Fuzzy mixture two warehouse inventory model involving fuzzy random variable lead time demand and fuzzy total demand

This paper considers a two-warehouse fuzzy-stochastic mixture inventory model involving variable lead time with backorders fully backlogged. The model is considered for two cases—without and with budget constraint. Here, lead-time demand is considered as a fuzzy random variable and the total cost is obtained in the fuzzy sense. The total demand is again represented by a triangular fuzzy number and the fuzzy total cost is derived. By using the centroid method of defuzzification, the total cost is estimated. For the case with fuzzy-stochastic budget constraint, surprise function is used to convert the constrained problem to a corresponding unconstrained problem in pessimistic sense. The crisp optimization problem is solved using Generalized Reduced Gradient method. The optimal solutions for order quantity and lead time are found in both cases for the models with fuzzy-stochastic/stochastic lead time and the corresponding minimum value of the total cost in all cases are obtained. Numerical examples are provided to illustrate the models and results in both cases are compared.     

Simulating Prosthetic Gait ‐ Lessons to Learn

Modern computer assisted above knee prostheses currently available on the market (e.g. C‐Leg from Otto‐Bock) allow the patients to choose their walking speed freely during level walking. Due to energy reasons it is only possible to change the passive mechanical characteristics of the prosthetic limb but no active actuators can be used to influence the knee joint movement. The control strategies for level walking act on the damping of the knee joint during swing phase of gait. During walking down ramps or descending stairs the control strategies are switched off in current designs. Based on a measured reference kinematics it should be possible to study the influence and behaving of technical devices in combination with the whole body biomechanical system. For this purpose we use a combined inverse‐forward‐dynamics approach. Kinematics parameters that are not critical in the inverse‐dynamics approach (leading to physiological reasonable forces and moments) and that are not influenced by the other parameters in the model are prescribed as in the classical inverse‐dynamics approach. To follow the other kinematics parameters (joint‐angles) control elements are used where the associated joint‐moments are determined by the goal trajectory and physiological boundary conditions. The so called “controlled joints” of the model and the applied technical device (prosthesis) are formulated as a forward‐dynamics system. In contrast to experimental studies a simulation study offers the advantage, that parameters can be varied in a wide range. Different ramp and stair inclinations can be implemented and different control strategies can be tested without any risk for the patient. The disadvantage lies in the fact, that there are only few reference sets for the kinematics of real patients. The modification of a reference kinematics for different situations (level walking, ramp or stair descent) is a big challenge and needs further investigation. On the other hand the combined inverse‐forward‐dynamics approach with its well balanced simplicity and complexity offers a flexible tool to reach this aim. This approach can also be used to investigate the interaction of the biomechanical system with other technical systems like sports equipment.     

On generalization of the analysis of variance

The standard analysis of variance procedures were developed and organized primarily in the context of the normal linear model; central to this organization is the orthogonality of components and the use of orthogonal projections. This paper examines two model-type generalizations of the normal linear model: the regression model with nonnormal error and the exponential linear model. Principles of conditioning and measurement are used to develop corresponding analysis-of-variance procedures. In each case a linear fibre or foliation structure replaces orthogonality; however, for the intersection of the two model-types, which is the normal linear model, the two quite-different fibre-foliation structures reduce to a product space structure, which with the appropriate inner product, is the usual orthogonality. For implementation, conditional-marginal densities are involved, the marginalization aspect being the restricting aspect: the marginalization degree is the number of nuisance parameters for the regression model-type and is the complement of the number of free parameters for the exponential model-type. Approximations are available and will be discussed subsequently.University of WaterlooUniversity of Toronto     

A perimeter-based angle measure in Minkowski planes

Measuring angles in the Euclidean plane is a well-known topic, but for general normed planes there exists a variety of different concepts. These can be of a special kind, e.g. also preserving special orthogonality types. But these concepts are no angle measures in the sense of measure theory since they are not additive. This motivates us to define a new angle measure for normed planes that is in fact a measure in the sense of measure theory. Furthermore, we look at related types of rotation and reflection.     

Modular solvers for image restoration problems using the discrepancy principle

Many problems in image restoration can be formulated as either an unconstrained non‐linear minimization problem, usually with a Tikhonov‐like regularization, where the regularization parameter has to be determined; or as a fully constrained problem, where an estimate of the noise level, either the variance or the signal‐to‐noise ratio, is available. The formulations are mathematically equivalent. However, in practice, it is much easier to develop algorithms for the unconstrained problem, and not always obvious how to adapt such methods to solve the corresponding constrained problem. In this paper, we present a new method which can make use of any existing convergent method for the unconstrained problem to solve the constrained one. The new method is based on a Newton iteration applied to an extended system of non‐linear equations, which couples the constraint and the regularized problem, but it does not require knowledge of the Jacobian of the irregularity functional. The existing solver is only used as a black box solver, which for a fixed regularization parameter returns an improved solution to the unconstrained minimization problem given an initial guess. The new modular solver enables us to easily solve the constrained image restoration problem; the solver automatically identifies the regularization parameter, during the iterative solution process. We present some numerical results. The results indicate that even in the worst case the constrained solver requires only about twice as much work as the unconstrained one, and in some instances the constrained solver can be even faster. Copyright © 2002 John Wiley & Sons, Ltd.     

Two-step generation of the equations of motion of closed-chains

This paper presents a two-step generation of the equations of motion of planar mechanisms using point and joint coordinates. First, the formulation replaces a rigid body by a dynamically equivalent constrained system of particles and uses Newton’s second law to study the motion of the particles without introducing any rotational coordinates. Then, the equations of motion are transformed to a reduced set in terms of selected relative joint variables using a velocity transformation matrix. For an open-chain, this process automatically eliminates all of the non-working constraint forces and leads to an efficient integration of the equations of motion. For a closed-chain, suitable joints should be cut and few cut-joints constraint equations are included. An example of a closed-chain is used to demonstrate the generality and efficiency of the proposed method.     

求解稀疏逻辑回归问题的嵌套BB算法的分裂增广拉格朗日算法

逻辑回归是经典的分类方法,广泛应用于数据挖掘、机器学习和计算机视觉.现研究带有程。模约束的逻辑回归问题.这类问题广泛用于分类问题中的特征提取,且一般是NP-难的.为了求解这类问题,提出了嵌套BB(Barzilai and Borwein)算法的分裂增广拉格朗日算法(SALM-BB).该算法在迭代中交替地求解一个无约束凸优化问题和一个带程。模约束的二次优化问题.然后借助BB算法求解无约束凸优化问题.通过简单的等价变形直接得到带程。模约束二次优化问题的精确解,并且给出了算法的收敛性定理.最后通过数值实验来测试SALM-BB算法对稀疏逻辑回归问题的计算精确性.数据来源包括真实的UCI数据和模拟数据.数值实验表明,相对于一阶算法SLEP,SALM-BB能够得到更低的平均逻辑损失和错分率.     

基于坐标的既有铁路曲线整正约束优化算法研究

既有铁路曲线整正是既有线改建设计中的重要部分,且结果直接影响最终设计质量和运营安全.基于最优化思想直接利用既有线上测点坐标进行曲线整正.构建了体现曲线整正成果优劣的目标函数,考虑了规范约束和控制点约束,建立了曲线整正约束最优化计算模型.基于罚函数的思想将曲线整正的非线性约束最优化问题转换为无约束最优化问题.根据目标函数的特点,采用N elder-M ead单纯形法迭代求解最优值.该算法逻辑简单,应用方便.应用结果表明算法可优化出拨距小,且满足约束条件的曲线整正成果,具有较强的实用性.     

A combined algorithm for identification and approximation

The optimum point of an unknown function which is measured with noise and the unknown parameters in the measurement noise are estimated by a combined algorithm. The almost sure convergence and the convergence rate are established for both unconstrained and constrained problems.Project supported by the National Natural Science Foundation of China.     

Correction of stiffness and mass matrices utilizing simulated measured modal data

Measured and analytical data are unlikely to be equal due to measured noise, model inadequacies and structural damage, etc. It is necessary to update the physical parameters of analytical models for proper simulation and design studies. Starting from simulated measured modal data such as natural frequencies and their corresponding mode shapes, this study presents the equations to update the physical parameters of stiffness and mass matrices simultaneously for analytical modelling by minimizing a cost function in the satisfaction of the dynamic constraints of orthogonality requirement and eigenvalue function. The proposed equations are straightforwardly derived by Moore–Penrose inverse matrix without using any multipliers. The cost function is expressed by the sum of the quadratic forms of both the difference between analytical and updated mass, and stiffness matrices. The results are compared with the updated mass matrix to consider the orthogonality requirement only and the updated stiffness matrix to consider the eigenvalue function only, respectively. Also, they are compared with Wei’s method which updates the mass and stiffness matrices simultaneously. The validity of the proposed method is illustrated in an application to correct the mass and stiffness matrices due to section loss of some members in a simple truss structure.     

Search algorithm for optimal damping parameters of transfemoral prosthetic limb

In this paper, we developed a mathematical model to find the parameters of prosthetic damper which will provide a similar trajectory for the prosthetic knee joint. Two popular searching methods namely grid searching and optimization are used to determine the damper's parameters. The proposed model is validated with a simulation process using the data of the able-body individuals. We utilized the ground reaction force of sound limb to determine the values of the damper parameters of a prosthetic knee joint for maximum symmetry. Symmetry between knee moments was also improved in the stance period with optimized parameters. Finally, optimization-based searching was observed to be more computationally efficient than the grid-based searching method. The present study will provide a virtual solution to set the prosthetic dampers parameters based on user needs. In the future, the present method can be used for adjusting damping of microprocessor prosthetic knee joint for symmetrical gait pattern.     

A continuous time,deterministic, nonstationary model of economic ordering

The paper considers the problem of economic ordering for a deterministic, nonstationary environment in continuous time. Previous work on the topic is reviewed. The specification of the cost criterion common in inventory theory is called in question for nonstationary situations as far as interest cost is concerned. It is proposed to account for interest by discounting rather than in a holding cost expression. The main interest of the paper is in three versions of the problem: First an unconstrained version, for which inventory is allowed to become negative (backlogging model), second a model in which inventory is constrained to be nonnegative (non-backlogging model), and third a nonbacklogging model with a storage space constraint. For the first two problems necessary optimality conditions are derived which are based on control theory for continuous time systems with jumps in the state trajectories, especially on Blaquière's impulsive maximum principle. These conditions reduce the problem of finding an optimal ordering plan, i.e. an unknown number of optimal ordering times and for each of them an optimal order size to a one parameter search problem. Due to the possibility of multiple solutions of the optimality conditions for each ordering time, one cannot in general identify a unique candidate ordering plan for each value of the search parameter, but only a tree-structured set of such plans. The optimality conditions for the first two problem versions and for a fourth one with a storage space constraint but without a non-backlogging constraint are eventually combined to yield a solution of the storage space constrained non-backlogging version.     

Noise analysis and suppression method in attitude determination using the global positioning system (GPS)

A noise suppression method is developed for attitude determination using the global positioning system. The influence of noise on attitude determination application is analyzed to determine the relationship of errors. In order to suppress the noise, the total least squares method is utilized for double difference carrier phase measurements and unit vectors between satellites and antennas. Experimental results indicate that compared to the traditional least squares method without noise suppression, the accuracy of the measurement of attitude angles is increased by about 30-50%. The increased computation time of this method does not significantly influence the real time performance for land vehicle application.     

Inventory control with an order-time constraint: optimality,uniqueness and significance

This paper analyzes a stochastic inventory problem with an order-time constraint that restricts the times at which a manufacturer places new orders to a supplier. This constraint stems from the limited upstream capacity in a supply chain, such as production capacity at a supplier or transportation capacity between a supplier and a manufacturer. Consideration of limited upstream capacity extends the classical inventory literature that unrealistically assumes infinite supplier/transporter capacity. But this consideration increases the complexity of the problem. We study the constraint under a Poisson demand process and allow for a fixed ordering cost. In presence of the constraint, we establish the optimality of an (s,S) policy under both the discounted and average cost objectives. Under the average cost objective, we show the uniqueness of the order-up-to level S. We numerically compare our model with the classical unconstrained model. We report significant savings in costs that can be achieved by using our model when the order time is constrained.     

Convergence analysis of a conforming adaptive finite element method for an obstacle problem

The adaptive algorithm for the obstacle problem presented in this paper relies on the jump residual contributions of a standard explicit residual-based a posteriori error estimator. Each cycle of the adaptive loop consists of the steps ‘SOLVE’, ‘ESTIMATE’, ‘MARK’, and ‘REFINE’. The techniques from the unrestricted variational problem are modified for the convergence analysis to overcome the lack of Galerkin orthogonality. We establish R-linear convergence of the part of the energy above its minimal value, if there is appropriate control of the data oscillations. Surprisingly, the adaptive mesh-refinement algorithm is the same as in the unconstrained case of a linear PDE—in fact, there is no modification near the discrete free boundary necessary for R-linear convergence. The arguments are presented for a model obstacle problem with an affine obstacle χ and homogeneous Dirichlet boundary conditions. The proof of the discrete local efficiency is more involved than in the unconstrained case. Numerical results are given to illustrate the performance of the error estimator.     

Nonnegative non-redundant tensor decomposition

Nonnegative tensor decomposition allows us to analyze data in their ‘native’ form and to present results in the form of the sum of rank-1 tensors that does not nullify any parts of the factors. In this paper, we propose the geometrical structure of a basis vector frame for sum-of-rank-1 type decomposition of real-valued nonnegative tensors. The decomposition we propose reinterprets the orthogonality property of the singularvectors of matrices as a geometric constraint on the rank-1 matrix bases which leads to a geometrically constrained singularvector frame. Relaxing the orthogonality requirement, we developed a set of structured-bases that can be utilized to decompose any tensor into a similar constrained sum-of-rank-1 decomposition. The proposed approach is essentially a reparametrization and gives us an upper bound of the rank for tensors. At first, we describe the general case of tensor decomposition and then extend it to its nonnegative form. At the end of this paper, we show numerical results which conform to the proposed tensor model and utilize it for nonnegative data decomposition.     

A Penalized Likelihood Approach to Rotation of Principal Components

A new paradigm for enhancing the interpretability of principal components through rotation is presented within the framework of penalized likelihood. The rotated components are computed as the maximizers of a Gaussian-based profile log-likelihood function plus a penalty term defined by a standard rotation criterion. This method enjoys a number of advantages over other methods for principal component rotation, notably (1) the rotation specifically targets ill-defined principal components, which may benefit the most from rotation, and (2) the connection with likelihood allows assessment of the fidelity of the rotated components to the data, thereby guiding the choice of penalty parameter. The method is illustrated with an application to a small functional dataset. Efficient computation of the penalized likelihood solution is possible using recently developed algorithms for optimization under orthogonality constraints.     

Optimal solutions to differential inclusions in presence of state constraints

Necessary conditions in terms of the Hamiltonian are given for optimal solutions to the differential inclusion problem when state constraints are present. This result extends a result of Clarke for the unconstrained problem. The data are nonsmooth, nonlinear, nonconvex. The method incorporates the state constraint in the cost functional as a penalty term for a sequence of unconstrained problems that approximate our problem. An application of Ekeland's variational principle, the known necessary conditions for the auxiliary problems, and a limiting process provide the necessary conditions.     

矩形到任意多边形区域的Schwarz-Christoffel变换数值解法

运用Schwarz-Christoffel变换方法，建立多边形区域到带状区域共形映射数学模型.对于模型中的约束条件和奇异积分问题，根据Riemann(黎曼)原理，建立复参数与实参数互逆变换，消除非线性系统的约束条件；经过合理积分路径的确定，模型中的奇异积分转化为Gauss-Jacobi(高斯 雅可比)型积分；采用Levenberg-Marquardt算法对非线性系统模型进行求解.根据第一类椭圆函数性质，建立了矩形区域到带状区域共形映射数学模型，通过复参数椭圆函数的计算，得到矩形边界与带状区域边界的关系.最后，对8点对称多边形区域与27点不规则条带状区域计算，将不规则封闭区域边界映射到矩形区域边界，矩形区域内的正交网格，通过变换之后在多边形区域内依然满足正交性，为研究不规则区域到规则区域映射的数值计算奠定基础.