Number Of Simulation Replications Research Articles

Computation of tolerance ellipses for bivariate and trivariate normal populations

We consider the computation of the critical constants c of a p-content -confidence tolerance ellipse for bivariate and trivariate normal populations which are probably the most used multivariate normal distributions in applications. There are only approximation methods available in the statistical literature for computing c. Without knowing the accurate value of c, it is impossible to assess the accuracy of the approximation methods. In this paper, a new method is given to compute c. Although the method is also based on Monte-Carlo simulation, it allows c to be computed as accurate as required if the number of simulation replications is sufficiently large. The R codes provided allow easy implementation of the method. Furthermore, the method allows the accuracy of the available approximation methods for bivariate and trivariate normal distributions to be assessed.

A dynamic sequential decision-making model on MRI real-time scheduling with simulation-based optimization.

Magnetic resonance imaging (MRI) is widely used in diagnostic medicine and contributes significantly to US health care spending. Scheduling MRI jobs involves uncertainties (e.g., patient arrival time, scanning time, and preparation time) that can lead to excessive delays and high costs in MRI operations. This study addresses real-time decision making in use of MRI scanners based on job assignment and sequencing decisions that override the appointment schedule. The decisions are made using real-time information of the waiting patients, the utilization status of the MRI scanners, and the partially revealed uncertainties of scanning times of current patients. A sequential decision-making framework and a simulation-based solution method are proposed to utilize massive real-time information and match the use of MRI rescheduling in practice. The results are then compared with a real case in a large midwestern academic medical center in the US. This study illustrates that the proposed method reduces patient waiting time by 21.7% and improves utilization of MRI scanners by 23.0%. An optimality gap of 13.6% is provided when compared to off line scheduling methods based on a mixed integer programming (MIP) model. The number of simulation replications in this approach uses the ranking and selection method, which not only reduces solution time, but also provides solution quality guarantees wherein the probability of errors in the proposed method for one day is less than 0.1%. In 100 randomly generated workday experiments, all of the scheduling decisions given by the proposed method perform better than current policy, with an average reduction of 17.93 minutes in each patient's waiting time and an improvement of scanner utilization by 7.20%.

On a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling

We consider a multistage stochastic discrete program in which constraints on any stage might involve expectations that cannot be computed easily and are approximated by simulation. We study a sample average approximation (SAA) approach that uses nested sampling, in which at each stage, a number of scenarios are examined and a number of simulation replications are performed for each scenario to estimate the next-stage constraints. This approach provides an approximate solution to the multistage problem. To establish the consistency of the SAA approach, we first consider a two-stage problem and show that in the second-stage problem, given a scenario, the optimal values and solutions of the SAA converge to those of the true problem with probability one when the sample sizes go to infinity. These convergence results do not hold uniformly over all possible scenarios for the second stage problem. We are nevertheless able to prove that the optimal values and solutions of the SAA converge to the true ones with probability one when the sample sizes at both stages increase to infinity. We also prove exponential convergence of the probability of a large deviation for the optimal value of the SAA, the true value of an optimal solution of the SAA, and the probability that any optimal solution to the SAA is an optimal solution of the true problem. All of these results can be extended to a multistage setting and we explain how to do it. Our framework and SAA results cover a large variety of resource allocation problems for which at each stage after the first one, new information becomes available and the allocation can be readjusted, under constraints that involve expectations estimated by Monte Carlo. As an illustration, we apply this SAA method to a staffing problem in a call center, in which the goal is to optimize the numbers of agents of each type under some constraints on the quality of service (QoS). The staffing allocation has to be decided under an uncertain arrival rate with a prior distribution in the first stage, and can be adjusted at some additional cost when better information on the arrival rate becomes available in later stages.

Optimal Computing Budget Allocation for Stochastic N–$k$ Problem in the Power Grid System

The $N$ − $k$ problem is very well known in the power industry and it tries to answer the question whether there exists a set of $k$ lines in a power network with $N$ elements whose removal would cause the failure of the system. In practice, it is common to evaluate a system according to an $N$ −1 criterion, i.e., $k = 1$ . While this problem has traditionally been considered in a deterministic setting, stochastic behavior within the system is important especially in the context of extreme events. A number of stochastic Monte Carlo models have been proposed to estimate the probability of cascading failures. In this paper, we deal with simulation budget allocation of the stochastic $N$ –1 problem. More specifically, we assume that a simulation model is able to provide us an estimate of the system failure rate when any line is tripped. It is not difficult to see how simulation of all configurations to some certain accuracy can become computationally expensive with the growth of $N$ . Under such a setting, we transform the $N$ –1 problem into a stochastic selection process with optimal computing budget allocation (OCBA): given $N$ configurations, we would like to sequentially allocate a certain number of simulation replications in order to answer the question whether the system is reliable or not. We show through theoretical analysis and numerical experiments that the probability of correctly identifying the system reliability state can be increased by applying OCBA allocation rules in the simulation budget allocation process.

Optimal splitting for rare-event simulation

Simulation is a popular tool for analyzing large, complex, stochastic engineering systems. When estimating rare-event probabilities, efficiency is a big concern, since a huge number of simulation replications may be needed in order to obtain a reasonable estimate of the rare-event probability. The idea of splitting has emerged as a promising variance reduction technique. The basic idea is to create separate copies (splits) of the simulation whenever it gets close to the rare event. Some splitting methods use an equal number of splits at all levels. This can compromise the efficiency and can even increase the estimation variance. This article formulates the problem of determining the number of splits as an optimization problem that minimizes the variance of an estimator subject to a constraint on the total computing budget. An optimal solution for a certain class of problems is derived that is then extended to the problem of choosing the better of two designs, where each design is evaluated via rare-event simulation. Theoretical results for the improvements that are achievable using the methods are provided. Numerical experiments indicate that the proposed approaches are efficient and robust.

Assessing the Relationship between Computational Speed and Precision

Simulation techniques are well suited to modelling diseases yet can be computationally intensive. This study explores the relationship between modelled effect size, statistical precision, and efficiency gains achieved using variance reduction and an executable programming language. A published simulation model designed to model a population with type 2 diabetes mellitus based on the UKPDS 68 outcomes equations was coded in both Visual Basic for Applications (VBA) and C++. Efficiency gains due to the programming language were evaluated, as was the impact of antithetic variates to reduce variance, using predicted QALYs over a 40-year time horizon. The use of C++ provided a 75- and 90-fold reduction in simulation run time when using mean and sampled input values, respectively. For a series of 50 one-way sensitivity analyses, this would yield a total run time of 2 minutes when using C++, compared with 155 minutes for VBA when using mean input values. The use of antithetic variates typically resulted in a 53% reduction in the number of simulation replications and run time required. When drawing all input values to the model from distributions, the use of C++ and variance reduction resulted in a 246-fold improvement in computation time compared with VBA - for which the evaluation of 50 scenarios would correspondingly require 3.8 hours (C++) and approximately 14.5 days (VBA). The choice of programming language used in an economic model, as well as the methods for improving precision of model output can have profound effects on computation time. When constructing complex models, more computationally efficient approaches such as C++ and variance reduction should be considered; concerns regarding model transparency using compiled languages are best addressed via thorough documentation and model validation.

Hybrid algorithm for discrete event simulation based supply chain optimization

Supply chain optimization, as a key determinant of strategic resources mobility along the value-added chain, allows each participant in the global network to capitalize on its particular strategic competency. Simulation is widely used to test the impact on supply chain performance for the strategic level decisions, such as the number of plants, the modes of transport, or the relocation of warehouses. However, the complexity of supply chain optimization problem and the stochastic nature of simulation cause the unaffordable computational load; the evaluation of a large number of alternatives for supply chain optimization is in a class of NP-hard problem and the number of simulation replications is required for accurately evaluating the performance of each alternative. The objective of the present work is to propose hybrid algorithm with the application of the nested partitioning (NP) method and the optimal computing budget allocation (OCBA) method to reduce the computational load, hence, to improve the efficiency of supply chain optimization via discrete event simulation. The NP method is a global sampling strategy that is continuously adapted via a partitioning of the feasible solution region. The number of candidate alternatives to be evaluated can be reduced by the application of NP. The OCBA method minimizes the number of samples (simulation replications) required to evaluate a particular alternative by allocating computing resources to potentially critical alternative. Carefully designed experiments show extensive numerical result to illustrate the benefits of the proposed approach.

Attribute Acceptance Sampling as a Tool for Verifying Requirements Using Monte Carlo Simulation

ABSTRACT Developed by quality engineers as an efficient statistical approach to verifying the acceptability of procured items within production environments, acceptance sampling can be generalized to other verification problems which similarly rely on the outcomes of stochastic sampling experiments. This article illustrates how the techniques and perspectives of attribute acceptance sampling can be adapted to the verification of probabilistic design requirements using Monte Carlo simulation. We consider requirements expressed as limit standards, specifying the performance indicator for conforming Monte Carlo trials; the minimum limiting proportion of conforming trials; and the maximum risk of accepting a nonconforming design. For such requirements, an attribute sampling plan prescribes the number of simulation replications that must be run and the number of nonconforming replications that can not be exceeded. The derivation and analysis of single-sample attribute acceptance plans for the mass-delivery requirement for NASA's Constellation Program (CxP) is provided as an example. More demanding (but potentially more efficient) alternatives to attribute acceptance sampling are suggested for applications with limited budgets for simulation trials.

Simulation Allocation for Determining the Best Design in the Presence of Correlated Sampling

We consider the problem of efficiently allocating simulation replications in order to maximize the probability of selecting the best design under the scenario in which system performances are sampled in the presence of correlation. In the case of two designs, we are able to derive the optimal allocation exactly, and find that in the presence of positive correlation, unless the variance of one design is significantly larger than that of the other, the number of simulation replications should be identical. In extending to a general number of competing designs, an approximation for the asymptotically optimal allocation is obtained. The approximation coincides with the independent case derived previously in the limit as the correlation vanishes and also agrees with the two-design exact solution. Furthermore, the allocations prescribed by the results seem to match intuition, in terms of the relationship to correlations and relative variances between designs, again suggesting that equal allocation is optimal for sufficiently high positive correlation. An allocation algorithm based on the approximation is proposed and tested on several numerical examples.

Optimization of discrete event systems via simultaneous perturbation stochastic approximation

We investigate the use of simultaneous perturbation stochastic approximation for the optimization of discrete-event systems via simulation. Application of stochastic approximation to simulation optimization is basically a gradient-based method, so much recent research has focused on obtaining direct gradients. However, such procedures are still not as universally applicable as finite-difference methods. On the other hand, traditional finite-difference-based stochastic approximation schemes require a large number of simulation replications when the number of parameters of interest is large, whereas the simultaneous perturbation method is a finite-difference-like method that requires only two simulations per gradient estimate, regardless of the number of parameters of interest. This can result in substantial computational savings for large-dimensional systems. We report simulation experiments conducted on a variety of discrete-event systems: a single-server queue, a queueing network, and a bus transit network. For the single-server queue, we also compare our work with algorithms based on finite differences and perturbation analysis.