A new enhancement of the Clarke and Wright savings heuristic for the capacitated vehicle routing problem

In this work we are concerned with the Clarke–Wright savings method for the classical capacitated vehicle routing problem. This is an NP-hard problem and numerous heuristic solution methods have been proposed. They can be classified as the classical ones and metaheuristics. Recent developments have shown that classical heuristics do not compare with the best metaheuristic implementations. However, some of them are very fast and simple to implement. This explains the popularity of the Clarke–Wright savings method in practice and the motivation behind its enhancements. We follow this line of research and propose a new enhancement which differs from the previous ones in its saving criterion: Customer demands are considered in addition to distances. Based on the extensive computational experiments we can say that the new method is not only very fast but also very accurate.     

Fine-tuning a parametric Clarke and Wright heuristic by means of EAGH (empirically adjusted greedy heuristics)

Alt?nel and Öncan (2005) (A new enhancement of the Clarke and Wrightsavings heuristic for the capacitated vehicle routing problem) proposed aparametric Clarke and Wright heuristic to solve the capacitated vehicle routingproblem (CVRP). The performance of this parametric heuristic is sensitive tofine-tuning. Antinel and Öncan used an enumerative parameter-settingapproach and improved on the results obtained with the original Clarke andWright heuristic, but their approach requires much more computation time tosolve an instance. Battarra et al (2008) (Tuning a parametricClarke–Wright heuristic through a genetic algorithm) proposed a geneticalgorithm to set the parameter values. They succeeded in reducing the timeneeded to solve an instance, but the quality of the solution was slightly worse.In this paper, we propose to use the EAGH (empirically adjusted greedyheuristics) procedure to set the parameter values. A computational experimentshows the efficiency of EAGH; in an even shorter time, we improve on the bestresults obtained with any parametric Clarke and Wright heuristic method proposedin the literature.     

Ant colony method to control variance reduction techniques in the Monte Carlo simulation of clinical electron linear accelerators of use in cancer therapy

The Monte Carlo simulation of clinical electron linear accelerators requires large computation times to achieve the level of uncertainty required for radiotherapy. In this context, variance reduction techniques play a fundamental role in the reduction of this computational time. Here we describe the use of the ant colony method to control the application of two variance reduction techniques: Splitting and Russian roulette. The approach can be applied to any accelerator in a straightforward way and permits the increasing of the efficiency of the simulation by a factor larger than 50.     

Imprecise random variables,random sets,and Monte Carlo simulation

The paper addresses the evaluation of upper and lower probabilities induced by functions of an imprecise random variable. Given a function g and a family $X_{\lambda }$ of random variables, where the parameter λ ranges in an index set Λ, one may ask for the upper/lower probability that $g(X_{\lambda })$ belongs to some Borel set B. Two interpretations are investigated. In the first case, the upper probability is computed as the supremum of the probabilities that $g(X_{\lambda })$ lies in B. In the second case, one considers the random set generated by all $g(X_{\lambda })$, $\lambda \in \mathrm{\Lambda }$, e.g. by transforming $X_{\lambda }$ to standard normal as a common probability space, and computes the corresponding upper probability. The two results are different, in general. We analyze this situation and highlight the implications for Monte Carlo simulation. Attention is given to efficient simulation procedures and an engineering application is presented.     

On the utility of graphics cards to perform massively parallel simulation of advanced Monte Carlo methods

We present a case-study on the utility of graphics cards to perform massively parallel simulation of advanced Monte Carlo methods. Graphics cards, containing multiple Graphics Processing Units (GPUs), are self-contained parallel computational devices that can be housed in conventional desktop and laptop computers and can be thought of as prototypes of the next generation of many-core processors. For certain classes of population-based Monte Carlo algorithms they offer massively parallel simulation, with the added advantage over conventional distributed multi-core processors that they are cheap, easily accessible, easy to maintain, easy to code, dedicated local devices with low power consumption. On a canonical set of stochastic simulation examples including population-based Markov chain Monte Carlo methods and Sequential Monte Carlo methods, we nd speedups from 35 to 500 fold over conventional single-threaded computer code. Our findings suggest that GPUs have the potential to facilitate the growth of statistical modelling into complex data rich domains through the availability of cheap and accessible many-core computation. We believe the speedup we observe should motivate wider use of parallelizable simulation methods and greater methodological attention to their design.     

On the Monte Carlo simulation of BSDEs: An improvement on the Malliavin weights

We propose a generic framework for the analysis of Monte Carlo simulation schemes of backward SDEs. The general results are used to re-visit the convergence of the algorithm suggested by Bouchard and Touzi (2004) [6]. By keeping the higher order terms in the expansion of the Skorohod integrals resulting from the Malliavin integration by parts in [6], we introduce a variant of the latter algorithm which allows for a significant reduction of the numerical complexity. We prove the convergence of this improved Malliavin-based algorithm, and derive a bound on the induced error. In particular, we show that the price to pay for our simplification is to use a more accurate localizing function.     

Improved estimators for the GMANOVA problem with application to Monte Carlo simulation

The problem of finding classes of estimators which improve upon the usual (e.g., ML, LS) estimator of the parameter matrix in the GMANOVA model under (matrix) quadratic loss is considered. Classes of improved estimators are obtained via combining integration-by-parts methods for normal and Wishart distributions. Also considered is the application of control variates to achieve better efficiency in multipopulation multivariate simulation studies.     

On iterative and Monte Carlo solutions of the Boltzmann equation

The two most commonly used techniques for solving the Boltzmann equation, with given boundary conditions, are first iterative equations (typically the BGK equation) and Monte Carlo methods. The present work examines the accuracy of two different iterative solutions compared with that of an advanced Monte Carlo solution for a one-dimensional shock wave in a hard sphere gas. It is found that by comparison with the Monte Carlo solution the BGK model is not as satisfactory as the other first iterative solution (Holway's) and that the BGK solution may be improved by using directional temperatures rather than a mean temperature.     

Estimation of the statistical error of the direct simulation Monte Carlo method

The statistical error of the direct simulation Monte Carlo method for numerical solution of the rarefied gas dynamics problems is investigated. Based on the central limit theorem for Markov processes, asymptotic confidence intervals for the errors connected with the number of time steps are obtained for estimates of the three main macroparameters of the flow (density, velocity, and temperature). For the quantities involved in the expressions for the confidence intervals, practical recommendations are given concerning their numerical evaluation simultaneously with the calculation of the flow macroparameters. The proposed approaches to constructing the confidence intervals are illustrated using the classical problem of heat transfer between two infinite parallel plates as an example.     

A Monte Carlo simulation approach to the characterization of uncertainties in cancer staging and radiation treatment decisions

Radiation treatment (RT) for cancer is a critical medical procedure that occurs in a complex environment that is subject to uncertainties and errors. We employed a simulation (a variant of Monte Carlo) model that followed a cohort of hypothetical breast cancer patients to estimate the probability of incorrect staging and treatment decisions. As inputs, we used a combination of literature information and expert judgement. Input variables were defined as probability distributions within the model. Uncertainties were propagated via simulation. Sensitivity and value-of-information analyses were then conducted to quantify the effect of variable uncertainty on the model outputs. We found a small but non-trivial probability that patients would be incorrectly staged and thus be subjected to inappropriate treatment. Some routinely used tests for staging and metastasis detection have very limited informational value. This work has implications for the methods used in cancer staging and subsequent risk assessment of treatment errors.     

Monte Carlo simulation and analytic approximation of epidemic processes on large networks

Low dimensional ODE approximations that capture the main characteristics of SIS-type epidemic propagation along a cycle graph are derived. Three different methods are shown that can accurately predict the expected number of infected nodes in the graph. The first method is based on the derivation of a master equation for the number of infected nodes. This uses the average number of SI edges for a given number of the infected nodes. The second approach is based on the observation that the epidemic spreads along the cycle graph as a front. We introduce a continuous time Markov chain describing the evolution of the front. The third method we apply is the subsystem approximation using the edges as subsystems. Finally, we compare the steady state value of the number of infected nodes obtained in different ways.     

On the optimization of approximate integration by Monte Carlo methods

We solve the problem of optimization of monte Carlo methods for approximate integration over an arbitrary absolutely continuous measure. We propose a convenient model of Monte Carlo methods which uses the notion of transition probability.     

On the optimal design of the randomized unbiased Monte Carlo estimators

We consider a class of unbiased Monte Carlo estimators and develop an efficient algorithm to produce the distribution of the optimal random truncation level. We establish the convergence and optimality results of the associated algorithm and also derive its exact complexity. We find this algorithm has a much lower complexity as compared to the existing one in the literature. The proposed algorithm is also applicable to optimization problems arising in supply chain management, such as economic reorder interval problem.     

On the Utility of Graphics Cards to Perform Massively Parallel Simulation of Advanced Monte Carlo Methods

We present a case study on the utility of graphics cards to perform massively parallel simulation of advanced Monte Carlo methods. Graphics cards, containing multiple Graphics Processing Units (GPUs), are self-contained parallel computational devices that can be housed in conventional desktop and laptop computers and can be thought of as prototypes of the next generation of many-core processors. For certain classes of population-based Monte Carlo algorithms they offer massively parallel simulation, with the added advantage over conventional distributed multicore processors that they are cheap, easily accessible, easy to maintain, easy to code, dedicated local devices with low power consumption. On a canonical set of stochastic simulation examples including population-based Markov chain Monte Carlo methods and Sequential Monte Carlo methods, we find speedups from 35- to 500-fold over conventional single-threaded computer code. Our findings suggest that GPUs have the potential to facilitate the growth of statistical modeling into complex data-rich domains through the availability of cheap and accessible many-core computation. We believe the speedup we observe should motivate wider use of parallelizable simulation methods and greater methodological attention to their design. This article has supplementary material online.     

Estimating the statistical error in calculating velocity and temperature components by the direct simulation Monte Carlo method

The direct simulation Monte Carlo (DSMC) method is widely used to solve problems in rarefied gas dynamics. In solving steady state problems, a special feature of the method is in using dependent sample values of random variables to calculate the macroparameters of gas flow. In this paper, the possibilities of methods of statistical physics to estimate the statistical error of the DSMC method are theoretically analyzed. A simple approach to approximate estimation of the statistical error in calculating temperature and velocity components is proposed. The approach is tested on a number of problems.