Plant location and procurement planning in knockdown production systems

We consider an integrated problem of plant location and capacity planning for components procurement in knockdown production systems. The problem is that of determining the schedule of opening components manufacturing plants, plans for acquisition of capacities in opened components manufacturing plants, and plans for components procurement in final assembly plants with the objective of minimizing the sum of fixed costs for opening plants, acquisition and operation costs of facilities, and delivery and subcontracting costs of components. The problem is formulated as a mixed integer linear program and solved by a two-stage solution procedure. In the solution procedure, the problem is decomposed into two tractable subproblems and these subproblems are solved sequentially. In the first stage, a dynamic plant location problem is solved using a cut and branch algorithm based on Gomory cuts, while a multiperiod capacity planning problem is solved in the second stage by a heuristic algorithm that uses a cut and branch algorithm and a variable reduction scheme. The solution procedure is tested on problems of a practical size and results show that the procedure gives reasonably good solutions.     

Real solutions of a problem in enumerative geometry

We study a 2-parameter family of enumerative problems over the reals. Over the complex field, these problems can be solved by Schubert calculus. In the real case the number of solutions can be different on the distinct connected components of the configuration space, resulting in a solution function. The cohomology calculation in the real case only gives the signed sum of the solutions, therefore in general it only gives a lower bound on the range of the solution function. We calculate the solution function for the 2-parameter family and we show that in the even cases the solution function is constant modulo 4. We show how to determine the sign of a solution and describe the connected components of the configuration space. We translate the problem to the language of quivers and also give a geometric interpretation of the sign. Finally, we discuss what aspects might be considered when solving other real enumerative problems.     

Scheduling of Products with Common and Product-Dependent Components Manufactured at a Single Facility

This paper analyses a scheduling problem concerned with the production of components at a single manufacturing facility where the manufactured components are subsequently assembled into a finite number of end products. Each product is composed of a common component and a product-dependent component, and completion time of a product is determined by the completion time of the last of two components. All the components are manufactured in a batch process at the single facility and, during the batch process, the manufactured components are individually moved to the next (assembly) station; switching from production of product-dependent components to common components only incurs a set-up cost. The solution properties are characterized subject to the mean flow time measure, based upon which an efficient branch-and-bound solution algorithm is exploited.     

解分数阶微分代数系统的Adomian分解方法

研究了利用Adomian分解求解分数阶微分代数系统的方法.分析了代数约束对Adomian方法求解的影响,指出直接解出代数约束变量,将原系统转化为微分系统进行Adomian分解的困难.提出确定代数变量级数解各分量的新方法,据此进行Adomian分解,得到整个系统的级数解.特别研究了代数约束为线性的分数阶微分代数系统的Adomian解法,证明了各变量间的线性代数约束关系可以转化为相应级数解中各分量的线性关系,从而方便求解,并结合具体例子证明了该方法简便有效.     

Stabilization strategies for some reaction–diffusion systems

A two-component reaction–diffusion system is considered. The question of stabilizing to zero one of the components of the solution via an internal control acting on a small subdomain and preserving nonnegativity of both components is investigated. Our results apply to predator–prey systems.     

Finding and identifying optimal inventory levels for systems with common components

In this article, we consider the problem of finding the optimal inventory level for components in an assembly system where multiple products share common components in the presence of random demand. Previously, solution procedures that identify the optimal inventory levels for components in a component commonality problem have been considered for two product or one common component systems. We will here extend this to a three products system considering any number of common components. The inventory problem considered is modeled as a two stage stochastic recourse problem where the first stage is to set the inventory levels to maximize expected profit while the second stage is to allocate components to products after observing demand. Our main contribution, and the main focus of this paper, is the outline of a procedure that finds the gradient for the stochastic problem, such that an optimal solution can be identified and a gradient based search method can be used to find the optimal solution.     

Two new algorithms for solving optimization problems with one linear objective function and finitely many constraints of fuzzy relation inequalities

This paper studies the optimization model of a linear objective function subject to a system of fuzzy relation inequalities (FRI) with the max-Einstein composition operator. If its feasible domain is non-empty, then we show that its feasible solution set is completely determined by a maximum solution and a finite number of minimal solutions. Also, an efficient algorithm is proposed to solve the model based on the structure of FRI path, the concept of partial solution, and the branch-and-bound approach. The algorithm finds an optimal solution of the model without explicitly generating all the minimal solutions. Some sufficient conditions are given that under them, some of the optimal components of the model are directly determined. Some procedures are presented to reduce the search domain of an optimal solution of the original problem based on the conditions. Then the reduced domain is decomposed (if possible) into several sub-domains with smaller dimensions that finding the components of the optimal solution in each sub-domain is very easy. In order to obtain an optimal solution of the original problem, we propose another more efficient algorithm which combines the first algorithm, these procedures, and the decomposition method. Furthermore, sufficient conditions are suggested that under them, the problem has a unique optimal solution. Also, a comparison between the recently proposed algorithm and the known ones will be made.     

Simple bounds for solutions of monotone complementarity problems and convex programs

For a solvable monotone complementarity problem we show that each feasible point which is not a solution of the problem provides simple numerical bounds for some or all components of all solution vectors. Consequently for a solvable differentiable convex program each primal-dual feasible point which is not optimal provides simple bounds for some or all components of all primal-dual solution vectors. We also give an existence result and simple bounds for solutions of monotone compementarity problems satisfying a new, distributed constraint qualification. This result carries over to a simple existence and boundedness result for differentiable convex programs satisfying a similar constraint qualification.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041. This material is based on work sponsored by National Science Foundation Grants MCS-8200632 and MCS-8102684.     

A technique to prove parameter-uniform convergence for a singularly perturbed convection–diffusion equation

A priori parameter explicit bounds on the solution of singularly perturbed elliptic problems of convection–diffusion type are established. Regular exponential boundary layers can appear in the solution. These bounds on the solutions and its derivatives are obtained using a suitable decomposition of the solution into regular and layer components. By introducing extensions of the coefficients to a larger domain, artificial compatibility conditions are not imposed in the derivation of these decompositions.     

Large Toeplitz Systems with Quasi-polynomial Right-hand Sides

We consider the asymptotics of the solutions of large linear systems with Toeplitz matrices generated by a complex valued symbol which is infinitely differentiable, has no zeros on the unit circle, and whose winding number about the origin is zero. The emphasis is on quasi-polynomials as right-hand sides, in which case we show that the central fragment of the solution is asymptotically also a quasi-polynomial. Moreover, we establish asymptotic formulas that give specific components of the solution independently of the other components. We are greatly indebted to the referees for suggesting substantial simplifications in our original proofs and the constructive advice which helped to improve the exposition.     

Final state problem for a system of nonlinear Schrödinger equations with three wave interaction

We consider the asymptotic behavior of a solution to a system of quadratic nonlinear Schrödinger equations with three wave interaction in two dimensions. We construct a particular solution which has a mass transition phenomenon among three components periodically in time. This is based on the analysis for a system of ordinary differential equations which approximates the solution of the system of nonlinear Schrödinger equations.     

A Parameter-Uniform Finite Difference Method for a Coupled System of Convection-Diffusion Two-Point Boundary Value Problems

A system of m (≥2) linear convection-diffusion two-point boundary value problems is examined,where the diffusion term in each equation is multiplied by a small parameterεand the equations are coupled through their convective and reactive terms via matrices B and A respectively.This system is in general singularly perturbed. Unlike the case of a single equation,it does not satisfy a conventional maximum princi- ple.Certain hypotheses are placed on the coupling matrices B and A that ensure exis- tence and uniqueness of a solution to the system and also permit boundary layers in the components of this solution at only one endpoint of the domain;these hypotheses can be regarded as a strong form of diagonal dominance of B.This solution is decomposed into a sum of regular and layer components.Bounds are established on these compo- nents and their derivatives to show explicitly their dependence on the small parameterε.Finally,numerical methods consisting of upwinding on piecewise-uniform Shishkin meshes are proved to yield numerical solutions that are essentially first-order conver- gent,uniformly inε,to the true solution in the discrete maximum norm.Numerical results on Shishkin meshes are presented to support these theoretical bounds.     

Numerical Homotopies to Compute Generic Points on Positive Dimensional Algebraic Sets

Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for four-bar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the components with linear subspaces in general position to obtain generic points of the components as the isolated solutions of an auxiliary system. Since this requires the solution of a number of larger overdetermined systems, the procedure is computationally expensive and also wasteful because many solution paths diverge. In this article an embedding of the original polynomial system is presented, which leads to a sequence of homotopies, with solution paths leading to generic points of all components as the isolated solutions of an auxiliary system. The new procedure significantly reduces the number of paths to solutions that need to be followed. This approach has been implemented and applied to various polynomial systems, such as the cyclic n-roots problem.     

Antiplane elastodynamic analysis of a strip weakened by multiple defects

The method of images is utilized to derive the solution of a screw dislocation under time-harmonic conditions for an elastic strip from the solution of infinite planes. The displacement and stress components are obtained for a strip under concentrated antiplane, time-harmonic traction. The dislocation solution is employed to formulate integral equation for a strip weakened by cracks and cavities. The effects of load frequency and crack orientation on the stress intensity factors are studied.     

Numerical analysis of a strongly coupled system of two singularly perturbed convection–diffusion problems

A system of two coupled singularly perturbed convection–diffusion ordinary differential equations is examined. The diffusion term in each equation is multiplied by a small parameter, and the equations are coupled through their convective terms. The problem does not satisfy a conventional maximum principle. Its solution is decomposed into regular and layer components. Bounds on the derivatives of these components are established that show explicitly their dependence on the small parameter. A numerical method consisting of simple upwinding and an appropriate piecewise-uniform Shishkin mesh is shown to generate numerical approximations that are essentially first order convergent, uniformly in the small parameter, to the true solution in the discrete maximum norm.      

非线性积分微分方程组解的振动性

本文给出了一类积分微分方程组存在和不存在正解的若干充分条件．     

Some inverse optimization problems under the Hamming distance

Given a feasible solution to a particular combinatorial optimization problem defined on a graph and a cost vector defined on the arcs of the graph, the corresponding inverse problem is to disturb the cost vector such that the feasible solution becomes optimal. The aim is to optimize the difference between the initial cost vector and the disturbed one. This difference can be measured in several ways. We consider the Hamming distance measuring in how many components two vectors are different, where weights are associated to the components. General algorithms for the bottleneck or minimax criterion are described and (after modification) applied to the inverse minimum spanning tree problem, the inverse shortest path tree problem and the linear assignment problem.     

Joint reliability importance of k-out-of-n systems

Joint Reliability Importance (JRI) is investigated to provide information on the type and the degree of interactions between two components by identifying the sign and the size of it. In specific, JRI is analyzed for k-out-of-n systems with (i) independently identically distributed (IID) components, (ii) non-identical and independent components and (iii) pair-wise dependent components. The closed form solution of JRI is derived for IID components. By using the solution, the variations of JRI are investigated with respect to the level of redundancy (n/k). In non-identical and independent case, the point of sign change of JRI is obtained. It is also shown that the sign of JRI can be determined by the relationships between the Schur-convexity (concavity) and the JRI. For dependent case, only the pair-wise dependence is considered. As a result, the error caused by assuming statistical independence between pair-wise dependent components is shown to be measured by their covariance and JRI.     

Worst scenario method in homogenization. Linear case

The paper deals with homogenization of a linear elliptic boundary problem with a specific class of uncertain coefficients describing composite materials with periodic structure. Instead of stochastic approach to the problem, we use the worst scenario method due to Hlaváček (method of reliable solution). A few criterion functionals are introduced. We focus on the range of the homogenized coefficients from knowledge of the ranges of individual components in the composite, on the values of generalized gradient in the places where these components change and on the average of homogenized solution in some critical subdomain. This research was supported by grant No. 201/03/0570 of the Grant Agency of the Czech Republic.     

Exact solution of the one-dimensional time-dependent Schrödinger equation with a rectangular well/barrier potential and its applications

We obtain an exact one-dimensional time-dependent solution for a wave function ψ(x, t) of a particle moving in the presence of a rectangular well or barrier. We present the solution, which holds for both the well and the barrier, in terms of the integrals of elementary functions; it is the sum of forward- and backward-moving components of the wave packet. We consider and numerically visualize the relative contribution of these components and of their interference to the probability density |ψ(x, t)|² and the particle arrival time and dwell time for the narrow and broad energy (momentum) distributions of the initial Gaussian wave packet. We show that in the case of a broad initial wave packet, the quantum mechanical counterintuitive effect of the influence of the backward-moving components on the considered quantities becomes essential.