Showing papers on "Optimal design published in 1978"

Hadamard Matrices and Their Applications

Abstract: An $n \times n$ matrix $H$ with all its entries $+1$ and $-1$ is Hadamard if $HH' = nI$. It is well known that $n$ must be 1, 2 or a multiple of 4 for such a matrix to exist, but is not known whether Hadamard matrices exist for every $n$ which is a multiple of 4. The smallest order for which a Hadamard matrix has not been constructed is (as of 1977) 268. Research in the area of Hadamard matrices and their applications has steadily and rapidly grown, especially during the last three decades. These matrices can be transformed to produce incomplete block designs, $t$-designs, Youden designs, orthogonal $F$-square designs, optimal saturated resolution III designs, optimal weighing designs, maximal sets of pairwise independent random variables with uniform measure, error correcting and detecting codes, Walsh functions, and other mathematical and statistical objects. In this paper we survey the existence of Hadamard matrices and many of their applications.

Optimality of Certain Asymmetrical Experimental Designs

Abstract: The problem of finding an optimal design for the elimination of one-way heterogeneity when a balanced block design does not exist is studied. A general result on the optimality of certain asymmetrical designs is proved and applied to the block design setting. It follows that if there is a group divisible partially balanced block design (GD PBBD) with 2 groups and $\lambda_2 = \lambda_1 + 1$, then it is optimal w.r.t. a very general class of criteria including all the commonly used ones. On the other hand, if there is a GD PBBD with 2 groups and $\lambda_1 = \lambda_2 + 1$, then it is optimal w.r.t. another class of criteria. Uniqueness of optimal designs and some other miscellaneous results are also obtained.

An algorithm for optimal designs on a design space

Abstract: Two variations of a simple monotunic algorithm for computing optimal designs on a finite design space are presented. Various properties are listed. Comparisons witn other algorithms are made.

Some Algorithmic Aspects of the Theory of Optimal Designs

Abstract: The approximate optimal design problem is treated as a constrained convex programming problem. A general class of optimal design algorithms is proposed from this point of view. Asymptotic convergence to optimal designs is also proved. Related problems like the implementability problem for the infinite support case and the general step-length algorithms are discussed.

Optimal centering, tolerancing, and yield determination via updated approximations and cuts

Abstract: This paper presents a new approach to optimal design centering, the optimal assignment of parameter tolerances and the determination and optimization of production yield. Based upon multidimensional linear cuts of the tolerance orthotope and uniform distributions of outcomes between tolerance extremes in the orthotope, exact formulas for yield and yield sensitivities, with respect to design parameters, are derived. The formulas employ the intersections of the cuts with the orthotope edges, the cuts themselves being functions of the original design constraints. Our computational approach involves the approximation of all the constraints by low-order multidimensional polynomials. These approximations are continually updated during optimization. Inherent advantages of the approximations which we have exploited are that explicit sensitivities of the design performance are not required, available simulation programs can be used, inexpensive function and gradient evaluations can be made, inexpensive calculations at vertices of the tolerance orthotope are facilitated during optimization and, subsequently, inexpensive Monte Carlo verification is possible. Simple circuit examples illustrate worst case design and design with yields of less then 100 percent. The examples also provide verification of the formulas and algorithms.

Optimal designs for exponential regression

Abstract: This paper is concerned with the optimal design problem for the particular case of non-linear parametrisation:the parameters to be estimated are included in exponents.Some properties of locally optimal designs as functions of estimated parameters are investigated and a table of such designs in given.We consider also designs to be optimal in the sense of minimax approach.

Optimal design measures with singular information matrices

Abstract: SUMMARY The problem of 'establishing a useful necessary and sufficient condition for a design measure with singular information matrix to be D,-optimal is well known. In this paper a sufficient condition involving a certain type of g-inverse is established for a more general convex criterion function to be minimized by such a design measure. The condition is necessary for DA-optimality and for a linear criterion function. There is some discussion about its necessity in general.

Optimal Designs for the Elimination of Multi-Way Heterogeneity

Abstract: The purpose of this paper is to study optimal designs for the elimination of multi-way heterogeneity. The $C$-matrix for the $n$-way heterogeneity setting when $n > 2$ is derived. It turns out to be a natural extension of the known formulas in the lower dimensional case. It is shown that under some regularity, the search for optimal designs can be reduced to that in a lower-way setting. Youden hyperrectangles are defined as higher dimensional generalizations of balanced block designs and generalized Youden designs. When all the sides are equal, they are called Youden hypercubes. It is shown that a Youden hyperrectangle is $E$-optimal and a Youden hypercube is $A$- and $D$-optimal. The latter is quite interesting since it is not always true in two-way settings.

Models and designs for experiments with mixtures

Abstract: Summary Properties such as the tensile strength of an alloy of. different metals and the freezing point of a mixture of liquid chemicals, depend on the proportions (by weight or volume) of the components present and not on the total amount of the mixture. In choosing a model to relate such a property to the proportions of the various components of the mixture, there arise intriguing difficulties due to the fact that proportions sum to unity. It is demonstrated how to construct models which allow for the possibility of inactive components (components that do not affect the property at all) or components with additive effects. The design of experiments to fit such models to data is then discussed with a view to determining whether a given component is inactive or has an additive effect. The optimal allocation of observations to simplex-lattice designs is considered for one of these models. The construction of D -optimal designs for these models is an open problem.

Two-dimensional shape optimal design†

Abstract: This paper treats shape optimal design of two-dimensional structures. Sensitivity of cost and constraint functions to changes in the shape of the structure are obtained by applying theorems from the calculus of variations and by using consistent first-order approximations of functions arising in an optimal design problem. A generalized steepest descent algorithm is presented for determination of the optimum shape. The method is applied to a load carrying shear plate that models a component of the high pressure seal in a gun launched high velocity projectile.

Treatment Contrasts in Paired Comparisons: Large-Sample Results, Applications, and Some Optimal Designs

Abstract: Treatment contrasts in paired comparison experiments are considered. Given that specified orthonormal treatment contrasts are null, remaining treatment parameters, orthonormal contrasts among them, and their logarithms are estimated. The estimators are shown to be consistent and, when suitably scaled and centered, to have distribution functions that are normal in the limit for large sample sizes. Hypothesis tests on treatment contrasts are obtained. The limiting distribution functions of related likelihood ratio tests are derived for defined local alternatives and their noncentrality parameters are shown. The theoretical results are applied to experiments with factorial treatment combinations and some optimal design results are obtained.

Design criteria for detecting model inadequacy

Abstract: Two criteria are proposed for choosing response surface designs for detecting inadequacy of the assumed model. They are both members of classes of criteria proposed by Atkinson & Fedorov (1975a), and are based on maximizing in some sense the expectation of the residual sum of squares over the region in the parameter space where the departure of the true surface from the assumed model is deemed unacceptable. Characterizations of these criteria are obtained and used to construct optimal designs for first-order versus second-order polynomial models. In the simple one-dimensional case, a simulation comparison of these designs with minimum bias designs and D-optimal designs for the higher-order model is made under a procedure in which the outcome of the lack of fit test determines the model to be used.

Optimal design of bending elements

Abstract: Derivation of an optimality criterion for finite element structural representations using constant-moment plate-bending triangular elements ispresented. Numerical examples of minimum-mass design of both simply-supported and clamped elastic plates for concentrated and distributed loading conditions under a single displacement constraint serve to illustrate the suggested procedure. Interpretation of the solution obtained and difficulties involved are discussed in the light of recent application of optimal control theory to structural optimization.

Optimal design of complex experiments with qualitative factors of influence

Abstract: A complex experiment with qualirarive factors influencing the outcome of the experiment can be seen as a general ANOVA setup. A design of such an experiment will be the assignment at which of the possible levels of the factors the actual experiment should be performed. In this paper optimal designs of such experiments will be characterized with respect to three different optimality criteria including the so called uniform optimality of a design. The possible applications of the main optimization result providing these characterizations can be used to more general experiments. The particular results on these generalizations will be indicated at the end of this paper.

Optimal design of beams with minimum compliance

Abstract: An optimization problem of non-linear elastic or viscous beams is discussed. To the beam an additional support is introduced whose location must be selected so as to minimize the compliance of the beam. The problem is solved with the aid of optimal control theory. Both rigid and flexible supports are considered. Some new optimization conditions, which are valid for arbitral compliance criterions, are deduced. A few illustrating examples are given.

A Geometric Construction of Generalized Youden Designs for $\nu$ A Power of a Prime

Abstract: A new method of construction of generalized Youden designs for $ u = s^m, s$ a power of a prime is introduced here. This generalizes the construction of Ruiz and Seiden which could be applied only to even powers of a prime. The number of experimental units required to carry out the design in the corresponding cases is the same. However, the present method can be used for construction of designs which could not be constructed previously even in the case of even powers. Moreover the present method presents a unified construction for even and odd powers of primes. For a fixed value of a prime it is noticed here that one can construct an infinite number of designs. This provides the experimenter with a choice of designs which may prove very useful in applications. A simpler method of construction is also presented. The price one has to pay for the simplicity is that more experimental units are required for carrying out the design.

Optimal Design of a Dry-Type Natural-Draft Cooling Tower by Geometric Programming

Abstract: In this paper, the optimal design of dry-type natural-draft cooling towers is investigated. Using physical laws and engineering design relations that govern the system, a rather detailed optimization model is developed. This model is then reformulated as a geometric programming problem. A primary consideration in this reformulation is how certain polynomial equations may be effectively replaced by inequalities. A numerical example follows.

Some aspects of optimization techniques for electromagnetic devices

Abstract: The optimized design of electromagnetic devices, such as transformers and electrical machines, is usually formulated as a general non-linear programming problem by its objective (cost function) and constraints (performance) functions. Direct search optimization methods are used to find the optimal design parameters of the device. With these methods, the solution is approximated in an iterative procedure, starting from an initial design and improvement is achieved with successive iterations. The paper deals with some aspects of three optimization techniques for electromagnetic devices: The SUMT (Sequential Unconstrained Minimization Technique), Least p-th approximation method and the Boundary search method. By clarifying these aspects and discussing some of the difficulties in applying these methods, the paper may assist the engineer who approaches the design of electromagnetic devices mathematically as a general programming problem.

Some extremal problems in designing discriminating experiments

Abstract: This paper is devoted to some properties of local T-optimaj. designs fox discriminating between two rival regression models. It is demonstrated that the problem of searching for an optimal design is closely connected with the problem of searching for a Tchebycheff extremal basis. The question raised concerns the number of supporting points in an optimal design.

Optimal Experimental Designs in Two Dimensions Using Minimum Bias Estimation

Abstract: The general problem of minimum bias estimation is reviewed for polynomial response surface models, where the true model, a polynomial of degree d + k — 1, is estimated by a polynomial of degree d — 1. Through the choice of estimator, the same minimum integrated squared bias B is achieved for any experimental design that satisfies a simple estimability condition. This design flexibility is used to construct D-optimal, V-optimal, and A-optimal experimental designs in two dimensions through a computerized simplex search procedure. The resulting optimal designs for both square and circular regions of interest are discussed and recommendations as to their use are made.

The robustness of chain block designs and coat-of-mail designs

Abstract: Chain block designs are relatively vulnerable to loss of information when missing values or outliers may occur An alternative class of designs, coat-of-mail designs, are proposed and the relative robustness of the two types of design are compared

Minimum-weight design of multi-purpose tie-column of solid construction

Abstract: The paper provides the minimum-weight design of an elastic pin-ended member of solid construction under prescribed bounds on the maximum permissible elongation under axial tension and on the minimum allowable Euler buckling load. The member is to act as a tie for a part of its design life and as a column for the rest. Optimization for more than one design requirement not only unifies the design of mass-produced structural/mechanical elements, but also provides, at times, a practically acceptable design in that the optimal design does not have zero cross-sectional area at simply-supported ends—a situation quite common in optimal designs for a single requirement. However, it is shown that the constraints on longitudinal elongation and buckling load are less severe than that on the maximum stress. The effectiveness of optimization is judged by comparing the volume (mass) of the optimally designed member with that of a prismatic bar having the same buckling load.

Optimal Design of Pitched Laminated Wood Beams

Abstract: The optimal design of a pitched laminated wood beam is considered. An engineering formulation is given in which the volume of the beam is minimized. The problem is then reformulated and solved as a generalized geometric (signomial) program. Sample designs are presented.