A divide and conquer approach to computing the mean first passage matrix for Markov chains via Perron complement reductions

Let M_T be the mean first passage matrix for an n‐state ergodic Markov chain with a transition matrix T. We partition T as a 2×2 block matrix and show how to reconstruct M_T efficiently by using the blocks of T and the mean first passage matrices associated with the non‐overlapping Perron complements of T. We present a schematic diagram showing how this method for computing M_T can be implemented in parallel. We analyse the asymptotic number of multiplication operations necessary to compute M_T by our method and show that, for large size problems, the number of multiplications is reduced by about 1/8, even if the algorithm is implemented in serial. We present five examples of moderate sizes (of orders 20–200) and give the reduction in the total number of flops (as opposed to multiplications) in the computation of M_T. The examples show that when the diagonal blocks in the partitioning of T are of equal size, the reduction in the number of flops can be much better than 1/8. Copyright © 2001 John Wiley & Sons, Ltd.     

A semi-analytic approach for the nonlinear dynamic response of circular plates

This paper presents a new semi-analytic perturbation differential quadrature method for geometrically nonlinear vibration analysis of circular plates. The nonlinear governing equations are converted into a linear differential equation system by using Linstedt–Poincaré perturbation method. The solutions of nonlinear dynamic response and the nonlinear free vibration are then sought through the use of differential quadrature approximation in space domain and analytical series expansion in time domain. The present method is validated against analytical results using elliptic function in several examples for both clamped and simply supported circular plates, showing that it has excellent accuracy and convergence. Compared with numerical methods involving iterative time integration, the present method does not suffer from error accumulation and is able to give very accurate results over a long time interval.     

A combined approach for segment-specific market basket analysis

Market baskets arise from consumers’ shopping trips and include items from multiple categories that are frequently chosen interdependently from each other. Explanatory models of multicategory choice behavior explicitly allow for such category purchase dependencies. They typically estimate own and across-category effects of marketing-mix variables on purchase incidences for a predefined set of product categories. Because of analytical restrictions, however, multicategory choice models can only handle a small number of categories. Hence, for large retail assortments, the issue emerges of how to determine the composition of shopping baskets with a meaningful selection of categories. Traditionally, this is resolved by managerial intuition. In this article, we combine multicategory choice models with a data-driven approach for basket selection. The proposed procedure also accounts for customer heterogeneity and thus can serve as a viable tool for designing target marketing programs. A data compression step first derives a set of basket prototypes which are representative for classes of market baskets with internally more distinctive (complementary) cross-category interdependencies and are responsible for the segmentation of households. In a second step, segment-specific cross-category effects are estimated for suitably selected categories using a multivariate logistic modeling framework. In an empirical illustration, significant differences in cross-effects and price elasticities can be shown both across segments and compared to the aggregate model.     

Marino—Prodi perturbation type results and Morse indices of minimax critical points for a class of functionals in Banach spaces

In this work, we study a class of Euler functionals defined in Banach spaces, associated with quasilinear elliptic problems involving p-Laplace operator (p > 2). First we obtain perturbation results in the spirit of the remarkable paper by Marino and Prodi (Boll. U.M.I. (4) 11(Suppl. fasc. 3): 1–32, 1975), using the new definition of nondegeneracy given in (Ann. Inst. H. Poincaré: Analyse Non Linéaire. 2:271–292, 2003). We also extend Morse index estimates for minimax critical points, introduced by Lazer and Solimini (Nonlinear Anal. T.M.A. 12:761–775, 1988) in the Hilbert case, to our Banach setting. Mathematics Subject Classification (1991) 58E05, 35B20, 35J60, 35J70     

A vague set based decision support approach for evaluating research funding programs

Scientific Research Assessment (SRA) is receiving increasing attention in both academic and industry. More and more organizations are recognizing the importance of SRA for the optimal use of scarce resources. In this paper, a vague set theory based decision support approach is proposed for SRA. Specifically, a family of parameterized S-OWA operator is developed for the aggregation of vague assessments. The proposed approach is introduced to evaluate the research funding programs of the National Natural Science Foundation of China (NSFC). It provides a soft and expansive way to help the decision maker in NSFC to make his decisions. The proposed approach can also be used for some other agencies to make similar assessment.     

A preconditioned and extrapolation-accelerated GMRES method for PageRank

In this paper, we propose and analyze GMRES-type methods for the PageRank computation. However, GMRES may converge very slowly or sometimes even diverge or break down when the damping factor is close to 1 and the dimension of the search subspace is low. We propose two strategies: preconditioning and vector extrapolation accelerating, to improve the convergence rate of the GMRES method. Theoretical analysis demonstrate the efficiency of the proposed strategies and numerical experiments show that the performance of the proposed methods is very much better than that of the traditional methods for PageRank problems.     

On the selection of auxiliary functions, operators, and convergence control parameters in the application of the Homotopy Analysis Method to nonlinear differential equations: A general approach

The Homotopy Analysis Method of Liao [Liao SJ. Beyond perturbation: introduction to the Homotopy Analysis Method. Boca Raton: Chapman & Hall/CRC Press; 2003] has proven useful in obtaining analytical solutions to various nonlinear differential equations. In this method, one has great freedom to select auxiliary functions, operators, and parameters in order to ensure the convergence of the approximate solutions and to increase both the rate and region of convergence. We discuss in this paper the selection of the initial approximation, auxiliary linear operator, auxiliary function, and convergence control parameter in the application of the Homotopy Analysis Method, in a fairly general setting. Further, we discuss various convergence requirements on solutions.     

A one-model approach based on relaxed combinations of inputs for evaluating input congestion in DEA

This paper provides a one-model approach of input congestion based on input relaxation model developed in data envelopment analysis (e.g. [G.R. Jahanshahloo, M. Khodabakhshi, Suitable combination of inputs for improving outputs in DEA with determining input congestion — Considering textile industry of China, Applied Mathematics and Computation (1) (2004) 263–273; G.R. Jahanshahloo, M. Khodabakhshi, Determining assurance interval for non-Archimedean ele improving outputs model in DEA, Applied Mathematics and Computation 151 (2) (2004) 501–506; M. Khodabakhshi, A super-efficiency model based on improved outputs in data envelopment analysis, Applied Mathematics and Computation 184 (2) (2007) 695–703; M. Khodabakhshi, M. Asgharian, An input relaxation measure of efficiency in stochastic data analysis, Applied Mathematical Modelling 33 (2009) 2010–2023]. This approach reduces solving three problems with the two-model approach introduced in the first of the above-mentioned reference to two problems which is certainly important from computational point of view. The model is applied to a set of data extracted from ISI database to estimate input congestion of 12 Canadian business schools.     

A combined branch-and-bound and genetic algorithm based approach for a flowshop scheduling problem

In this paper, we study the application of a meta-heuristic to a two-machine flowshop scheduling problem. The meta-heuristic uses a branch-and-bound procedure to generate some information, which in turn is used to guide a genetic algorithm's search for optimal and near-optimal solutions. The criteria considered are makespan and average job flowtime. The problem has applications in flowshop environments where management is interested in reducing turn-around and job idle times simultaneously. We develop the combined branch-and-bound and genetic algorithm based procedure and two modified versions of it. Their performance is compared with that of three algorithms: pure branch-and-bound, pure genetic algorithm, and a heuristic. The results indicate that the combined approach and its modified versions are better than either of the pure strategies as well as the heuristic algorithm.     

A heuristic approach for deriving the priority vector in AHP

This paper proposes an approach for deriving the priority vector from an inconsistent pair-wise comparison matrix through the nearest consistent matrix and experts judgments, which enables balancing the consistency and experts judgments. The developed algorithm for achieving a nearest consistent matrix is based on a logarithmic transformation of the pair-wise comparison matrix, and follows an iterative feedback process that identifies an acceptable level of consistency while complying with experts preferences. Three numerical examples are examined to illustrate applications and advantages of the developed approach.     

A Taylor expansion approach for solving partial differential equations with random Neumann boundary conditions

Nonlinear partial differential equation with random Neumann boundary conditions are considered. A stochastic Taylor expansion method is derived to simulate these stochastic systems numerically. As examples, a nonlinear parabolic equation (the real Ginzburg-Landau equation) and a nonlinear hyperbolic equation (the sine-Gordon equation) with random Neumann boundary conditions are solved numerically using a stochastic Taylor expansion method. The impact of boundary noise on the system evolution is also discussed.     

A model for resistance welding including phase transitions and Joule heating

In this paper, we introduce a new model for solid–liquid phase transitions triggered by Joule heating as they arise in the case of resistance welding of metal parts. The main novelties of the paper are the coupling of the thermistor problem with a phase‐field model and the consideration of phase‐dependent physical parameters through a mixture ansatz. The PDE system resulting from our modeling approach couples a strongly nonlinear heat equation, a non‐smooth equation for the the phase parameter (standing for the local proportion of one of the two phases) with a quasistatic electric charge conservation law. We prove the existence of weak solutions in the three‐dimensional (3D) case, whereas the regularity result and the uniqueness of solution is stated only in the two‐dimensional case. Indeed, uniqueness for the 3D system is still an open problem. Copyright © 2011 John Wiley & Sons, Ltd.     

A heuristic approach for combined equipment-planning and routing in multi-layer SDH/WDM networks

The paper deals with a multi-layer network design problem for a high-speed telecommunication network based on Synchronous Digital Hierarchy (SDH) and Wavelength Division Multiplex (WDM) technology. The network has to carry a certain set of demands with the objective of minimizing the investment in the equipment. The different layers are the fiber-layer, 2.5 Gbit/s-, 10 Gbit/s- and WDM-systems. Several variations of the problem including path-protected demands and specific types of cross-connect equipment are considered. The problem is described as a mixed integer linear programming model and some results for small networks are presented. Two greedy heuristics, a random start heuristic and a GRASP-like approach are implemented to solve large real world problems.     

Perturbation, extraction and refinement of invariant pairs for matrix polynomials

Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefits can be expected for polynomial eigenvalue problems, for which the concept of an invariant subspace needs to be replaced by the concept of an invariant pair. Little has been known so far about numerical aspects of such invariant pairs. The aim of this paper is to fill this gap. The behavior of invariant pairs under perturbations of the matrix polynomial is studied and a first-order perturbation expansion is given. From a computational point of view, we investigate how to best extract invariant pairs from a linearization of the matrix polynomial. Moreover, we describe efficient refinement procedures directly based on the polynomial formulation. Numerical experiments with matrix polynomials from a number of applications demonstrate the effectiveness of our extraction and refinement procedures.     

A survey of some estimates of eigenvalues and condition numbers for certain preconditioned matrices

Eigenvalue and condition number estimates for preconditioned iteration matrices provide the information required to estimate the rate of convergence of iterative methods, such as preconditioned conjugate gradient methods. In recent years various estimates have been derived for (perturbed) modified (block) incomplete factorizations. We survey and extend some of these and derive new estimates. In particular we derive upper and lower estimates of individual eigenvalues and of condition number. This includes a discussion that the condition number of preconditioned second order elliptic difference matrices is O(h⁻¹). Some of the methods are applied to compute certain parameters involved in the computation of the preconditioner.     

Existence results for strong vector equilibrium problems and their applications

New existence results for the strong vector equilibrium problem are presented, relying on a well-known separation theorem in infinite-dimensional spaces. The main results are applied to strong cone saddle-points and strong vector variational inequalities providing new existence results, and furthermore they allow recovery of an earlier result from the literature.     

A combined statistical approach and ground movement model for improving taxi time estimations at airports

With the expected continued increases in air transportation, the mitigation of the consequent delays and environmental effects is becoming more and more important, requiring increasingly sophisticated approaches for airside airport operations. Improved on-stand time predictions (for improved resource allocation at the stands) and take-off time predictions (for improved airport-airspace coordination) both require more accurate taxi time predictions, as do the increasingly sophisticated ground movement models which are being developed. Calibrating such models requires historic data showing how long aircraft will actually take to move around the airport, but recorded data usually includes significant delays due to contention between aircraft. This research was motivated by the need to both predict taxi times and to quantify and eliminate the effects of airport load from historic taxi time data, since delays and re-routing are usually explicitly considered in ground movement models. A prediction model is presented here that combines both airport layout and historic taxi time information within a multiple linear regression analysis, identifying the most relevant factors affecting the variability of taxi times for both arrivals and departures. The promising results for two different European hub airports are compared against previous results for US airports.     

A new population seeding technique for permutation-coded Genetic Algorithm: Service transfer approach

Genetic Algorithm (GA) is a popular heuristic method for dealing complex problems with very large search space. Among various phases of GA, the initial phase of population seeding plays an important role in deciding the span of GA to achieve the best fit w.r.t. the time. In other words, the quality of individual solutions generated in the initial population phase plays a critical role in determining the quality of final optimal solution. The traditional GA with random population seeding technique is quite simple and of course efficient to some extent; however, the population may contain poor quality individuals which take long time to converge with optimal solution. On the other hand, the hybrid population seeding techniques which have the benefit of good quality individuals and fast convergence lacks in terms of randomness, individual diversity and ability to converge with global optimal solution. This motivates to design a population seeding technique with multifaceted features of randomness, individual diversity and good quality. In this paper, an efficient Ordered Distance Vector (ODV) based population seeding technique has been proposed for permutation-coded GA using an elitist service transfer approach. One of the famous combinatorial hard problems of Traveling Salesman Problem (TSP) is being chosen as the testbed and the experiments are performed on different sized benchmark TSP instances obtained from standard TSPLIB [54]. The experimental results advocate that the proposed technique outperforms the existing popular initialization methods in terms of convergence rate, error rate and convergence time.