Distribution Of Uncertain Parameters Research Articles

An accelerated Benders decomposition method for distributionally robust sustainable medical waste location and transportation problem

This study addresses the sustainable medical waste location and transportation (SMWLT) problem from the viewpoint of social risk, environmental impact, and economic performance, where model uncertainty includes risk and transportation costs. In practice, it is usually hard to obtain the exact probability distribution of uncertain parameters. To address this challenge, this study first constructs an ambiguity set to model the partial distribution information of uncertain parameters. Based on the constructed ambiguity set, this study develops a new multi-objective distributionally robust chance-constrained (DRCC) model for the SMWLT problem. Subsequently, this study adopts the robust counterpart (RC) approximation method to reformulate the proposed DRCC model as a computationally tractable mixed-integer linear programming (MILP) model. Furthermore, an accelerated Benders decomposition (BD) enhanced by valid inequalities is designed to solve the resulting MILP model, which significantly improves the solution efficiency compared with the classical BD algorithm and CPLEX solver. Finally, a practical case in Chongqing, China, is addressed to illustrate the effectiveness of our DRCC model and the accelerated BD solution method.

A holistic sustainability framework for remanufacturing under uncertainty

The manufacturing and remanufacturing sectors are increasingly embracing sustainability as a critical aspect of their operations. However, existing sustainability frameworks often fall short of capturing the multifaceted nature of sustainability and addressing uncertainties. To address these limitations, this paper proposes a novel holistic sustainability assessment framework specifically tailored for remanufacturing systems. By integrating economic, environmental, and social dimensions, the framework provides a comprehensive approach to decision-making under uncertainty. The framework incorporates a flexible weighting scheme, allowing customization based on organizational priorities, and addresses uncertainties through stochastic optimization techniques. The applicability and effectiveness of the framework are demonstrated through case studies in diverse industries, including consumer electronics, automotive, and industrial machinery remanufacturing. Sensitivity analyses provide insights into the robustness of the framework and the impact of varying sustainability indicator weights, uncertain parameter distributions, and environmental regulations. The proposed framework offers a valuable tool for remanufacturing companies, enhancing their sustainability performance and navigating the complexities of uncertain operating environments.

Stochastic First-Order Algorithms for Constrained Distributionally Robust Optimization

We consider distributionally robust optimization (DRO) problems, reformulated as distributionally robust feasibility (DRF) problems, with multiple expectation constraints. We propose a generic stochastic first-order meta-algorithm, where the decision variables and uncertain distribution parameters are each updated separately by applying stochastic first-order methods. We then specialize our results to the case of using two specific versions of stochastic mirror descent (SMD): (i) a novel approximate version of SMD to update the decision variables, and (ii) the bandit mirror descent method to update the distribution parameters in the case of [Formula: see text]-divergence sets. For this specialization, we demonstrate that the total number of iterations is independent of the dimensions of the decision variables and distribution parameters. Moreover, the cost per iteration to update both sets of variables is nearly independent of the dimension of the distribution parameters, allowing for high-dimensional ambiguity sets. Furthermore, we show that the total number of iterations of our algorithm has a logarithmic dependence on the number of constraints. Experiments on logistic regression with fairness constraints, personalized parameter selection in a social network, and the multi-item newsvendor problem verify the theoretical results and show the usefulness of the algorithm, in particular when the dimension of the distribution parameters is large. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms—Continuous. Funding: This work was supported by the National Science Foundation [Grant 2112533]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.0167 .

Improving crop yield estimation by unified model parameters and state variable with Bayesian inference

Data assimilation techniques integrating crop growth models and remote sensing technologies offer a feasible approach for large-scale crop yield estimation. Previous research has primarily focused on either recalibrate the uncertain model parameters or updating model state variables independently using remotely sensed observations. In this study, we developed a two-step inference algorithm that couples the parameter inference and the state update, to solve the joint posterior distribution of uncertain parameters and state variables given the remote sensing observations. An Observing Simulation System (OSS) experiment was first performed based on the WOFOST crop model to validate the effectiveness of the parameter inference method. The results indicate that, the parameter inference method successfully improved the estimation of different types of model parameters and enhanced yield estimation. Furthermore, leaf area index (LAI) retrieved from Sentinel-2 was assimilated into the WOFOST model to simulate winter wheat yield at the plot scale in the northeastern part of Henan Province. The results demonstrated that the proposed two-step inference algorithm can more effectively correct model simulation biases and improve winter wheat yield estimation accuracy (R²=0.58, MAPE=12.75 %, and RMSE=1112 kg·ha⁻¹), outperforming the standard EnKF algorithm (R²=0.51, MAPE=14.62 %, RMSE=1328 kg·ha⁻¹). Overall, attributed to its unified approach to estimating both model parameters and state variables, the proposed two-step inference algorithm shows promising application prospects for data assimilation-based crop yield estimation.

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.