Probability Distribution Of Random Parameters Research Articles

Computation of nonparametric, mixed effects, maximum likelihood, biosensor data based-estimators for the distributions of random parameters in an abstract parabolic model for the transdermal transport of alcohol.

The existence and consistency of a maximum likelihood estimator for the joint probability distribution of random parameters in discrete-time abstract parabolic systems was established by taking a nonparametric approach in the context of a mixed effects statistical model using a Prohorov metric framework on a set of feasible measures. A theoretical convergence result for a finite dimensional approximation scheme for computing the maximum likelihood estimator was also established and the efficacy of the approach was demonstrated by applying the scheme to the transdermal transport of alcohol modeled by a random parabolic partial differential equation (PDE). Numerical studies included show that the maximum likelihood estimator is statistically consistent, demonstrated by the convergence of the estimated distribution to the "true" distribution in an example involving simulated data. The algorithm developed was then applied to two datasets collected using two different transdermal alcohol biosensors. Using the leave-one-out cross-validation (LOOCV) method, we found an estimate for the distribution of the random parameters based on a training set. The input from a test drinking episode was then used to quantify the uncertainty propagated from the random parameters to the output of the model in the form of a 95 error band surrounding the estimated output signal.

An optimization approach for sustainable and resilient supply chain design with regional considerations

Regional and geographical differences between facilities is of paramount importance in supply chain design. However, the impact of the regions' performance on supply chain design decisions remains a relatively under-researched subject in the literature. This paper presents a hybrid methodology for designing a sustainable supply chain that is resilient to random disruptions. We propose a multi-period multi-objective optimization model that utilizes a k-means clustering method to evaluate the regions' sustainability performance. The model aims to determine sourcing and network design decisions as well as resilience strategies. To manage the operational risks associated with the supply chain, we employ a new robustness measure that eliminates the need to estimate the probability distribution of random parameters. Finally, a Benders decomposition algorithm is developed to solve the model. Practical insights are drawn from an actual case study of a downstream petrochemical industry in Iran.

A distributionally robust optimization approach for stochastic elective surgery scheduling with limited intensive care unit capacity

In this paper, we study the decision process of assigning elective surgery patients to available surgical blocks in multiple operating rooms (OR) under random surgery durations, random postoperative length-of-stay in the intensive care unit (ICU), and limited capacity of ICU. The probability distributions of random parameters are assumed to be ambiguous, and only the mean values and ranges are known. We propose a distributionally robust elective surgery scheduling (DRESS) model that seeks optimal surgery scheduling decisions to minimize the cost of performing and postponing surgeries and the worst-case expected costs associated with overtime and idle time of ORs and lack of ICU capacity (which causes premature discharges or transfers). We evaluate the worst-case over a family of distributions characterized by the known mean values and ranges of random parameters. We leverage the separability of DRESS formulation in deriving an exact mixed-integer nonlinear programming reformulation. We linearize and derive a family of symmetry breaking inequalities to improve the solvability of the reformulation using an adapted column-and-constraint generation algorithm. Finally, we conduct extensive numerical experiments that demonstrate the superior performance of our DR approach as compared to the stochastic programming approach, and provide insights into DRESS.

Статистическое моделирование ядра вероятностного распределения и его применение к решению задачи квантильной оптимизации с билинейной функцией потерь

The article considers a plane quantile optimization problem with a bilinear loss function, which, using suffi cient optimality conditions, is reduced to a linear programming problem. The reduction is based on the use of a polyhedral model of the kernel of the probability distribution of the vector of random parameters. To build this model, an algorithm based on the method of statistical modeling is proposed. A description of the software package for constructing a kernel model for a number of probability distributions of random parameters is given.

Data-driven probabilistic small signal stability analysis for grid-connected PV systems

Probabilistic small signal stability analysis is useful to quantify the effect of uncertain parameters, such as the solar and wind generations, on system stability margin. In the literature, a semi-analytic method based on general polynomial chaos expansion (gPCE) has been proposed for probabilistic small signal stability analysis. The gPCE method can achieve high order accuracy with lower computation burden, but it needs a predefined probability distribution of random parameters, which may not be available in practical engineering applications. To reduce such dependency on the full information of probability distribution, this paper presents a data-driven polynomial chaos expansion (DD-PCE) method for assessing the impact of uncertain solar irradiance on the stability margin of a grid-connected photovoltaic (PV) system. In the proposed method, the system stability margin is characterized in terms of its critical mode damping ratio, and the functional relationship between the critical mode damping ratio and the uncertain solar irradiance is estimated by orthonormal polynomial expansion. The orthonormal polynomial basis is constructed using the moments of solar irradiance, and the coefficients of the orthonormal polynomial expansion are calculated by the collocation points of polynomial basis. The efficiency of the proposed method is validated by comparisons with gPCE method and Monte Carlo method.

Bayesian generalized structured component analysis.

Generalized structured component analysis (GSCA) is a component-based approach to structural equationmodelling, which adopts components of observed variables as proxies for latent variables and examines directional relationships among latent and observed variables. GSCA has been extended to deal with a wider range of data types, including discrete, multilevel or intensive longitudinal data, as well as to accommodate a greater variety of complex analyses such as latent moderation analysis, the capturing of cluster-level heterogeneity, and regularized analysis. To date, however, there has been no attempt to generalize the scope of GSCA into the Bayesian framework. In this paper, a novel extension of GSCA, called BGSCA, is proposed that estimates parameters within the Bayesian framework. BGSCA can be more attractive than the original GSCA for various reasons. For example, it can infer the probability distributions of random parameters, account for error variances in the measurement model, provide additional fit measures for model assessment and comparison from the Bayesian perspectives, and incorporate external information on parameters, which may be obtainable from past research, expert opinions, subjective beliefs or knowledge on the parameters. We utilize a Markov chain Monte Carlo method, the Gibbs sampler, to update the posterior distributions for the parameters of BGSCA. We conduct a simulation study to evaluate the performance of BGSCA. We also apply BGSCA to real data to demonstrate its empirical usefulness.

Data-driven chance constrained stochastic program

In this paper, we study data-driven chance constrained stochastic programs, or more specifically, stochastic programs with distributionally robust chance constraints (DCCs) in a data-driven setting to provide robust solutions for the classical chance constrained stochastic program facing ambiguous probability distributions of random parameters. We consider a family of density-based confidence sets based on a general $$\phi $$ź-divergence measure, and formulate DCC from the perspective of robust feasibility by allowing the ambiguous distribution to run adversely within its confidence set. We derive an equivalent reformulation for DCC and show that it is equivalent to a classical chance constraint with a perturbed risk level. We also show how to evaluate the perturbed risk level by using a bisection line search algorithm for general $$\phi $$ź-divergence measures. In several special cases, our results can be strengthened such that we can derive closed-form expressions for the perturbed risk levels. In addition, we show that the conservatism of DCC vanishes as the size of historical data goes to infinity. Furthermore, we analyze the relationship between the conservatism of DCC and the size of historical data, which can help indicate the value of data. Finally, we conduct extensive computational experiments to test the performance of the proposed DCC model and compare various $$\phi $$ź-divergence measures based on a capacitated lot-sizing problem with a quality-of-service requirement.

An enhanced robustness approach for managing supply and demand uncertainties

Managing supply and demand uncertainties is a topic that receives increasing management attention due to (1) more price-based competitions forcing firms purchase from cheaper but less-reliable or unproven suppliers, and (2) undesirable consequences of unaddressed demand fluctuations such as reduced service level, financial loss, reputational damage, and loss of market share. This paper presents a realistic production–distribution planning model that is robust to common supply interruptions and demand variations. A robustness approach, named “Elastic p-Robustness”, is introduced that obviates the need to estimate probability distribution of random parameters when managing operational uncertainties of the supply chain. The application of the proposed approach is investigated in an actual organization from discrete, durable parts manufacturing sector. Our analyses of numerical results focus on (1) exploring different tactics for managing supply and demand variations, (2) examining the benefits of concurrent consideration of supply and demand uncertainties, (3) benchmarking the performance of the proposed approach against the popular robustness algorithms, and (4) investigating the price of robustness under various supply and demand scenarios.

Reliability Analysis of Alumina-Based Ceramic Cutting Tool's Wear Life by Saddle Point Approximation

Under continuous cutting conditions, the reliability of alumina-based ceramic cutting tools wear life with abrasion wear was discussed extensively. An ultimate state equation of alumina-based ceramic cutting tools wear life was proposed according to stress-strength interference theory. Based on the premise that the probability distribution of random parameters has been known, the fracture toughness KIC and the hardness H which obey Weibull distribution were transformed into normal distribution by means of the equivalent normal distribution method, and then the reliability of alumina-based ceramic cutting tools wear life was analyzed by saddle point approximation method. The probability density function and cumulative distribution function of the ultimate state equation were accurately and quickly obtained by way of saddle point approximation, as a result, the saddle point approximation method was proved accurate with high computing speed in comparison with the Monte-Carlo method. Therefore, the application of the saddle point approximation method developed the reliability analysis theory of alumina-based ceramic cutting tools wear life.

Comparative analysis of different probability distributions of random parameters in the assessment of water distribution system reliability

During recent years, several methods based on the probabilistic approach have been proposed for the analysis of the performance of water distribution systems (WDSs). Uncertain elements are described by probabilistic laws chosen and parameterised on the basis of the network characteristics. However, the choice of the most suitable probabilistic distribution and of the statistical parameters can be difficult because of the lack of information about the WDSs. Among the stochastic parameters that affect the network performance, a fundamental role is played by the times to failure and repair of the system components. The impact of the chosen probability distributions of these fundamental variables on the evaluation of water distribution network reliability is analysed. The study is performed by using a technique capable of considering the mechanical failure of the network components, the spatial and temporal variations of the water demand and the uncertain distribution of the pipe roughness. This analysis allows quantification of the effect of any inaccuracy that may occur in the probabilistic characterisation of the random parameters.

Reliability-Based Sensitivity Analysis for Machining Precision by Saddle-Point Approximation

Based on the saddle-point approximation theory, the reliability sensitivity formula is derived systematically. Then, combining the reliability theory with the sensitivity analysis method, the reliability-based sensitivity of machining precision for ball-end milling cutter is analyzed. Based on the premise that the probability distribution of random parameters is known, firstly, the saddle-point approximation theory is employed to solve the reliability of machining precision for ball-end milling cutter and, as a result, the saddle-point approximation method is proved accurate with higher speed in comparison with the Monte-Carlo method. Secondly, the reliability-based sensitivity is calculated. Finally, the effect of changes in machining parameters on the reliability of machining precision is analyzed.

Reliability Sensitivity Analysis for Differential Expansion of Steam Turbine by Saddlepoint Approximation

Based on Saddlepoint Approximation method and sensitivity analysis method, reliability sensitivity analysis for differential expansion of steam turbine with random parameters are studied. On the premise of the probability distribution of random parameters, using Saddlepoint Approximation method, probability density function of limit state function of differential expansion of steam turbine is obtained. The result of Saddlepoint Approximation method is very close to the one of Monte-Carlo, and the computing speed is fast. Then, the sensitivity analysis method and probability density function were employed to discuss the variation regularities of reliability sensitivity and the effect of design parameters on reliability of differential expansion of steam turbine is analyzed.

Multi-objective stochastic integer linear programming with fixed recourse

The real life decision problems have three main properties. The first one is to have conflicting objectives in the problem structure, the second one is the stochasticity in the description of problem parameters in contexts where the probability distribution of random parameters is known and the last one is due to involvement of integer decision variables which increased dimension of the problem. Multi-objective nature with discrete variables and imprecise parameters make the mathematical expression of the problem harder to solve with the traditional approaches. An efficient algorithm is developed, via extending the well-known L-shaped method using generalised benders decomposition to efficiently handle the integer variables in the first stage and the integer recourse in the second stage of the model formulation. The proposed solution method able to identify all efficient integer feasible solutions converge in a finite number of iterations. A numerical example and a computational implementation are also included for illustration.

Forecast on Payback Period of Restaurant Projects Investment by Computer Simulation

Investment projects of the civil engineering have many characteristics,such as large scale of construction,tremendous investment cost,long payback period of investment, and bad risk. For investors, the investment payback period of construction projects is an important index that evaluate and measure economic effect of project investment. The investment capital can be recovered in the operation of actual projects by the influence of random factors, it is difficult that the investment payback period of construction projects is calculated generally using analytic method. Combined with restaurant project in this paper, set up the mathematical model of the payback period of investment, find out random factors that influence its cash flow of restaurant project, to research and measure the data of random factors on existing similar projects, to determine the probability distribution of random parameters,and Statistical data processing, make use of the computer simulation, set up the simulation model of the payback period of investment, to forecast the static and dynamic payback period of construction projects,provided decision-making with scientific base and meaningful reference.

Reliability Analysis for Differential Expansion of Steam Turbine by Saddlepoint Approximation

The reliability analysis for differential expansion of steam turbine is discussed extensively. The ultimate state equation of differential expansion of steam turbine is proposed according to stress-strength interference theory. Based on the premise that the probability distribution of random parameters has been known, the differential expansion of steam turbine is analyzed by saddlepoint approximation method. The probability density function and cumulative distribution function of the ultimate state equation are accurately and quickly obtained by the way of saddlepoint approximation and, as a result, the saddlepoint approximation method is proved accurate with high computing speed in comparison with the Monte-Carlo method. Therefore, the application of the saddlepoint approximation method is accurate and efficient.

Applying the Minimum Risk Criterion in Stochastic Recourse Programs

In the setting of stochastic recourse programs, we consider the problem of minimizing the probability of total costs exceeding a certain threshold value. The problem is referred to as the minimum risk problem and is posed in order to obtain a more adequate description of risk aversion than that of the accustomed expected value problem. We establish continuity properties of the recourse function as a function of the first-stage decision, as well as of the underlying probability distribution of random parameters. This leads to stability results for the optimal solution of the minimum risk problem when the underlying probability distribution is subjected to perturbations. Furthermore, an algorithm for the minimum risk problem is elaborated and we present results of some preliminary computational experiments.

Distributional efficiency in multiobjective stochastic linear programming

Several concepts of distributional efficiency are proposed for the Multiobjective Stochastic Linear Programming (MSLP) problem, in contexts where the probability distribution of random parameters is known and the decision maker (DM) has an unknown multi-attribute utility function belonging to a given glass U . We present a general efficient set, the U -admissible solutions, and two subsets, the U -unanimous and U -advocated solutions, the latter being particularly relevant to the case of a single DM. We show how advocated solutions can be generated and/or tested when U is the class of non-decreasing additive concave functions.