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Multiobjective Meta-Heuristic Product Scheduling for Multi-Machine Manufacturing Systems
Get PDFFlow line are one of the most commonly encountered layouts in manufacturing industries, where several product types (grades) are manufactured using a sequence of sub-systems or machinery with different tasks. With increasing prices of energy and speci�c customer demands employing effective product scheduling strategies has become essential for manufacturing industries to maintain their business viability. In this paper, a new product scheduling method is proposed for multi-machine, multi-product �ow lines. The objective here is to control the production start time for each grade so that the product delivery time errors are minimised. It is also desired to minimise the overall makespan variability caused by non-Gaussian uncertainties formulated by the entropy of the delivery time errors. Therefore, the proposed product scheduling strategy is a nonlinear multi-objective optimisation problem with non-Gaussian uncertainties. To solve this problem, the nonlinear dynamic �ow line model is converted to a linear dynamic equivalent using a (Max,+) algebraic approach. Then, a Proportional-Integral (PI) scheduling controller is used to control the production start time for each grade. The scheduling controller coef�cients are tuned by a MultiObjective Differential Evolution (MODE) algorithm. Simulation
results show the effectiveness of the proposed technique and a comparison is made between MODE, Genetic Algorithm (GA) and Particle Swarm Optimisation (PSO)
iGen 0.1: a program for the automated generation of models and parameterisations
Get PDFComplex physical systems can often be simulated using very high resolution models but this is not always practical because of computational restrictions. In this case the model must be simplified or parameterised in order to make it computationally tractable. A parameterised model is created using an ad-hoc selection of techniques which range from the formal to the purely intuitive, and as a result it is very difficult to objectively quantify the fidelity of the model to the physical system. It is rare that a parameterised model can be formally shown to simulate a physical system to within some bounded error. Here we introduce a new approach to parameterising models which allows error to be formally bounded. The approach makes use of a newly developed computer program, which we call iGen, that analyses the source code of a high-resolution model and formally derives a much faster, parameterised model that closely approximates the original, reporting bounds on the error introduced by any approximations. These error bounds can be used to formally justify conclusions about a physical system based on observations of the model's behaviour. Using increasingly complex physical systems as examples we illustrate that iGen has the ability to produce parameterisations that run typically orders of magnitude faster than the underlying, high-resolution models from which they are derived
NLEVP: A Collection of Nonlinear Eigenvalue Problems. Users' Guide
No full textThis is the Users' Guide for NLEVP: a collection of nonlinear eigenvalue problems provided in the form of a MATLAB toolbox. A separate paper describes the collection and its organization
Roots of Stochastic Matrices and Fractional Matrix Powers
Get PDFIn Markov chain models in finance and healthcare a transition matrix over a certain time interval is needed
but only a transition matrix over a longer time interval may be available. The problem arises of determining a stochastic th root of a stochastic matrix (the given transition matrix). By exploiting the theory of functions of matrices, we develop results on the existence and characterization of stochastic th roots. Our contributions include characterization of when a real matrix hasa real th root, a classification of th roots of a possibly singular matrix, a sufficient condition for a th root of a stochastic matrix to have
unit row sums, and the identification of two classes of
stochastic matrices that have stochastic th roots for all . We also delineate a wide variety of possible configurations as regards existence, nature (primary or nonprimary), and number of stochastic roots, and develop a necessary condition for existence of a stochastic root in terms of the spectrum of the given matrix.
On the computational side, we emphasize finding an approximate stochastic root: perturb the principal root or the principal logarithm to the nearest stochastic matrix or the nearest intensity matrix, respectively, if they are not valid ones; minimize the residual over all stochastic matrices and also over stochastic matrices that are primary functions
of . For the first two nearness problems, the global minimizers are found in the Frobenius norm. For the last two nonlinear programming problems, we derive explicit formulae for the gradient and Hessian of the objective function and investigate Newton's method, a spectral projected gradient method (SPGM) and the sequential quadratic programming method to solve the problem as well as various matrices to start the iteration.
Numerical experiments show that SPGM starting with the perturbed to minimize over all stochastic matrices is method of choice.
Finally, a new algorithm is developed for computing arbitrary real powers of a matrix . The algorithm starts with a Schur decomposition,
takes square roots of the triangular factor , evaluates an Pad\'e approximant of at , and squares the result times.
The parameters and are chosen to minimize the cost subject to achieving double precision accuracy in the valuation of the Pad\'e approximant, making use of a result that bounds the error in the matrix Pad\'e approximant by the error in the scalar Pad\'e approximant with argument the norm of the matrix. The Pad\'e approximant is evaluated from the continued fraction representation in bottom-up fashion, which is shown to be numerically stable. In the squaring phase the diagonal and first superdiagonal are computed from explicit formulae for , yielding increased accuracy. Since the basic algorithm is designed for , a criterion for reducing an arbitrary real to this range is developed, making use of bounds for the condition number of the problem. How best to compute for a negative integer is also investigated. In numerical experiments the new algorithm is found to be superior in accuracy and stability to several alternatives, including the use of an eigendecomposition, a method based on the Schur--Parlett algorithm with our new algorithm applied to the diagonal blocks and approaches based on the formula
iGen: The automated generation of a parameterisation of entrainment in marine stratocumulus
No full textIn a previous paper we described a new technique for automatically generating parameterisations using a program called iGen. iGen generates parameterisations by analysing the source code of a high resolution model that resolves the physics to be parameterised. In order to demonstrate that this technique works with a model of realistic complexity we have used iGen to generate a parameterisation of entrainment in marine stratocumulus. We present details of our technique in which iGen was used to analyse the source code of a cloud resolving model and generate a parameterisation of the mean and standard deviation of entrainment velocity in marine stratocumulus in terms of the large-scale state of the boundary layer. The parameterisation was tested against results from the DYCOMS-II intercomparison of cloud resolving models and the parameterisation of mean entrainment velocity was found to be compared to for the ensemble of cloud resolving model
Dynamic Optimal Scheduling Method and Its Application for Converter Fault in Steelmaking and Continuous Casting Production Process
No full textIn steelmaking and continuous casting (SMCC) production process, converter fault can lead to unexpected changes to the pre-specified converter-continuous caster production mode so that the original scheduling plan becomes unrealizable. In this paper, the dynamic scheduling problem in response to converter fault is firstly analyzed. This is then followed by the establishment of a novel multi-objective nonlinear programming model (MONPM) by introducing the production mode parameter α, production schedule parameter β and �. The proposed method considers changes in production mode, production schedule of charge, the interval characteristics of processing time. In specific, a two-stage dynamic optimal scheduling method is proposed including the production path planning of charges (PPP) and the production time scheduling (PTS). As a result, a dynamic optimal scheduling software system (DOSSS)
is developed and is successfully applied to the scheduling of the largest iron and steel company (BaoSteel)
in China. The real-time application shows that the proposed method can efficiently reduce scheduling time, significantly increase the outputs of converters and dramatically shorten the redundant waiting time for molten steel
An Optimal Iterative Solver for Symmetric Indefinite Systems stemming from Mixed Approximation
No full textWe discuss the design and implementation of a suite of functions for solving symmetric indefinite linear systems associated with mixed approximation of systems of PDEs. The novel feature of our iterative solver is the incorporation of error control in the natural "energy" norm in combination with an a posteriori estimator for the PDE approximation error. This leads to a robust and optimally efficient stopping criterion: the iteration is terminated as soon as the algebraic error is insignificant compared to the approximation error. We describe a "proof of concept" MATLAB implementation of this algorithm and we illustrate its effectiveness when integrated into the Incompressible Flow Iterative Solution Software (IFISS) package (cf. ACM Transactions on Mathematical Software 33, Article 14, 2007)
The Hausdorff dimension of some random invariant graphs
No full textWeierstrass� example of an everywhere continuous but nowhere
differentiable function is given by w(x) =
P1
n=0 n cos 2bnx where
2 (0, 1), b 2, b > 1. There is a well-known and widely accepted,
but as yet unproven, formula for the Hausdorff dimension of the graph
of w. Hunt [H] proved that this formula holds almost surely on the
addition of a random phase shift. The graphs of Weierstrass-type functions
appear as repellers for a certain class of dynamical system; in this
note we prove formulae analogous to those for random phase shifts of
w(x) but in a dynamic context. Let T : S1 ! S1 be a uniformly expanding
map of the circle. Let : S1 ! (0, 1), p : S1 ! R and define
the function w(x) =
P1
n=0 (x)(T(x)) · · · (Tn�1(x))p(Tn(x)). The
graph of w is a repelling invariant set for the skew-product transformation
T(x, y) = (T(x), (x)�1(y�p(x))) on S1�R and is continuous but
typically nowhere differentiable. With the addition of a random phase
shift in p, and under suitable hypotheses including a partial hyperbolicity
assumption on the skew-product, we prove an almost sure formula
for the Hausdorff dimension of the graph of w using a generalisation
of techniques from [H] coupled with thermodynamic formalism
Universal scaling of rotation intervals for quasi-periodically forced circle maps
No full textWe introduce a simplifying assumption which makes it possible to approximate the rotation number of an invertible quasi-periodically forced circle map by an integral in the limit of large forcing. We use this to describe universal scaling laws for the width of the non-trivial rotation interval of non-invertible quasi-periodically forced circle maps in this limit, and compare the results with numerical simulations