Numerically Computing Real Points on Algebraic Sets

Given a polynomial system f, a fundamental question is to determine if f has real roots. Many algorithms involving the use of infinitesimal deformations have been proposed to answer this question. In this article, we transform an approach of Rouillier, Roy, and Safey El Din, which is based on a classical optimization approach of Seidenberg, to develop a homotopy based approach for computing at least one point on each connected component of a real algebraic set. Examples are presented demonstrating the effectiveness of this parallelizable homotopy based approach.     

A linearization‐based approach of homotopy analysis method for non‐linear time‐fractional parabolic PDEs

In this paper, a novel approach, namely, the linearization‐based approach of homotopy analysis method, to analytically treat non‐linear time‐fractional PDEs is proposed. The presented approach suggests a new optimized structure of the homotopy series solution based on a linear approximation of the non‐linear problem. A comparative study between the proposed approach and standard homotopy analysis approach is illustrated by solving two examples involving non‐linear time‐fractional parabolic PDEs. The performed numerical simulations demonstrate that the linearization‐based approach reduces the computational complexity and improves the performance of the homotopy analysis method.     

On the algebraic solution of fuzzy linear systems based on interval theory

In this paper, fuzzy linear systems involving a crisp square matrix and a fuzzy right-hand side vector are considered. A new approach to solve such systems based on interval theory and the new concept “interval inclusion linear system” is proposed. Also, new necessary and sufficient conditions are derived for obtaining the unique algebraic solution. Numerical examples are given to illustrate the efficiency of the proposed method.     

Fuzzy clusterwise quasi-likelihood generalized linear models

The quasi-likelihood method has emerged as a useful approach to the parameter estimation of generalized linear models (GLM) in circumstances where there is insufficient distributional information to construct a likelihood function. Despite its flexibility, the quasi-likelihood approach to GLM is currently designed for an aggregate-sample analysis based on the assumption that the entire sample of observations is taken from a single homogenous population. Thus, this approach may not be suitable when heterogeneous subgroups exist in the population, which involve qualitatively distinct effects of covariates on the response variable. In this paper, the quasi-likelihood GLM approach is generalized to a fuzzy clustering framework which explicitly accounts for such cluster-level heterogeneity. A simple iterative estimation algorithm is presented to optimize the regularized fuzzy clustering criterion of the proposed method. The performance of the proposed method in recovering parameters is investigated based on a Monte Carlo analysis involving synthetic data. Finally, the empirical usefulness of the proposed method is illustrated through an application to actual data on the coupon usage behaviour of a sample of consumers.     

Degree of cure-dependent modelling for polymer curing processes at small strains

In this paper, a thermodynamically consistent small strain constitutive model is formulated that is directly based on the degree of cure, a key parameter in the curing (reaction) kinetics. The new formulation is also in line with the earlier proposed hypoelastic approach, cf. Hossain et al., 2010. The curing process of polymers is a complex phenomenon involving a series of chemical reactions which transform a viscoelastic fluid into a viscoelastic solid during which the temperature, the chemistry and the mechanics are coupled. Some representative numerical examples conclude the paper and show the capability of the newly proposed constitutive formulation to capture major phenomena observed during the curing processes of polymers. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A method of fuzzy preference relation to authorize the worldwide interoperability for microwave access license in Taiwan

This study proposes a preference relation based evaluation framework to help the National Communication Commission (NCC) in Taiwan authorize a worldwide interoperability for microwave access (WiMAX) license under a fuzzy environment where the uncertainty, subjectivity and vagueness are dealt with linguistic variables parameterized by triangular fuzzy numbers. This study applies the fuzzy multi-criteria decision making approach to determine the importance weights of evaluation criteria and consolidate the performance ratings of possible alternatives. Aggregated the evaluators’ opinions toward the criteria and alternatives, the fuzzy preference relation approach is utilized to obtain the non-dominated degree of each alternative for the decision makers to make a final decision. Simultaneously, an empirical case involving sixteen quantitative and fifteen qualitative evaluation criteria, thirteen telecommunication applicants assessed by twelve specialists from various fields of telecommunication industry in Taiwan is solicited to demonstrate the proposed approach.     

The principle of minimum of partial local variations for determining convective flows in the numerical solution of one-dimensional nonlinear scalar hyperbolic equations

For the CABARET finite difference scheme, a new approach to the construction of convective flows for the one-dimensional nonlinear transport equation is proposed based on the minimum principle of partial local variations. The new approach ensures the monotonicity of solutions for a wide class of problems of a fairly general form including those involving discontinuous and nonconvex functions. Numerical results illustrating the properties of the proposed method are discussed.     

Hybrid control variates-based simulation method for structural reliability analysis of some problems with low failure probability

The safety analysis of systems with nonlinear performance function and small probability of failure is a challenge in the field of reliability analysis. In this study, an efficient approach is presented for approximating small failure probabilities. To meet this aim, by introducing Probability Density Function (PDF) control variates, the original failure probability integral was reformulated based on the Control Variates Technique (CVT). Accordingly, using the adaptive cooperation of the subset simulation (SubSim) and the CVT, a new formulation was offered for the approximation of small failure probabilities. The proposed formulation involves a probability term (resulting from a fast-moving SubSim) and an adaptive weighting term that refines the obtained probability. Several numerical and engineering problems, involving nonlinear performance functions and system-level reliability problems, are solved by the proposed approach and common reliability methods. Results showed that the proposed simulation approach is not only more efficient, but is also robust than common reliability methods. It also presents a good potential for application in engineering reliability problems.     

Fair transfer price and inventory holding policies in two-enterprise supply chains

A key issue in supply chain optimisation involving multiple enterprises is the determination of policies that optimise the performance of the supply chain as a whole while ensuring adequate rewards for each participant.In this paper, we present a mathematical programming formulation for fair, optimised profit distribution between echelons in a general multi-enterprise supply chain. The proposed formulation is based on an approach applying the Nash bargaining solution for finding optimal multi-partner profit levels subject to given minimum echelon profit requirements.The overall problem is first formulated as a mixed integer non-linear programming (MINLP) model. A spatial and binary variable branch-and-bound algorithm is then applied to the above problem based on exact and approximate linearisations of the bilinear terms involved in the model, while at each node of the search tree, a mixed integer linear programming (MILP) problem is solved. The solution comprises inter-firm transfer prices, production and inventory levels, flows of products between echelons, and sales profiles.The applicability of the proposed approach is demonstrated by a number of illustrative examples based on industrial processes.     

An adjustable approach to fuzzy soft set based decision making

Molodtsov’s soft set theory was originally proposed as a general mathematical tool for dealing with uncertainty. Recently, decision making based on (fuzzy) soft sets has found paramount importance. This paper aims to give deeper insights into decision making based on fuzzy soft sets. We discuss the validity of the Roy-Maji method and show its true limitations. We point out that the choice value designed for the crisp case is no longer fit to solve decision making problems involving fuzzy soft sets. By means of level soft sets, we present an adjustable approach to fuzzy soft set based decision making and give some illustrative examples. Moreover, the weighted fuzzy soft set is introduced and its application to decision making is also investigated.     

On some conjectures proposed by Haı̈m Brezis

Druet (Ann. Inst. H. Poincaré Anal. Non Linèaire 19(2) (2002) 125) solved two conjectures proposed by Haı̈m Brezis (Comm. Pure Appl. Math. 39 (1986) 17) about “low”-dimension phenomena for some elliptic problem with critical Sobolev exponent. In Druet (Ann. Inst. H. Poincaré Anal. Non Linèaire 19(2) (2002) 125), the proof of one of the two conjectures is reduced to an asymptotic analysis which is carried over with very general techniques involving pointwise estimates. We propose here a different and simpler approach in the blow-up analysis based on integral estimates and on a careful expansion of the energy functional.     

A web-based methodology for product design evaluation and optimisation

The authors have developed a methodology that takes advantages of the World Wide Web to analyse and develop optimal new product designs. This paper describes the methodology and illustrates its application to a case study involving the design of an actual Web site where music CDs are sold. The proposed methodology has the following features: (a) it is based on a design inspired by conjoint analysis; (b) it involves unobtrusive electronic measurement of the actual behavior of Web users who remain undisturbed by experimental factors; and (c) it utilises an integer programming approach to seek optimal Web site configurations. The methodology uses limited dependent variable methods to develop response models that provide the basis for the development of objective functions for an optimisation model. The optimisation model can consider either single or multiple objective functions by using a Pareto optimum approach.     

用Gibbs抽样算法计算定数截尾时Weibull分布的贝叶斯估计

本文假设产品寿命服从两参数 Weibull分布 ,在定数截尾寿命试验的情况下 ,利用近年来发展起来的 Gibbs抽样算法 ,设计了计算参数贝叶斯估计的 Gibbs抽样方案 ,计算了一个实例并与经典的 BL UE和 BL IE估计结果进行比较。结果表明 ,较之传统的数值积分法 ,Gibbs抽样算法使用起来方便直接 ,更适合于计算可靠性指标的贝叶斯估计     

Optimal aggregation of fuzzy preference relations with an application to broadband internet service selection

This paper investigates the aggregation of multiple fuzzy preference relations into a collective fuzzy preference relation in fuzzy group decision analysis and proposes an optimization based aggregation approach to assess the relative importance weights of the multiple fuzzy preference relations. The proposed approach that is analytical in nature assesses the weights by minimizing the sum of squared distances between any two weighted fuzzy preference relations. Relevant theorems are offered in support of the proposed approach. Multiplicative preference relations are also incorporated into the approach using an appropriate transformation technique. An eigenvector method is introduced to derive the priorities from the collective fuzzy preference relation. The proposed aggregation approach is tested using two numerical examples. A third example involving broadband internet service selection is offered to illustrate that the proposed aggregation approach provides a simple, effective and practical way of aggregating multiple fuzzy preference relations in real-life situations.     

Heuristic approach on dynamic lot-sizing model for durable products with end-of-use constraints

A version of the dynamic lot-sizing (DLS) problem involving durable products with end-of-use constraints is analyzed in this paper. First, we mathematically formulate this problem, then certain properties are derived to construct the structure of the optimal solution. Next, based on these properties, a recursive optimization algorithm is proposed for a single-item problem. Moreover, an approximate algorithm is designed on the basis of the optimization algorithm, with linear computational complexity. A heuristic approach is proposed for solving the two-item DLS problem. The difficulty in solving this problem lies in its decomposition into item-level subproblems while ensuring the feasibility of the solution. The proposed technique aims to resolve this issue by combining the capabilities of Lagrangian relaxation to decompose the problem into smaller subproblems, and a genetic algorithm (GA) is used to update the Lagrangian multipliers. Further, the computational results obtained using the proposed approach are enumerated to demonstrate its effectiveness. Finally, the conclusion and remarks are given to discuss the possible future works.     

Internal Optimal Controller Synthesis for Navier–Stokes Equations

Barbu and Triggiani (Indiana Univ. Math. J. 2004; 53:1443–1494) have proposed a solution of the internal feedback stabilization problem of Navier–Stokes equations with no-slip boundary conditions. They have shown that any unstable steady-state solution can be exponentially stabilized by a finite-dimensional feedback controller with support in an arbitrary open subset of positive measure. The finite dimension of the feedback controller is minimal and is related to the largest algebraic multiplicity of the unstable eigenvalues of the linearized equation. The feedback law is obtained as a solution of a linear-quadratic control problem. In this paper, we formulate a practical algorithm implementation of the proposed stabilization approach, based on the finite element method, and demonstrate its applicability and effectiveness using an example involving the stabilization of two-dimensional Navier–Stokes equations.     

Short Communication: A note on optimal hybrid V-cycle multilevel algorithms for mixed finite element systems with penalty term

In this paper, we investigate some cost-effective hybrid V-cycle multilevel algorithms for the discrete systems that arise when a mixed finite element approach is used to solve certain second-order elliptic boundary value problems. By introducing a small penalty parameter, the perturbed indefinite system can be reduced to a symmetric positive definite system involving the unknown flux alone. We study the numerical behaviour of some hybrid V-cycle multilevel algorithms with optimal computational complexity based on the hierarchical spatial decomposition approach proposed by Cai, Goldstein and Pasciaks for the reduced system. © 1997 by John Wiley & Sons, Ltd.     

An Inverse Reliability-based Approach for Designing under Uncertainty with Application to Robust Piston Design

In this work, we propose an optimization framework for designing under uncertainty that considers both robustness and reliability issues. This approach is generic enough to be applicable to engineering design problems involving nonconvex objective and constraint functions defined in terms of random variables that follow any distribution. The problem formulation employs an Inverse Reliability Strategy that uses percentile performance to address both robustness objectives and reliability constraints. Robustness is achieved through a design objective that evaluates performance variation as a percentile difference between the right and left trails of the specified goals. Reliability requirements are formulated as Inverse Reliability constraints that are based on equivalent percentile performance levels. The general proposed approach first approximates the formulated problem via a Gaussian Kriging model. This is then used to evaluate the percentile performance characteristics of the different measures inherent in the problem formulation for various design variable settings via a Most Probable Point of Inverse Reliability search algorithm. By using these percentile evaluations in concert with the response surface methodology, a polynomial programming approximation is generated. The resulting problem formulation is finally solved to global optimality using the Reformulation–Linearization Technique (RLT) approach. We demonstrate this overall proposed approach by applying it to solve the problem of reducing piston slap, an undesirable engine noise due to the secondary motion of a piston within a cylinder.     

Real (investment) options with multiple sources of rare events

We value real (investment) options when the underlying asset follows a mixed jump-diffusion process involving various types (sources) of rare events (jumps). These jumps are assumed independent of each other, with each type having a log-normally distributed jump size and a random (Poisson-distributed) arrival time. They may represent uncertainties about the arrival and impact (on the underlying investment) of new information concerning technological innovation, competition, political risk, regulatory effects and other sources. An analytic solution is presented for European claims (call or put options) with multiple sources of jumps. A discrete-time (Markov-chain) methodology (implemented within a finite-difference scheme) is proposed for the valuation of American as well as European options. The approach is also applicable to financial options with multiple types of rare events. The approach is illustrated through valuing complex real options with compound features involving interactions between optimal investment and subsequent operating decisions. Specifically, we value a growth option and an extension option.     

A time-based pheromone approach for the ant system

The ant system (AS) is a metaheuristic approach originally developed for solving the traveling salesman problem. AS has been successfully applied to various hard combinatorial optimization problems and different variants have been proposed in the literature. In this paper, we introduce a time-based pheromone approach for AS (TbAS). Due to this nature TbAS is applicable to routing problems involving time-windows. The novelty in TbAS is the multi-layer pheromone network structure which implicitly utilizes the service time information associated with the customers as a heuristic information. To investigate the performance of TbAS, we use the well-known vehicle routing problem with time-windows as our testbed and we conduct an extensive computational study using the Solomon (Algorithms for the vehicle routing and scheduling problems with time window constraints 35:254?C265, 1987) instances. Our results reveal that the proposed time-based pheromone approach is effective in obtaining good quality solutions.