Block Diagonal Structure Research Articles

An improved neural network leader algorithm for part-machine grouping in group technology

In this paper, a robust neural network-based leader algorithm is proposed for the part-machine grouping problem in group technology. This clustering method involves a modification to the normal use of Carpenter and Grossberg's ART1 neural network. The robustness of the modified algorithm to random ordering of the input data is tested using three data sets. The data sets include an industry-size problem consisting of 1000 parts and 100 machine types. The experiments revealed that the method results in the identification of cluster and block diagonal structures rapidly and to a good degree of perfection, even for large, industry-size data sets. The solutions obtained were also found to be robust to the order of presentation of the input data. The proposed method offers a promising solution to a cellular manufacturing problem that is yet to be solved satisfactorily.

Increasing the efficiency of pooled estimation with a block-diagonal covariance structure

A small number of time-series observations relative to regions precludes estimation of the entire structure of regional dependence in a pooled regression model. The resulting need for parsimonious models of regional dependence can be satisfied through the use of spatial autocorrelation structures. This article explores an alternative methodology that allows the researcher to estimate disturbance covariances for regions that are closely linked, even when the number of time-series observations is relatively low. The approach presented here shares the advantage of spatial autocorrelation structures in being parsimonious, but offers the additional advantage of relying more completely on sample information to provide estimates of dependence between regions within specified regional groups. Monte Carlo experiments suggest that block-covariance models offer substantial efficiency gains over simple heteroskedastic models. The experiments also suggest that when the number of time-series observations is limited and the correlations of disturbances between regions are small, block structures yield efficiency gains over a full-information model.

Exploiting special structure in a primal—dual path-following algorithm

A primal-dual path-following algorithm that applies directly to a linear program of the form, min{ctxźAx = b, Hx źu, x ź 0,x ź źn}, is presented. This algorithm explicitly handles upper bounds, generalized upper bounds, variable upper bounds, and block diagonal structure. We also show how the structure of time-staged problems and network flow problems can be exploited, especially on a parallel computer. Finally, using our algorithm, we obtain a complexity bound of O( $$\sqrt n $$ ds2 log(nk)) fortransportation problems withs origins,d destinations (s

A diagonal quadratic approximation method for large scale linear programs

An augmented Lagrangian method is proposed for handling the common rows in large scale linear programming problems with block-diagonal structure and linking constraints. Using a diagonal quadratic approximation of the augmented Lagrangian one obtains subproblems that can be readily solved in parallel by a nonlinear primal-dual barrier method for convex separable programs. The combined augmented Lagrangian/barrier method applies in a natural way to stochastic programming and multicommodity networks.

A New Stability Robustness Test for Additively Perturbed Interconnected Systems

Abstract This paper considers the stability robustness of additively perturbed systems using the small-μ methodology. The uncertainty Δ(s) in the plant may perturb all the blocks of the moninal value PO(s), and the norm of any block in Δ(s) is bounded by σ ¯ Δ i j j ω ≤ δ j ω for some RH∞ -function δ(s). The system is first transformed into an equivalent form in which the uncertainty has a block diagonal structure. The stability robustness is then characterized in terms of the structured singular value of a transfer function matrix H. Some computational aspect of μ(H) will be considered.

Machine-component cell formation in group technology: a neural network approach

SUMMARY This paper presents a neural network clustering method for the part-machine grouping problem in group technology. Among the several neural networks, a Carpenter-Grossberg network is selected due to the fact that this clustering method utilizes binary-valued inputs and it can be trained without supervision. It is shown that this adaptive leader algorithm offers the capability of handling large, industry-size data sets due to the computational efficiency. The algorithm was tested on three data sets from prior literature, and solutions obtained were found to result in block diagonal forms. Some solutions were also found to be identical to solutions presented by others. Experiments on larger data sets, involving 10000 parts by 100 machine types, revealed that the method results in the identification of clusters with fast execution times. If a block diagonal structure existed in the input data, it was identified to a good degree of perfection. It was also found to be efficient with some imperfections i...

A preconditioned conjugate gradient Uzawa‐type method for the solution of the stokes problem by mixed Q1–P0 stabilized finite elements

AbstractWe study the behaviour of a conjugate gradient Uzawa‐type method for a stabilized finite element approximation of the Stokes problem. Many variants of the Uzawa algorithm have been described for different finite elements satisfying the well‐known Inf‐Sup condition of Babuška and Brezzi, but it is surprising that developments for unstable ‘low‐order’ discretizations with stabilization procedures are still missing. Our paper is presented in this context for the popular (so‐called) Q1–P0 element.First we show that a simple stabilization technique for this element permits us to retain the property of a convergence factor bounded independently of the discretization mesh size. The second contribution of this work deals with the construction of a less costly preconditioner taking full advantages of the block diagonal structure of the stabilization matrix. Its efficiency is supported by 2D and. 3D numerical results.

A close neighbour algorithm for designing cellular manufacturing systems

The first step in creating a cellular manufacturing system is to identify machine groups and form part families. Clustering and data organization (CDR) algorithms (such as the bond energy algorithm) and array sorting (ARS) methods (such as the rank order clustering algorithm) have been proposed to solve the machine and part grouping problem. However, these methods do not always produce a solution matrix that has a block diagonal structure, making visual identification of machine groups and part families extremely difficult. This paper presents a ‘close neighbour algorithm’ to solve this problem. The algorithm overcomes many deficiencies of the CDR and ASM methods. The algorithm is tested against ten existing algorithms in solving test problems from the literature. Test results show that the algorithm is very reliable and efficient.

Simulation investigation of the stochastic behaviour of selected group technology similarity coefficients

Abstract Abstract. The problem of restructuring manufacturing facilities into a cellular format has been addressed using a variety of approaches, including block-diagonalization of the machine-component matrix, mathematical programming formulations and cluster analyses. This paper addresses the problem of assessing whether or not the machine-component mix of a particular facility is amenable to cellular restructuring. The focus is cluster analytic; specifically a number of recently developed similarity coefficients are applied to problems generated via Monte Carlo simulation. The results show that, in a number of cases, the quality of the original block diagonal structure had a significant effect on the shape (in particular, skewncss and kurtosis) of the distribution of similarity coefficient values.

Efficient Algorithms for Solving Systems of Nonlinear Equations with a Block Diagonal Structure

The purpose of this paper is to present efficient algorithms for computing solutions to systems of nonlinear equations with a block diagonal structure. The special structure of the systems, which arises in economic models, allows us to use the decomposition of the problems.

Multivariate outlier tests with structured co variance matrices

Outlier tests are developed for multivariate data where there is a structure to the covariance or correlation matrix. Particular structures considered are the block diagonal structure where there are reasons to assume that one set of variables is independent of another, and the equicorrelation structure where it may be assumed that all pairs of variables have the same correlation. Likelihood ratio tests for an outlier are derived for these situations and critical values, under the null hypothesis of no outliers present, are determined for selected sample sizes and dimensions, using Bonferroni bounds or simulation. The powers of the tests are compared with those of the Wilks′ statistic for a variety of situations. It is shown that the test procedures which incorporate knowledge of the correlation structure have considerably greater power than the usual tests particularly in relatively small samples with several dimensions.

A matrix geometric solution to a resequencing problem

Consider a M/ M/2/ B queue with heterogenous servers which operates under the resequencing constraint that customers should leave the system in the order in which they entered it. A matrix geometric solution to the steady state buffer occupation probabilities of this system is provided by noting that the infinitesmal generator matrix possesses a block diagonal structure.

Decentralized stabilization and pole assignment for general proper systems

Considered is the problem of finding existence conditions and a controller synthesis procedure, using decentralized control, for assigning the poles of a linear time-invariant proper system described by a state-space model (C, A, B, D), where no assumption is made regarding the structure of D. This problem has direct application to the decentralized stabilization problem, decentralized robust servomechanism problem, etc., and is a nontrivial extension to the standard decentralized problem where it is assumed that the direct feedthrough terms either are absent or have a block-diagonal structure. >

Block diagonal structure in discrete transforms

The author investigates and summarises some of the computational tasks of discrete transforms in which block diagonal structure plays a dominant role. Walsh-Hadamard transform (WHT) based algorithm designs for various well known discrete transforms are presented; it can be proved that, owing to their block diagonal structure, the WHT based discrete transforms are more efficient than those of the conventional radix-r algorithms for transforms of length N<or=64. It is proved that block diagonal structures exist in the running Walsh-Hadamard transform, the running discrete Hartley transform (DHT), and the running discrete cosine transform (OCT). With regard to block diagonal structure in the transform conversions between DHT and DCT, some existing research results are summarised, and an efficient architecture for generating multiple discrete transform simultaneously is proposed. The two-stage DFT algorithm proposed by O.K. Ersoy (1987) is extended to that of the DHT and it is proved that two stage DHTs possess a somewhat more interesting 'balanced-block-diagonal' structure. In the context of VLSI system design, two factors are of particular importance: the regularity of processor cells and local communication between processors. The hardware implementation of the block diagonal algorithm, for moderate N, just meets the above requirements. An example of the WHT/DHT is also included.

The Walsh-Hadamard/discrete Hartley transform

A new fast algorithm is proposed here to compute the discrete Hartley transform (DHT) via the natural-ordered Walsh-Hadamard transform. The processing is carried out on an intraframe basis in (N × N) data blocks, where N is an integer power of 2. The Walsh-Hadamard transform (WHT)W coefficients are computed directly, and then used to obtain the DHT coefficients. This is achieved by an (N × N) transformation matrix (H transform matrix) which is orthonormal and has a block-diagonal structure. As such, it results in substantial saving in the number of multiplications required to obtain the DHT, relative to direct computation. Its total operation is almost equal to that of fast radix 2 DHT, but with a much more simple data transfer path. Using the binary fraction technique, we could get the result within 3 significant digits. The simple, regular and high immunity of round-off noise properties make it practical to be used in signal processing. Above all, it provides a good conversion for these two useful trans...

Maximum Entropy Convex Decompositions of Doubly Stochastic and Nonnegative Matrices

Two maximum entropy convex decompositions are computed with the use of the iterative proportional fitting procedure. First, a doubly stochastic version of a 5 × 5 British social mobility table is represented as the sum of 120 5 × 5 permutation matrices. The most heavily weighted permutations display a bandwidth form, indicative of relatively strong movements within social classes and between neighboring classes. Then the mobility table itself is expressed as the sum of 6 985 5 × 5 transportation matrices—possessing the same row and column sums as the mobility table. A particular block-diagonal structure is evident in the matrices assigned the greatest weight. The methodology can be applied as well to the representation of other nonnegative matrices in terms of their extreme points, and should be extendable to higher-order mathematical structures—for example, operators and functions.

Fast computation of discrete Hartley transform via Walsh–Hadamard transform

A new fast algorithm is proposed to compute the discrete Hartley transform (DHT) via the Walsh–Hadamard transform (WHT). The processing is carried out on an interframe basis in (N × N) data blocks, where N is an integer power of two. The WHT coefficients are obtained directly, and then used to obtain the DHT coefficients. This is achieved by a transform matrix, the H-transform matrix, which is ortho-normal and has a block-diagonal structure. A complete derivation of the block-diagonal structure for the H-transform matrix is given.

Block-diagonal structure of Walsh-Hadamard/discrete cosine transform

The A-matrix, the conversion matrix for Walsh-Hadamard/discrete cosine transform, is known for its efficient block-diagonal structure. This associates with the even/odd structure of the transform kernels. In the letter we present a direct matrix derivation by using the intrinsic properties of the discrete cosine transform and the Walsh-Hadamard transform.

Efficient solving of the group technology problem

Abstract In this paper, two formulations of the group technology problem are considered: the standard formulation and the augmented formulation. The standard formulation is based on the 0–1 machine-incidence matrix and does not consider any costs. In the augmented formulation with each partj, cost c j is associated and the number of machines in each cell is limited to N. This formulation allows the creation of machine cells and part families with a low degree of interaction (or without any interaction, if required) by removing parts with low values of the corresponding costs from the incidence matrix. To solve these formulations, tow very efficient algorithms are presented. The cluster identification algorithm with the computational time complexity 0(2mn) finds optimal machine cells and part families provided that the machine-part incidence matrix has the block diagonal structure embedded. This appears to be the most efficient algorithm developed to date. The cost analysis algorithm is designed to solve the augmented group technology problem. Its computational complexity is 0(2 mn + nlogn). The two algorithms are illustrated in numerical examples.

Industrial application of the frequency-domain aspects of the Kalman filter

Several classical industrial filtering applications have used notch filters to reject the unwanted dominant frequencies. For a class of industrial systems, it has been shown that the filter gain between the observations and the signal estimates is unity. While the gain between the observations and the disturbance state estimates is zero, the zeros of the Kalman filter are determined by the poles of the disturbance model. These results are generalised to a more representative case of a block-diagonal structure with feedback coupling.