Distribution Of Uncertain Parameters Research Articles

Solving Contextual Stochastic Optimization Problems through Contextual Distribution Estimation

Stochastic optimization models always assume known probability distributions about uncertain parameters. However, it is unrealistic to know the true distributions. In the era of big data, with the knowledge of informative features related to uncertain parameters, this study aims to estimate the conditional distributions of uncertain parameters directly and solve the resulting contextual stochastic optimization problem by using a set of realizations drawn from estimated distributions, which is called the contextual distribution estimation method. We use an energy scheduling problem as the case study and conduct numerical experiments with real-world data. The results demonstrate that the proposed contextual distribution estimation method offers specific benefits in particular scenarios, resulting in improved decisions. This study contributes to the literature on contextual stochastic optimization problems by introducing the contextual distribution estimation method, which holds practical significance for addressing data-driven uncertain decision problems.

Intelligent optimal layout of drainage pipe network monitoring points based on information entropy theory

The rapid expansion of urban drainage pipe networks, driven by economic development, poses significant challenges for efficient monitoring and management. The complexity and scale of these networks make it difficult to effectively monitor and manage the discharge of urban domestic sewage, rainwater, and industrial effluents, leading to illegal discharges, leakage, environmental pollution, and economic losses. Efficient management relies on a rational layout of drainage pipe network monitoring points. However, existing research on optimal monitoring point layout is limited, primarily relying on manual analysis and fuzzy clustering methods, which are prone to human bias and ineffective monitoring data. To address these limitations, this study proposes a coupled model approach for the automatic optimization of monitoring point placement in drainage pipe networks. The proposed model integrates the information entropy index, Bayesian reasoning, the Monte Carlo method, and the stormwater management model (SWMM) to optimize monitoring point placement objectively and measurably. The information entropy algorithm is utilized to quantify the uncertainty and complexity of the drainage pipe network, facilitating the identification of optimal monitoring point locations. Bayesian reasoning is employed to update probabilities based on observed data, while the Monte Carlo method generates probabilistic distributions for uncertain parameters. The SWMM is utilized to simulate stormwater runoff and pollutant transport within the drainage pipe network. Results indicate that (1) the relative mean error of the parameter inversion simulation results of the pollution source tracking model is linearly fitted with the information entropy. The calculation shows that there is a good positive linear correlation between them, which verifies the feasibility of the information entropy algorithm in the field of monitoring node optimization; (2) the information entropy algorithm can be well applied to the optimal layout of a single monitoring node and multiple monitoring nodes, and it can correspond well to the inversion results of the tracking model parameters; (3) the constructed monitoring point optimization model can well realize the optimal layout of monitoring points of a drainage pipe network. Finally, the pollution source tracking model is used to verify the effectiveness of the optimal layout of monitoring points, and the whole process has less human participation and a high degree of automation. The automated monitoring point optimization layout model proposed in this study has been successfully applied in practical cases, significantly improving the efficiency of urban drainage network monitoring and reducing the degree of manual participation, which has important practical significance for improving the level of urban water environment management.

Efficient metamodel-based importance sampling coupled with single-loop estimation method for parameter global reliability sensitivity analysis

To efficiently estimate the main effects and total effects of uncertain distribution parameters on the uncertainty of failure probability, we construct single-loop estimation formulas by introducing auxiliary variables through the equal probability transformation. This approach circumvents the original nested triple-loop process. For generating samples used in the derived single-loop estimation formulas, direct Monte Carlo simulation can be employed. To reduce the number of samples in Monte Carlo simulation, the important sampling technique can be integrated into the proposed single-loop estimation formulas. Additionally, to enhance the efficiency of identifying the states (failure or safety) of all used samples, an adaptive Kriging model can be introduced. Subsequently, the adaptive Kriging model coupled with Monte Carlo simulation, and the adaptive Kriging model coupled with the importance sampling technique, are integrated into the derived single-loop formulas to concurrently and efficiently estimate the main effects and total effects of uncertain distribution parameters. The results of three case studies validate the accuracy and efficiency of the proposed method.

Designing robust green sustainable supply chain network by bi-objective optimization method

Designing a green supply chain is becoming popular in the context of sustainable development. To address this academic concern, this paper designs a multi-product, multi-echelon green supply chain network (GSCN) from economic and environmental aspects. During the modeling process, the main challenge is to access the accurate probability distributions of uncertain parameters from limited historical data. To overcome this difficulty, this paper develops a distributionally robust design framework for bi-objective GSCN, where the distribution information of uncertain parameter is partially available and characterized by ambiguity sets. For the tractability, this paper discusses robust counterpart reformulation under Wasserstein-distance-based ambiguity sets. Finally, the obtained mixed integer programming model is resolved via commercial optimization software. To validate the proposed optimization framework, we design a meat supply chain network for a Chinese realistic food enterprise. The computational results demonstrate that the proposed distributionally robust model can provide reliable solutions compared with stochastic optimization method.

Optimization of unmanned aerial vehicle inspection strategy for infrastructure based on model-enabled diagnostics and prognostics

The use of unmanned aerial vehicles (UAVs) for structural health inspection has become a promising technique to perform labor-intensive, accessibility-challenged, and sometimes dangerous inspection tasks. This paper presents a novel physics-informed UAV inspection planning framework for infrastructure structural health assessment based on model-based diagnostics and prognostics enabled by physics-based probabilistic analysis. It bridges the gap between UAV mission planning and inspection with model-based probabilistic analysis, by allowing bidirectional information exchange, namely (1) structural damage state diagnostics using UAV inspection data and (2) UAV inspection optimization through model-based failure prognostics. Based on the bidirectional communication, the impacts of the three key UAV inspection parameters (i.e., inspection distance, inspection interval, and critical maintenance threshold) on structural life-cycle cost are analyzed. The optimization of key UAV inspection parameters is achieved by minimizing the cost per unit time (CPUT) through model-based pre-posterior analysis. In this analysis, synthetic observations are generated using predictive models according to the prior distributions of various uncertainty sources (e.g., detection rate, damage state evolution, etc.). The generated synthetic observations are then used to obtain the posterior distributions of uncertain parameters, enabling the integration of prior information and Bayesian model updating into inspection optimization through a cost function. The proposed model offers a robust method to accommodate the inherent uncertainties in failure prognostics, leading to a more effective optimization of the UAV inspection parameters. The practical application of the framework is demonstrated through a miter gate example. The results show that the proposed method is able to efficiently determine the optimal UAV inspection parameters and continuously update the information model.

An Adjustable Scenario Optimisation Approach in Operating PV-Rich Distribution Systems

The embrace of intermittent distributed energy resources (DER) on the demand side increases the concerns around the safe operation of distribution networks. Existing solutions either neglect the available information around the distribution of uncertain parameters or underestimate DER capability in providing real-time recourse actions. In this paper, we propose an adjustable scenario optimisation-based solution approach that maximises the usage of DER generation within the operation horizon, while providing the required level of network security. Our modelling i) guarantees hard constraints satisfaction, ii) allows fast-acting devices, e.g., inverters, to adjust in response to live realisations, and iii) enables the operators to maintain the soft constraints with a required probability using a joint chance-constrained (JCC) program. We reformulate the JCCs using the scenario optimisation approach that directly takes empirical data and provides a pre-specified bound on the risk of constraint violation. To increase the scalability of our proposal, we introduce an approximation of JCCs to be used within the scenario optimisation approach. Through extensive comparisons and simulations, we show that our approach outperforms the conventional scenario optimisation and adjustable robust approaches respectively by 83% and 80% in our experiments.

A data-driven robust EVaR-PC with application to portfolio management.

We investigate the robust chance constrained optimization problem (RCCOP), which is a combination of the distributionally robust optimization (DRO) and the chance constraint (CC). The RCCOP plays an important role in modeling uncertain parameters within a decision-making framework. The chance constraint, which is equivalent to a constraint of Value-at-risk (VaR), is approximated by risk measures such as Entropic Value-at-risk (EVaR) or Conditional Value-at-risk (CVaR) due to its difficulty to be evaluated. An excellent approximation requires both tractability and non-conservativeness. In addition, the DRO assumes that we know partial information about the distribution of uncertain parameters instead of their known true underlying probability distribution. In this article, we develop a novel approximation EVaR- PC based on EVaR for CC. Then, we evaluate the proposed approximation EVaR- PC using a discrepancy-based ambiguity set with the wasserstein distance. From a theoretical perspective, the EVaR- PC is less conservative than EVaR and the wasserstein distance possesses many good theoretical properties; from a practical perspective, the discrepancy-based ambiguity set can make full use of the data to estimate the nominal distribution and reduce the sensitivity of decisions to priori knowledges. To show the advantages of our method, we show its application in portfolio management in detail and give the relevant experimental results.

Data-Driven Distributionally Robust Optimization over Time

Stochastic optimization (SO) is a classical approach for optimization under uncertainty that typically requires knowledge about the probability distribution of uncertain parameters. Because the latter is often unknown, distributionally robust optimization (DRO) provides a strong alternative that determines the best guaranteed solution over a set of distributions (ambiguity set). In this work, we present an approach for DRO over time that uses online learning and scenario observations arriving as a data stream to learn more about the uncertainty. Our robust solutions adapt over time and reduce the cost of protection with shrinking ambiguity. For various kinds of ambiguity sets, the robust solutions converge to the SO solution. Our algorithm achieves the optimization and learning goals without solving the DRO problem exactly at any step. We also provide a regret bound for the quality of the online strategy that converges at a rate of [Formula: see text], where T is the number of iterations. Furthermore, we illustrate the effectiveness of our procedure by numerical experiments on mixed-integer optimization instances from popular benchmark libraries and give practical examples stemming from telecommunications and routing. Our algorithm is able to solve the DRO over time problem significantly faster than standard reformulations. Funding: This work was supported by Deutsche Forschungsgemeinschaft (DFG): Projects B06 and B10 in CRC TRR 154 and Project-ID 416229255 - SFB 1411 and Federal Ministry for Economic Affairs and Energy, Germany [Grant 03EI1036A]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoo.2023.0091 .

Two‐stage distributionally robust optimization model for a pharmaceutical cold supply chain network design problem

AbstractPharmaceutical safety has received increasing attention from governments and corporations, and building a safe and effective pharmaceutical cold supply chain network has become an important issue. This paper proposes a two‐stage pharmaceutical cold supply chain network design problem considering drug safety, in which the drug demand, transportation costs, and drug safety risk costs are assumed to be random variables. Due to the presence of uncertainty and the fact that information about the distribution of uncertain parameters is often only partially known, a distributionally robust optimization method is used to handle the uncertainty. A two‐stage distributionally robust optimization model is constructed, in which the reliability of the estimation of the demand necessary to satisfy the entire cold chain network is ensured to be greater than a certain predetermined level by introducing an ambiguous joint chance constraint. The decision process for the problem can be divided into strategic and operational decisions with the goal of minimizing the total costs related to facility construction, the purchase of raw materials, drug production, transportation, and safety risks. By introducing an ambiguity set with mean and covariance information to describe the uncertain parameters, the two‐stage model is eventually reformulated as a standard second‐order cone program, thus making it computationally tractable. Finally, numerical experiments are presented to demonstrate the effectiveness of the proposed models and optimization methods.

An Interval Probabilistic Energy Flow Calculation Method for CCHP Campus Microgrids

Probabilistic models are used to describe the uncertainty of injected power in traditional probabilistic energy flow (PEF) calculations. Owing to the difficulty of obtaining the accurate distribution parameters of the probability models, the uncertainty of the distribution parameters needs to be considered in the PEF calculation of a combined cooling, heating, and power campus microgrid (CCHP-CMG). In this article, the parameterized probability box (p-box) model is applied for describing the uncertain distribution parameters of the probability model of injected power, and the interval PEF (IPEF) calculation model of a CCHP-CMG is developed. An affine arithmetic-based interval point estimation method (AIPEM) is put forward to solve the IPEF calculation model. The location intervals of injected powers are calculated from their p-boxes, and the interval energy flow calculation is performed with each location interval to estimate the moment intervals of the output variables. Affine arithmetic is used to reduce the interval expansion in the interval calculation, and to further reduce the interval expansion caused by affine multiplication/division operation, a method of removing the accumulation of extra noise terms is proposed. In addition, based on the Nataf inverse transformation and the p-box model, an AIPEM considering the correlation of injected powers is proposed that considers the uncertain correlation coefficients. A case study on a CCHP-CMG shows that the proposed AIPEM has higher computational accuracy than the existing interval cumulant method, especially when the uncertain fluctuation range of the injected power is large, although it consumes more CPU time.

An efficient Kriging model-based importance sampling method for estimating the failure probability-based parameter global sensitivity index with uncertain distribution parameters

The failure probability-based parameter global sensitivity index measures the importance of each uncertain distribution parameter on the uncertainty of failure probability. The crude Monte Carlo simulation for estimating the failure probability-based parameter global sensitivity index involves a complicated nested triple-loop process, and the huge amount of computational burden restricts the application of the failure probability-based parameter global sensitivity. For efficiently analyzing the failure probability-based parameter global sensitivity index, this paper proposes an efficient Kriging model-based importance sampling method. First, the standard normal auxiliary variables are inducted by using the equivalent probability transformation, in which the equivalent limit state function with respect to the uncertain distribution parameters and the standard normal auxiliary variables are constructed, so that the importance sampling samples of uncertain distribution parameters and the standard normal auxiliary variables can be generated based on the equivalent limit state function. Secondly, based on the importance sampling technique, the formula of single-loop method for estimating the failure probability-based parameter global sensitivity index is derived in which the importance sampling samples can be reused to obtain all uncertain distribution parameters' failure probability-based global sensitivity indices. Thirdly, the equivalent probability transformation is used again to transform the importance sampling samples in the high-dimensional space into the relatively low-dimensional input space, and the Kriging model of the original limit state function with relatively low-dimensional input space is adaptively constructed to recognize the states (failure or safety) of all high-dimensional importance sampling samples. Then, the failure probability-based parameter global sensitivity indices can be efficiently estimated. The results of three case studies verify the accuracy and efficiency of the proposed method.

Construction of adaptive Kriging metamodel for failure probability estimation considering the uncertainties of distribution parameters

A critical problem in engineering reliability analysis is obtaining an accurate failure probability with a high computational efficiency. This study aims to present failure probability estimation under the conditions of uncertain input variables and their uncertain distribution parameters. An adaptive Kriging model of failure probability with respect to distribution parameters (FP-DP model) is developed, which avoids coupling modeling among the distribution parameters, input variables, and failure probability. An improved U-learning function that simultaneously considers the statistical information of uncertain distribution parameters and failure probability is proposed to select new sampling points for the FP-DP model. The stopping criteria based on sample distances and relative errors of the predicted failure probability are constructed to improve the convergence performance around the limit state function. Three numerical and four engineering examples with different complexities are considered to verify the effectiveness of the proposed adaptive FP-DP Kriging metamodel. The results show that the proposed method can obtain an accurate failure probability with fewer sampling points of uncertain distribution parameters than some existing methods, indicating that the proposed method can be efficiently integrated into reliability-based design optimization problems considering both the uncertainties of input variables and their distribution parameters.

Distributionally robust optimisation model for multi-objective hub location problem via considering ambiguousness

We formulate a new multi-objective hub location problem by considering economic, customer satisfaction and environmental objectives. The economic objective includes loss cost in addition to transportation cost and hub construction cost. The environmental objective covers noise pollution cost and carbon emissions trading cost. The customer satisfaction objective is innovatively defined as the sum of the transportation time satisfaction and transportation quality satisfaction. Transportation costs, noise levels and carbon emissions are assumed as uncertain parameters. In practice, the probability distributions of uncertain parameters are often ambiguous. To characterise this ambiguity, we first construct an ambiguity set to propose a distributionally robust multi-objective hub location model and derive its safe approximation under mean and dispersion ambiguity set. Then, from the perspective of the government, a goal programming model is established. Finally, we apply the proposed model to design China's super logistics hub network to verify the model's validity and better performance.

Distributionally robust optimization with multivariate second-order stochastic dominance constraints with applications in portfolio optimization

In this paper, we study the stochastic optimization problem with multivariate second-order stochastic dominance (MSSD) constraints where the distribution of uncertain parameters is unknown. Instead, only some historical data are available. Using the Wasserstein metric, we construct an ambiguity set and develop a data-driven distributionally robust optimization model with multivariate second-order stochastic dominance constraints (DROMSSD). By utilizing the linear scalarization function, we transform MSSD constraints into univariate constraints. We present a stability analysis focusing on the impact of the variation of the ambiguity set on the optimal value and optimal solutions. Moreover, we carry out quantitative stability analysis for the DROMSSD problems as the sample size increases. Specially, in the context of the portfolio, we propose a convex lower reformulation of the corresponding DROMSSD models under some mild conditions. Finally, some preliminary numerical test results are reported. We compare the DROMSSD model with the sample average approximation model through out-of-sample performance, certificate and reliability. We also use real stock data to verify the effectiveness of the DROSSM model.

Two-stage robust power cost minimization in a natural gas compressor station

Optimal operation of a compressor station is important since it accounts for 25 % to 50 % of a company's total operating budget. In short-term management of a compressor station, handling demand uncertainty is important yet challenging. Previous studies either require precise information about the distribution of uncertain parameters or greatly simplify the compressor model. We build a two-stage robust optimization framework of power cost minimization in a natural gas compressor station with non-identical compressors. In the first stage, decision variables are the ON/OFF state of each compressor and discharge pressure. The worst-case cost of the second stage is incorporated in the first stage. First-stage decision variables feasibility is discussed and proper feasibility cuts are also proposed for the first stage. We employ a piece-wise approximation and propose accelerate methods. Our numerical results highlight two advantages of robust approach when managing uncertainty in practical settings: (1) the feasibility of first-stage decision can be increased by up to 45 % , and (2) the worst-case cost can be reduced by up to 25 % compared with stochastic programming models. Furthermore, our numerical experiments show that the designed accelerate algorithm has time improvements of 1518.9% on average (3785.9% at maximum).

POSA315 A Computationally Efficient Alternative Method for Probabilistic One-Way Sensitivity Analysis

McCabe (2020) proposed probabilistic one-way sensitivity analysis (POSA) to replace traditional one-way sensitivity analysis (OWSA). The POSA approach is also considered for the NICE guidance update as it incorporates the whole distribution of uncertain parameters, not only , and better reflects the uncertainty of cost-effectiveness outcomes than deterministic sensitivity analysis. POSA is implemented by fixing each uncertain parameter to selected quantiles one-by-one, and running a traditional probabilistic sensitivity analysis (PSA) on all other parameters.

Scenario-robust pre-disaster planning for multiple relief items.

The increasing vulnerability of the population from frequent disasters requires quick and effective responses to provide the required relief through effective humanitarian supply chain distribution networks. We develop scenario-robust optimization models for stocking multiple disaster relief items at strategic facility locations for disaster response. Our models improve the robustness of solutions by easing the difficult, and usually impossible, task of providing exact probability distributions for uncertain parameters in a stochastic programming model. Our models allow decision makers to specify uncertainty parameters (i.e., point and probability estimates) based on their degrees of knowledge, using distribution-free uncertainty sets in the form of ranges. The applicability of our generalized approach is illustrated via a case study of hurricane preparedness in the Southeastern United States. In addition, we conduct simulation studies to show the effectiveness of our approach when conditions deviate from the model assumptions.

Robustness of building energy optimization with uncertainties using deterministic and stochastic methods: Analysis of two forms

Building performance optimization is effective in searching for optimal design solutions, but its result may be unrobust with uncertainties in input parameters, especially for form design. Stochastic optimization is a promising approach to handle this problem, but its effectiveness in improving robustness is unclear. In this study, stochastic optimization is compared to the traditional deterministic method under holistic uncertainties in 13 input parameters. Energy optimizations of two office building forms, a shoe-box and an irregular quadrilateral, were inspected. The results indicate that robustness problems exist in both deterministic and stochastic optimization with uncertainties, but utilizing optimization results is still a good strategy, as the average chance of the optimized form outperforming random designs exceeds 0.7 for all cases involved. Using stochastic optimization helps obtain more robust results, but the expected increase in the average chance is only 3.2 % and 1.2 % for the two forms, and thus its effect should not be overstressed. We suggest that it be used only in major decision-making events considering its hefty demands in time and computation power. Reducing the uncertainty in input parameters is a more effective strategy. By reducing the variance in input parameters by 44 % compared with the “blind guess” scenario, the average chance is expected to increase by 7.2 % and 1.4 %, respectively. Making the assumed distribution closer to the real distribution of uncertain parameters is also helpful, with expected increases of 2.2 % and 0.7 %, respectively, if the variance difference between the two distributions is reduced by 44 % relative to a “blind guess.”

State-Variable Modeling for a Class of Two-Stage Stochastic Optimization Problems

This paper considers a class of two-stage stochastic mixed-integer optimization problems where, for a given first-stage solution, we can determine the optimal values of recourse variables sequentially. This class of problems arises in a wide variety of applications. In the case of multivariate discrete distributions for uncertain parameters, a standard stochastic programming formulation of these problems involves an exponential number of scenarios, therefore an exponential number of variables and constraints. We propose a new mixed-integer programming modeling approach where the number of variables and constraints is independent of the number of scenarios and scales at most pseudopolynomially with the problem size. The proposed modeling approach relies on state variables that track the system’s state as the uncertainty realizes sequentially. We demonstrate the advantages of the proposed approach in two applications arising in project scheduling and operating room allocation. Summary of Contribution: This paper proposes a new modeling approach for a class of two-stage stochastic optimization problems that is computationally more efficient than the traditional scenario-based stochastic integer programming models. The proposed modeling approach relies on state variables that track the system's state as the uncertainty realizes sequentially. We demonstrated the efficiency of the proposed approach by computational results on two applications in project scheduling and operating room allocation.

Wasserstein distributionally robust chance-constrained optimization for energy and reserve dispatch: An exact and physically-bounded formulation

In the context of transition towards sustainable, cost-efficient and reliable energy systems, the improvement of current energy and reserve dispatch models is crucial to properly cope with the uncertainty of weather-dependent renewable power generation. In contrast to traditional approaches, distributionally robust optimization offers a risk-aware framework that provides performance guarantees when the distribution of uncertain parameters is not perfectly known. In this paper, we develop a distributionally robust chance-constrained optimization with a Wasserstein ambiguity set for energy and reserve dispatch, and provide an exact reformulation. While preserving the exactness, we then improve the model by enforcing physical bounds on the uncertainty space, resulting in a bilinear program. We solve the resulting bilinear model with an iterative algorithm which is computationally efficient and has convergence guarantee. A thorough out-of-sample analysis is performed to compare the proposed model against a scenario-based stochastic program. We also compare the performance of the proposed exact reformulation against an existing approximate technique in the literature, built upon a conditional-value-at-risk measure. We eventually show that the proposed physically-bounded exact reformulation outperforms the other methods by achieving a cost-optimal yet reliable trade-off between reserve procurement and load curtailment.