Non-myopic vehicle and route selection in dynamic DARP with travel time and workload objectives

In dial-a-ride problems, a fleet of n vehicles is routed to transport people between pick-up and delivery locations. We consider an elementary version of the problem where trip requests arrive in time and require an immediate vehicle assignment (which triggers an appropriate route update of the selected vehicle). In this context, a relatively general objective can be stated as a weighted sum of the system's effort and the customers' inconvenience. However, optimizing almost any objective in this immensely complex stochastic system is prohibitively difficult. Thus the earlier work has largely resorted to heuristic cost functions that arise, e.g., from the corresponding static systems. By using the framework of Markov decision processes and the classical M/M/1 queue as a highly abstract model for a single vehicle, we explain why certain intuitive cost functions indeed give satisfactory results in the dynamic system, and also give an explicit interpretation of different components appearing in a general cost function. The resulting family of heuristic control policies is demonstrated to offer a desired type of performance thus justifying the assumed analogy between a multi-queue and dial-a-ride systems.     

A computational study on the quadratic knapsack problem with multiple constraints

The quadratic knapsack problem (QKP) has been the subject of considerable research in recent years. Despite notable advances in special purpose solution methodologies for QKP, this problem class remains very difficult to solve. With the exception of special cases, the state-of-the-art is limited to addressing problems of a few hundred variables and a single knapsack constraint.In this paper we provide a comparison of quadratic and linear representations of QKP based on test problems with multiple knapsack constraints and up to eight hundred variables. For the linear representations, three standard linearizations are investigated. Both the quadratic and linear models are solved by standard branch-and-cut optimizers available via CPLEX. Our results show that the linear models perform well on small problem instances but for larger problems the quadratic model outperforms the linear models tested both in terms of solution quality and solution time by a wide margin. Moreover, our results demonstrate that QKP instances larger than those previously addressed in the literature as well as instances with multiple constraints can be successfully and efficiently solved by branch and cut methodologies.     

Numerical stability of symmetric solitary-wave-like waves of a two-layer fluid—Forced modified KdV equation

Forced internal waves at the interface of a two-layer incompressible fluid in a two-dimensional domain with rigid horizontal boundaries are studied. The lower boundary is assumed to have a small obstruction. We derive a time-dependent forced modified KdV equation when the KdV theory fails and study the stabilities of four types of symmetric time-independent solitary-wave-like solutions numerically.     

Constructing G 1 continuous curve on a free-form surface with normal projection

In this paper, we develop a new method for G ¹ continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with a specified tangent direction at every point. Based on the normal projection method, we design a G ¹ continuous curve in three-dimensional space and then project orthogonally the curves onto the given surface. With the techniques in classical differential geometry, we derive a system of differential equations characterizing the projection curve. The resulting interpolation curve is obtained by numerically solving the initial-value problems for a system of first-order ordinary differential equations in the parametric domain associated to the surface representation for a parametric case or in three-dimensional space for an implicit case. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification such that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface, and numerical experiments demonstrate that it is effective and potentially useful in surface trim, robot, patterns design on surface and other industrial and research fields.     

Parameter Estimation Algorithms for Hierarchical Distributed Systems

There has been a great deal of research activity in the area of identification of distributed parameter systems over the past two decades. An extensive treatment of off-line schemes ( e.g. , output least squares, estimation error, etc. ) together with a comprehensive survey of the literature can be found in the monograph by Banks and Kunisch [4] . In the case of on-line, or adaptive, schemes, the available literature is less extensive and more recent ( Isermann et al . [7] ). The on-line methods give estimates recursively as the measurements are obtained within the time limit imposed by the sampling period. These include recursive projection algorithm ( Baumeister et al. [5] ), recursive least squares algorithm ( Glentis et al . [6] ), on-line excitation algorithms ( Ludwig et al. [8] ), etc . In this paper an equivalent 2nd order dynamical system is formulated from a given trajectory representing the pattern to be recognised and simulated in order to estimate the parameters for hierarchical distributed systems using 1st and 2nd order dynamics. Recommendations for the best estimation strategy are given.     

New generalized method to construct new non-travelling wave solutions and travelling wave solutions of K–D equations

With the aid of computerized symbolic computation, we obtain new types of general solution of a first-order nonlinear ordinary differential equation with six degrees of freedom and devise a new generalized method and its algorithm, which can be used to construct more new exact solutions of general nonlinear differential equations. The (2+1)-dimensional K–D equation is chosen to illustrate our algorithm such that more families of new exact solutions are obtained, which contain non-travelling wave solutions and travelling wave solutions.     

Unconditionally stable C1-cubic spline collocation method for solving parabolic equations

This article aims to present a new approach based on C¹-cubic splines introduced by Sallam and Naim Anwar [Sallam, S. and Naim Anwar, M. (2000). Stabilized cubic C¹-spline collocation method for solving first-order ordinary initial value problems, Int. J. Comput. Math., 74, 87–96.], which is A-stable, for the time integration of parabolic equations (diffusion or heat equation). The introduced method is an example of the so-called method of lines (the solution is thought to consist of space discretization and time integration), which is an extension of the 1/3-Simpson's finite-difference scheme. Our main objective is to prove the unconditional stability of the proposed method as well as to show that the method is convergent and is of order O (h ²)?+?O (k ⁴) i.e. it is a fourth-order in time and second-order in space. Computational results also show that the method is relevant for long time interval problems.     

A weighted H1 seminorm regularization method for Fredholm integral equations of the first kind

Many problems in mathematics and engineering lead to Fredholm integral equations of the first kind, e.g. signal and image processing. These kinds of equations are difficult to solve numerically since they are ill-posed. Therefore, regularization is required to obtain a reasonable approximate solution. This paper presents a new regularization method based on a weighted H¹ seminorm. Details of numerical implementation are given. Numerical examples, including one-dimensional and two-dimensional integral equations, are presented to illustrate the efficiency of the proposed approach. Numerical results show that the proposed regularization method can restore edges as well as details.     

Geoadditive expectile regression

Quantile regression has emerged as one of the standard tools for regression analysis that enables a proper assessment of the complete conditional distribution of responses even in the presence of heteroscedastic errors. Quantile regression estimates are obtained by minimising an asymmetrically weighted sum of absolute deviations from the regression line, a decision theoretic formulation of the estimation problem that avoids a full specification of the error term distribution. Recent advances in mean regression have concentrated on making the regression structure more flexible by including nonlinear effects of continuous covariates, random effects or spatial effects. These extensions often rely on penalised least squares or penalised likelihood estimation with quadratic penalties and may therefore be difficult to combine with the linear programming approaches often considered in quantile regression. As a consequence, geoadditive expectile regression based on minimising an asymmetrically weighted sum of squared residuals is introduced. Different estimation procedures are presented including least asymmetrically weighted squares, boosting and restricted expectile regression. The properties of these procedures are investigated in a simulation study and an analysis on rental fees in Munich is provided where the geoadditive specification allows for an analysis of nonlinear effects of the size of flats or the year of construction and the spatial distribution of rents simultaneously.     

Functional linear regression after spline transformation

Functional linear regression has been widely used to model the relationship between a scalar response and functional predictors. If the original data do not satisfy the linear assumption, an intuitive solution is to perform some transformation such that transformed data will be linearly related. The problem of finding such transformations has been rather neglected in the development of functional data analysis tools. In this paper, we consider transformation on the response variable in functional linear regression and propose a nonparametric transformation model in which we use spline functions to construct the transformation function. The functional regression coefficients are then estimated by an innovative procedure called mixed data canonical correlation analysis (MDCCA). MDCCA is analogous to the canonical correlation analysis between two multivariate samples, but is between a multivariate sample and a set of functional data. Here, we apply the MDCCA to the projection of the transformation function on the B-spline space and the functional predictors. We then show that our estimates agree with the regularized functional least squares estimate for the transformation model subject to a scale multiplication. The dimension of the space of spline transformations can be determined by a model selection principle. Typically, a very small number of B-spline knots is needed. Real and simulation data examples are further presented to demonstrate the value of this approach.