that, the solutions obtained with the McRow model are of better robustness.

A stochastic optimal energy management strategy considering battery health for hybrid electric bus

The problem of battery health coupled with energy management brings a considerable challenge to the hybrid electric bus. To address this challenge, three contributions are made to realize optimal energy management control while prolonging battery life. First, a semi-empirical aging model of lithium iron phosphate battery is built and identified by the data fitting method, based on the battery cycling test. Besides, a severity factor map is constructed by employing the proposed aging model to characterize the relative aging of the battery under different operating conditions. Second, to make the driver demand torque more appropriate for statistical prediction, a Markov chain is formulated to predict driving behavior and also a stochastic vehicle mass estimation method is proposed to assist the prediction of required torque. Then, a stochastic multi-objective optimization problem is formulated by taking the severity factor map as a battery degradation criterion, where minimized battery degradation and fuel consumption can be simultaneously realized. Finally, a stochastic model predictive control strategy that considers battery health is established. Both simulation and hardware-in-loop tests are performed. The results demonstrate that fuel economy and battery degradation can be improved by 16.73% and 13.8% compared with rule-based strategy, respectively.

A Stochastic Multi-Objective Approach to Pilot Bus Selection for Secondary Voltage Regulation

This letter presents a multi-objective probabilistic model for the optimal selection of pilot buses in secondary voltage regulation (SVR) of transmission grids. The pilot-bus selection (PS) problem is formulated in the form of a stochastic multi-objective optimization problem (SMOP). The resultant SMOP is solved toward the non-dominated sorting genetic algorithm (NSGA-II) together with a fuzzy set theory. The applicability of the proposed PS model is examined on a classical test system as well as a practical transmission grid.

Monte Carlo Simulation-Based Bat Algorithm for Solving Stochastic Multi-Objective Optimization Problems

Monte Carlo Simulation-Based Bat Algorithm for Solving Stochastic Multi-Objective Optimization Problems

Bayesian model calibration and optimization of surfactant-polymer flooding

The physical models governing surfactant-polymer (SP) flooding process are subject to parametric uncertainties, accurate quantification of which is crucial for improved decision making. Moreover, history matching of SP flooding is an ill-posed problem, typically characterized by a multimodal posterior distribution of these model parameters. This paper presents a systematic approach for Bayesian history matching and uncertainty quantification in the model calibration stage of SP flooding using coreflood experimental data. The approach is as follows. First, we construct a surrogate of the computationally expensive physics-based model using a polynomial chaos expansion (PCE-proxy). Second, we formulate a Bayesian calibration problem for inferring the model parameters from a single coreflood experiment that measures pressure drop and oilcut profiles. Third, we solve the Bayesian calibration problem by sampling directly from the posterior using Markov chain Monte Carlo (MCMC). We validate the calibrated parameters by successfully predicting the result of two other coreflood experiments. Then, we extend this framework to stochastic multiobjective optimization of injection slug size design under uncertainties in model parameters (captured by the posterior of Bayesian calibration problem) and oil price (modeled as a geometric random walk with constant drift and volatility). To identify the Pareto frontier of the stochastic multiobjective optimization problem, we employ a variant of Bayesian global optimization (BGO), a class of algorithms capable of optimizing black-box, gradient-free, computationally expensive functions. In particular, we use the extended expected improvement over the dominated hypervolume to sequentially select simulations that seek to reveal the Pareto frontier. An addendum of the implemented BGO is that it quantifies the epistemic uncertainty about the Pareto frontier as induced by the limited number of simulations used to construct it.

Design under uncertainty of carbon capture, utilization and storage infrastructure considering profit, environmental impact, and risk preference

This study presents a decision making tool for risk management of a carbon capture utilization and storage (CCUS) network under uncertainty among conflicting objectives. A two-phase-two-stage stochastic multi-objective optimization problem solving algorithm is formulated to balance environmental impact and various sources of uncertainty and corresponding risk by installing and operating a CCUS network. The algorithm allows decision makers to choose their own tolerance on risk. By conducting case studies that have different target profits for CCUS networks, the algorithm provides optimal results based on the decision maker’s attitude to risk. To evaluate risks imposed by uncertain parameters, a concept of downside risk is introduced. By setting different target profit levels, the suggested tool enables decision makers to choose their own tolerance and preference for risk. In the model, the life cycle assessment is applied to evaluate all environmental contributions caused by installation and operations of the CCUS network. The model provides the trade-off relationship between total annual benefit with financial risk as well as corresponding environmental impact. The aim of this model to optimize CCUS supply chain networks is to provide an intuitive decision making algorithm to balance conflicting objectives within a single framework. This problem is formulated as a mixed integer linear program model. To illustrate the applicability of the model, four optimal CCUS network models for the various types of industrial complex of Korea in 2030 are presented. Results indicate that risk-averse cases with a low profit target are more reliable in stochastic uncertainty, and that risk-taking decision makers tend to invest more on capture facility and produce more product than do, risk-averse decision makers.

Asymptotic convergence of stationary points of stochastic multiobjective programs with parametric variational inequality constraint via SAA approach

We consider the sample average approximation method for a stochastic multiobjective programming problem constrained by parametric variational inequalities. The first order necessary conditions for the original problem and its sample average approximation (SAA) problem are established under constraint qualifications. By graphical convergence of set-valued mappings, the stationary points of the SAA problem converge to the stationary points of the true problem when the sample size tends to infinity. Under proper assumptions, the convergence of optimal solutions of SAA problems is also obtained. The numerical experiments on stochastic multiobjective optimization problems with variational inequalities are given to illustrate the efficiency of SAA estimators.

STOCHASTIC MULTIOBJECTIVE OPTIMIZATION ON A BUDGET: APPLICATION TO MULTIPASS WIRE DRAWING WITH QUANTIFIED UNCERTAINTIES

Design optimization of engineering systems with multiple competing objectives is a painstakingly tedious process especially when the objective functions are expensive-to-evaluate computer codes with parametric uncertainties. The effectiveness of the state-of-the-art techniques is greatly diminished because they require a large number of objective evaluations, which makes them impractical for problems of the above kind. Bayesian global optimization (BGO), has managed to deal with these challenges in solving single-objective optimization problems and has recently been extended to multi-objective optimization (MOO). BGO models the objectives via probabilistic surrogates and uses the epistemic uncertainty to define an information acquisition function (IAF) that quantifies the merit of evaluating the objective at new designs. This iterative data acquisition process continues until a stopping criterion is met. The most commonly used IAF for MOO is the expected improvement over the dominated hypervolume (EIHV) which in its original form is unable to deal with parametric uncertainties or measurement noise. In this work, we provide a systematic reformulation of EIHV to deal with stochastic MOO problems. The primary contribution of this paper lies in being able to filter out the noise and reformulate the EIHV without having to observe or estimate the stochastic parameters. An addendum of the probabilistic nature of our methodology is that it enables us to characterize our confidence about the predicted Pareto front. We verify and validate the proposed methodology by applying it to synthetic test problems with known solutions. We demonstrate our approach on an industrial problem of die pass design for a steel wire drawing process.

Scalable modeling and solution of stochastic multiobjective optimization problems

Abstract We present a scalable computing framework for the solution stochastic multiobjective optimization problems. The proposed framework uses a nested conditional value-at-risk (nCVaR) metric to find compromise solutions among conflicting random objectives. We prove that the associated nCVaR minimization problem can be cast as a standard stochastic programming problem with expected value (linking) constraints. We also show that these problems can be implemented in a modular and compact manner using PLASMO (a Julia -based structured modeling framework) and can be solved efficiently using PIPS-NLP (a parallel nonlinear solver). We apply the framework to a CHP design study in which we seek to find compromise solutions that trade-off cost, water, and emissions in the face of uncertainty in electricity and water demands.

Design under uncertainty of carbon capture and storage infrastructure considering cost, environmental impact, and preference on risk

We present a stochastic decision-making algorithm for the design and operation of a carbon capture and storage (CCS) network; the algorithm incorporates the decision-maker’s tolerance of risk caused by uncertainties. Given a set of available resources to capture, store, and transport CO2, the algorithm provides an optimal plan of the CCS infrastructure and a CCS assessment method, while minimizing annual cost, environmental impact, and risk under uncertainties. The model uses the concept of downside risk to explicitly incorporate the trade-off between risk and either economic or environmental objectives at the decision-making level. A two-phase-two-stage stochastic multi-objective optimization problem (2P2SSMOOP) solving approach is implemented to consider uncertainty, and the ε-constraint method is used to evaluate the interaction between total annual cost with financial risk and an Eco-indicator 99 score with environmental risk. The environmental impact is measured by Life Cycle Assessment (LCA) considering all contributions made by operation and installation of a CCS infrastructure. A case study of power-plant CO2 emission in Korea is presented to illustrate the application of the proposed modeling and solution method.

Solving stochastic multiobjective vehicle routing problem using probabilistic metaheuristic

In this paper we propose a new metaheuristic algorithm for solving stochastic multiobjective combinatorial optimization(SMOCO) problems. Indeed, we find that the various initiatives that have been launched recently on this subject, they propose the classical metaheuristics to solve a stochastic multi-objective problems,but when the stochasticity effects is not taken into account, the choice of an arbitrary value leading to a particular configuration and a high loss of information. To conserve the stochastic nature of SMOCO problems,the pareto ranking should be defined on the random objectives functions directly rather than converted deterministic objectives. From these considerations, the scope of this research should consist of: (i)Proposed novel methodology of stochastic optimality for ranking objective functions characterized by non-continuous and no closed form expression. This novel approach is based on combinatorial probability and can be incorporated in a multiobjective evolutionary algorithm. (ii)Provide probabilistic approaches to elitism and diversification in multiobjective evolutionary algorithms. Finally, The behavior of the resulting Probabilistic Multi-objective Evolutionary Algorithms (PrMOEAs) is empirically investigated on the multi-objective stochastic VRP problem.

Multi-objective optimization for planning liner shipping service with uncertain port times

This paper considers a joint tactical planning problem for the number of ships, the planned maximum sailing speed, and the liner service schedule in order to simultaneously optimize the expected cost, the service reliability and the shipping emission in the presence of port time uncertainty. The problem is formulated into a stochastic multi-objective optimization problem at the operational level. The relationships between the objectives and the decision variables are established. A simulation-based non-dominated sorting genetic algorithm is then presented to solve this problem. A case study is provided to illustrate the results and the application of the model.

Stochastic Optimization over a Pareto Set Associated with a Stochastic Multi-Objective Optimization Problem

We deal with the problem of minimizing the expectation of a real valued random function over the weakly Pareto or Pareto set associated with a Stochastic Multi-objective Optimization Problem, whose objectives are expectations of random functions. Assuming that the closed form of these expectations is difficult to obtain, we apply the Sample Average Approximation method in order to approach this problem. We prove that the Hausdorff---Pompeiu distance between the weakly Pareto sets associated with the Sample Average Approximation problem and the true weakly Pareto set converges to zero almost surely as the sample size goes to infinity, assuming that our Stochastic Multi-objective Optimization Problem is strictly convex. Then we show that every cluster point of any sequence of optimal solutions of the Sample Average Approximation problems is almost surely a true optimal solution. To handle also the non-convex case, we assume that the real objective to be minimized over the Pareto set depends on the expectations of the objectives of the Stochastic Optimization Problem, i.e. we optimize over the image space of the Stochastic Optimization Problem. Then, without any convexity hypothesis, we obtain the same type of results for the Pareto sets in the image spaces. Thus we show that the sequence of optimal values of the Sample Average Approximation problems converges almost surely to the true optimal value as the sample size goes to infinity.

Runtime Analysis of an Evolutionary Algorithm for Stochastic Multi-Objective Combinatorial Optimization

For stochastic multi-objective combinatorial optimization (SMOCO) problems, the adaptive Pareto sampling (APS) framework has been proposed, which is based on sampling and on the solution of deterministic multi-objective subproblems. We show that when plugging in the well-known simple evolutionary multi-objective optimizer (SEMO) as a subprocedure into APS, ε-dominance has to be used to achieve fast convergence to the Pareto front. Two general theorems are presented indicating how runtime complexity results for APS can be derived from corresponding results for SEMO. This may be a starting point for the runtime analysis of evolutionary SMOCO algorithms.

Stochastic Multiobjective Optimization: Sample Average Approximation and Applications

We investigate one stage stochastic multiobjective optimization problems where the objectives are the expected values of random functions. Assuming that the closed form of the expected values is difficult to obtain, we apply the well known Sample Average Approximation (SAA) method to solve it. We propose a smoothing infinity norm scalarization approach to solve the SAA problem and analyse the convergence of efficient solution of the SAA problem to the original problem as sample sizes increase. Under some moderate conditions, we show that, with probability approaching one exponentially fast with the increase of sample size, an ϵ-optimal solution to the SAA problem becomes an ϵ-optimal solution to its true counterpart. Moreover, under second order growth conditions, we show that an efficient point of the smoothed problem approximates an efficient solution of the true problem at a linear rate. Finally, we describe some numerical experiments on some stochastic multiobjective optimization problems and report preliminary results.