A vehicle routing problem with time windows and stochastic demands

Abstract

This study presents an approach for considering a vehicle routing problem where customers’ pickup demands are uncertain and require serving within some settled time windows. Customers’ demands are assumed to follow given discrete probability distributions. This study proposes a nonlinear stochastic integer program with recourse to formulate the vehicle routing problem with stochastic demands and time windows (VRPTW‐SD, for short). The objective of the VRPTW‐SD is to minimize the total cost of the first‐stage solution and expected recourse cost of the second‐stage solution. The total cost of the first‐stage problem includes the total travel cost for all links and the total waiting cost at all nodes. When a vehicle capacity is attained or exceeded, recourse actions are needed and recourse costs incurred in order to finish the planned route schedules. Two categories of schedule failure are introduced in this work; the recourse costs derive from the variations in travel time travel time, waiting time, and penalties of late arrival for time windows. In addition, an optimization algorithm is developed for solving the VRPTW‐SD, according to the framework of the L‐shaped method. Numerical results are given to demonstrate its validity.     

Prediction of apparent properties with uncertain material parameters using high‐order fictitious domain methods and PGD model reduction

This contribution presents a numerical strategy to evaluate the effective properties of image‐based microstructures in the case of random material properties. The method relies on three points: (1) a high‐order fictitious domain method; (2) an accurate spectral stochastic model; and (3) an efficient model‐reduction method based on the proper generalized decomposition in order to decrease the computational cost introduced by the stochastic model. A feedback procedure is proposed for an automatic estimation of the random effective properties with a given confidence. Numerical verifications highlight the convergence properties of the method for both deterministic and stochastic models. The method is finally applied to a real 3D bone microstructure where the empirical probability density function of the effective behaviour could be obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

New minimum error rate algorithms for DFE

Abstract

We propose new adaptive minimum symbol error rate algorithms (MSER) for decision feedback equalization over M‐ary PAM channels. In addition, we take into consideration biased as well as unbiased estimates leading to two major versions respectively called biased MSER (BMSER) and unbiased MSER (UMSER). The exact forms of these algorithms are computationally complex and require channel parameter information and thus must be processed off‐line. We thus modify the exact forms into stochastic and simplified versions to reduce computation load. The stochastic version requires no channel information and hence can be processed on‐line, but at the cost of convergence rate. Merits and characteristics of various versions are discussed and compared.     

A polynomial dimensional decomposition for stochastic computing

This article presents a new polynomial dimensional decomposition method for solving stochastic problems commonly encountered in engineering disciplines and applied sciences. The method involves a hierarchical decomposition of a multivariate response function in terms of variables with increasing dimensions, a broad range of orthonormal polynomial bases consistent with the probability measure for Fourier‐polynomial expansion of component functions, and an innovative dimension‐reduction integration for calculating the coefficients of the expansion. The new decomposition method does not require sample points as in the previous version; yet, it generates a convergent sequence of lower‐variate estimates of the probabilistic characteristics of a generic stochastic response. The results of five numerical examples indicate that the proposed decomposition method provides accurate, convergent, and computationally efficient estimates of the tail probability of random mathematical functions or the reliability of mechanical systems. Copyright © 2008 John Wiley & Sons, Ltd.     

Three‐dimensional stochastic finite element method for elasto‐plastic bodies

A new stochastic finite element method (SFEM) is formulated for three‐dimensional softening elasto‐plastic bodies with random material properties. The method is based on the Karhunen–Loeve and polynomial chaos expansions, and able to efficiently estimate complete probabilistic characteristics of the response, such as moments or PDFs. To reduce the computational complexity in the three‐dimensional setting, two alterations are made with respect to the two‐dimensional SFEM proposed earlier by the authors. First, a variability preserving modification of the Karhunen–Loeve expansion is rigorously derived and applied in the stochastic discretization of random fields representing material properties. Second, an efficient algorithm for parallel processing is developed, with time consumption being the same order as for an ordinary FEM, rendering the proposed SFEM an effective alternative to Monte‐Carlo simulation. The applicability of the proposed method to stochastic analysis of strain localization is examined using Monte‐Carlo simulation. Then, it is applied to a fault formation problem which is a recent concern of earthquake engineering. Ground surface layers are modelled by a softening elasto‐plastic body, and the evolution of probabilistic characteristics of the rupture process is analysed in detail. Some practical observations are made regarding the nature of the fault formation from the stochastic viewpoint. Copyright © 2001 John Wiley & Sons, Ltd.     

Lifetime cost optimization with time-dependent reliability

Product lifetime cost is largely determined by product lifetime reliability. In product design, the former is minimized while the latter is treated as a constraint and is usually estimated by statistical means. In this work, a new lifetime cost optimization model is developed where the product lifetime reliability is predicted with computational models derived from physical principles. With the physics-based reliability method, the state of a system is indicated by computational models, and the time-dependent system reliability is then predicted for a given set of distributions and stochastic processes in the model input. A sampling approach to extreme value distributions of input stochastic processes is employed to make the system reliability analysis efficient and accurate. The physics-based reliability analysis is integrated with the lifetime cost model. The integration enables the minimal lifetime costs including those of maintenance and warranty. Two design examples are used to demonstrate the proposed model.     

Reduced chaos expansions with random coefficientsin reduced‐dimensional stochastic modeling of coupled problems

We address the curse of dimensionality in methods for solving stochastic coupled problems with an emphasis on stochastic expansion methods such as those involving polynomial chaos expansions. The proposed method entails a partitioned iterative solution algorithm that relies on a reduced‐dimensional representation of information exchanged between subproblems to allow each subproblem to be solved within its own stochastic dimension while interacting with a reduced projection of the other subproblems. The proposed method extends previous work by the authors by introducing a reduced chaos expansion with random coefficients. The representation of the exchanged information by using this reduced chaos expansion with random coefficients enables an expeditious construction of doubly stochastic polynomial chaos expansions that separate the effect of uncertainty local to a subproblem from the effect of statistically independent uncertainty coming from other subproblems through the coupling. After laying out the theoretical framework, we apply the proposed method to a multiphysics problem from nuclear engineering. Copyright © 2013 John Wiley & Sons, Ltd.     

Simulation of a non-Gaussian stochastic process based on a combined distribution of the UHPM and the GBD

To simulate non-Gaussian stochastic processes based on the first four moments, various simulation methods are presented, in which the determination of the transformation model and the calculation of the correlation coefficients between non-Gaussian stochastic processes and Gaussian stochastic processes are the critical procedures in these simulation methods. However, some existing simulation methods are limited to specific ranges. Furthermore, their practical applications are affected negatively due to the expensive cost of determining the transformation model and the correlation coefficients between non-Gaussian and Gaussian stochastic processes. Therefore, an accurate and efficient simulation method of a non-Gaussian stochastic process with a broader range is proposed in this article. Since the simulation of non-Gaussian processes and the Nataf transformation of non-Gaussian variables have some similar characteristics, a new combined distribution is proposed based on the unified Hermite polynomial model (UHPM) and the generalized beta distribution (GBD). Then, the combined distribution is employed in the simulation of non-Gaussian stochastic processes, in which the transformation model is deduced by the combined distribution. The correlation coefficient transformation function (CCTF) between the Gaussian and non-Gaussian stochastic processes can be evaluated through the interpolation method. Furthermore, numerical examples are presented to show the accuracy and effectiveness of the proposed simulation method for non-Gaussian stochastic processes.     

Reliability Analysis Considering the Component Collision Behavior for a Large‐scale Open Source Solution

Open source software systems that serve as key components of critical infrastructures in the society are still ever‐expanding now. Many open source software systems are developed in all parts of the world, that is, Firefox, Apache HTTP server, Linux, Android, and so on. Especially, a large‐scale open source solution composed of several open source software is now attracting attention as a next‐generation software development paradigm because of the cost reduction, quick delivery, and work saving. In this paper, we propose a new approach to software reliability assessment based on stochastic differential equations and a hierarchical Bayesian model in order to consider the interesting aspect of the collision status in the binding phase of open source software. Also, we analyze actual software fault‐count data to show numerical examples of software reliability assessment considering the component collision for several open source software. Moreover, we show that the proposed reliability analysis can assist improvement of quality for the large‐scale open source solution. Copyright © 2013 John Wiley & Sons, Ltd.     

Boundary value problems defined on stochastic self‐similar multiscale geometries

A method for solving elasticity problems defined on composite bodies with a stochastic multiscale microstructure is presented. It is considered that the composite is made from two types of materials with different elastic moduli. One of these is taken as the matrix, while the other forms the inclusions. The inclusions form a stochastic fractal with a finite, but potentially large, number of scales and are randomly distributed within the matrix. The method presented here leads to the statistics of the solution, i.e. the mean and the variance of the stress and displacement fields. It is based on the stochastic finite element method (spectral approach, second‐order technique) and on scaling properties of the spatial distribution of inclusions over the problem domain. This scaling allows for a simple formulation of the multiscale problem and leads to significant computation cost savings, especially when the fractal has a large number of relevant scales. Several examples are presented and used to verify the proposed method against computationally intensive classical finite element models in which the mesh is refined down to the scale of the finest inclusions. Copyright © 2007 John Wiley & Sons, Ltd.     

Modelling and fatigue life assessment of complex structures

This paper presents some of the motivations and main conclusions from a series of joint Nordic research initiatives in which an integrated research approach to the development of future generations of advanced fabricated structures have been employed. The integrated research approach includes coordinated efforts in several key technologies: high‐speed welding processes, high strength materials, cost‐effective NDE, post‐weld treatments and FE‐based design assessment tools. Traditionally, fatigue assessment methods for welded structures have been developed based on small‐scale test specimens and verification studies for large structures are rarely published. Applications on complex structures have led to several new assessment concepts and areas for future work. A modified structural stress method that proposes a multi‐linear stress distribution through the plate thickness is introduced. Also, a crack growth assessment method in which the constraint equations of a sub‐model are linked to the global model is presented. Both these new methods are promising for complex structures. The crucial role of boundary conditions for complex structures is highlighted as is the future challenge of understanding and making use of the residual stress state for welded structures.     

Exploring Sodium‐Ion Storage Mechanism in Hard Carbons with Different Microstructure Prepared by Ball‐Milling Method

Hard carbon is considered as one of the most promising anodes in sodium‐ion batteries due to its high capacity, low cost, and abundant resources. However, the available capacity and low initial Coulombic efficiency (ICE) limits the practical application of hard carbon anode. This issue results from the unclear understanding of the Na⁺ storage mechanism in hard carbon. In this work, a series of hard carbons with different microstructures are synthesized through an “up to down” approach by using a simple ball‐milling method to illustrate the sodium‐ion storage mechanism. The results demonstrate that ball‐milled hard carbon with more defects and smaller microcrystalline size shows less low‐potential‐plateau capacity and lower ICE, which provides further evidence to the “adsorption–insertion” mechanism. This work might give a new perspective to design hard carbon material with a proper structure for efficient sodium‐ion storage to develop high‐performance sodium‐ion batteries.     

Sparse grids‐based stochastic approximations with applications to aerodynamics sensitivity analysis

This work compares sample‐based polynomial surrogates, well suited for moderately high‐dimensional stochastic problems. In particular, generalized polynomial chaos in its sparse pseudospectral form and stochastic collocation methods based on both isotropic and dimension‐adapted sparse grids are considered. Both classes of approximations are compared, and an improved version of a stochastic collocation with dimension adaptivity driven by global sensitivity analysis is proposed. The stochastic approximations efficiency is assessed on multivariate test function and airfoil aerodynamics simulations. The latter study addresses the probabilistic characterization of global aerodynamic coefficients derived from viscous subsonic steady flow about a NACA0015 airfoil in the presence of geometrical and operational uncertainties with both simplified aerodynamics model and Reynolds‐Averaged Navier‐Stokes (RANS) simulation. Sparse pseudospectral and collocation approximations exhibit similar level of performance for isotropic sparse simulation ensembles. Computational savings and accuracy gain of the proposed adaptive stochastic collocation driven by Sobol' indices are patent but remain problem‐dependent. Copyright © 2015 John Wiley & Sons, Ltd.     

Bayesian Metamodeling for Computer Experiments Using the Gaussian Kriging Models

In the past two decades, more and more quality and reliability activities have been moving into the design of product and process. The design and analysis of computer experiments, as a new frontier of the design of experiments, has become increasingly popular among modern companies for optimizing product and process conditions and producing high‐quality yet low‐cost products and processes. This article mainly focuses on the issue of constructing cheap metamodels as alternatives to the expensive computer simulators and proposes a new metamodeling method on the basis of the Gaussian stochastic process model or Gaussian Kriging. Rather than a constant mean as in ordinary Kriging or a fixed mean function as in universal Kriging, the new method captures the overall trend of the performance characteristics of products and processes through a more accurate mean, by efficiently incorporating a scheme of sparseness prior–based Bayesian inference into Kriging. Meanwhile, the mean model is able to adaptively exclude the unimportant effects that deteriorate the prediction performance. The results of an experiment on empirical applications demonstrate that, compared with several benchmark methods in the literature, the proposed Bayesian method is not only much more effective in approximation but also very efficient in implementation, hence more appropriate than the widely used ordinary Kriging to empirical applications in the real world. Copyright © 2011 John Wiley & Sons, Ltd.     

Flexible,Planar‐Integrated,All‐Solid‐State Fiber Supercapacitors with an Enhanced Distributed‐Capacitance Effect

Flexible and highly efficient energy storage units act as one of the key components in portable electronics. In this work, by planar‐integrated assembly of hierarchical ZnCo₂O₄ nanowire arrays/carbon fibers electrodes, a new class of flexible all‐solid‐state planar‐integrated fiber supercapacitors are designed and produced via a low‐cost and facile method. The as‐fabricated flexible devices exhibit high‐efficiency, enhanced capacity, long cycle life, and excellent electrical stability. An enhanced distributed‐capacitance effect is experimentally observed for the device. This strategy enables highly flexible new structured supercapacitors with maximum functionality and minimized size, thus making it possible to be readily applied in flexible/portable photoelectronic devices.     

An adaptive spectral Galerkin stochastic finite element method using variability response functions

A methodology is proposed in this paper to construct an adaptive sparse polynomial chaos (PC) expansion of the response of stochastic systems whose input parameters are independent random variables modeled as random fields. The proposed methodology utilizes the concept of variability response function in order to compute an a priori low‐cost estimate of the spatial distribution of the second‐order error of the response, as a function of the number of terms used in the truncated Karhunen–Loève (KL) expansion. This way the influence of the response variance to the spectral content (correlation structure) of the random input is taken into account through a spatial variation of the truncated KL terms. The criterion for selecting the number of KL terms at different parts of the structure is the uniformity of the spatial distribution of the second‐order error. This way a significantly reduced number of PC coefficients, with respect to classical PC expansion, is required in order to reach a uniformly distributed target second‐order error. This results in an increase of sparsity of the coefficient matrix of the corresponding linear system of equations leading to an enhancement of the computational efficiency of the spectral stochastic finite element method. Copyright © 2015 John Wiley & Sons, Ltd.     

The stochastic Galerkin scaled boundary finite element method on random domain

The research work extends the scaled boundary finite element method to non‐deterministic framework defined on random domain wherein random behaviour is exhibited in the presence of the random‐field uncertainties. The aim is to blend the scaled boundary finite element method into the Galerkin spectral stochastic methods, which leads to a proficient procedure for handling the stress singularity problems and crack analysis. The Young's modulus of structures is considered to have random‐field uncertainty resulting in the stochastic behaviour of responses. Mathematical expressions and the solution procedure are derived to evaluate the statistical characteristics of responses (displacement, strain, and stress) and stress intensity factors of cracked structures. The feasibility and effectiveness of the presented method are demonstrated by particularly chosen numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Sustainable Hydrothermal Carbonization Synthesis of Iron/Nitrogen‐Doped Carbon Nanofiber Aerogels as Electrocatalysts for Oxygen Reduction

It is urgent to develop new kinds of low‐cost and high‐performance nonprecious metal (NPM) catalysts as alternatives to Pt‐based catalysts for oxygen reduction reaction (ORR) in fuel cells and metal–air batteries, which have been proved to be efficient to meet the challenge of increase of global energy demand and CO₂ emissions. Here, an economical and sustainable method is developed for the synthesis of Fe, N codoped carbon nanofibers (Fe–N/CNFs) aerogels as efficient NPM catalysts for ORR via a mild template‐directed hydrothermal carbonization (HTC) process, where cost‐effective biomass‐derived d (+)‐glucosamine hydrochloride and ferrous gluconate are used as precursors and recyclable ultrathin tellurium nanowires are used as templates. The prepared Fe/N‐CNFs catalysts display outstanding ORR activity, i.e., onset potential of 0.88 V and half‐wave potential of 0.78 V versus reversible hydrogen electrode in an alkaline medium, which is highly comparable to that of commercial Pt/C (20 wt% Pt) catalyst. Furthermore, the Fe/N‐CNFs catalysts exhibit superior long‐term stability and better tolerance to the methanol crossover effect than the Pt/C catalyst in both alkaline and acidic electrolytes. This work suggests the great promise of developing new families of NPM ORR catalysts by the economical and sustainable HTC process.     

Stochastic inverse heat conduction using a spectral approach

An adjoint‐based functional optimization technique in conjunction with the spectral stochastic finite element method is proposed for the solution of an inverse heat conduction problem in the presence of uncertainties in material data, process conditions and measurement noise. The ill‐posed stochastic inverse problem is restated as a conditionally well‐posed L₂ optimization problem. The gradient of the objective function is obtained in a distributional sense by defining an appropriate stochastic adjoint field. The L₂ optimization problem is solved using a conjugate‐gradient approach. Accuracy and effectiveness of the proposed approach is appraised with the solution of several stochastic inverse heat conduction problems. Copyright © 2004 John Wiley & Sons, Ltd.