An Optimal Reliability and Maintainability Design of a Searching System

In this article, we deal with an optimal reliability and maintainability design problem of a searching system with complex structures. The system availability and life cycle cost are used as optimization criteria and estimated by simulation. We want to determine MTBF (Mean Time between Failures) and MTTR (Mean Time to Repair) for all components and ALDT (Administrative and Logistics Delay Times) of the searching system in order to minimize the life cycle cost and to satisfy the target system availability. A hybrid genetic algorithm with a heuristic method is proposed to find near-optimal solutions and compared with a general genetic algorithm.     

Optimal designs for non-competitive enzyme inhibition kinetic models

In this paper, we present a new method for determining optimal designs for enzyme inhibition kinetic models, which are used to model the influence of the concentration of a substrate and an inhibition on the velocity of a reaction. The approach uses a nonlinear transformation of the vector of predictors such that the model in the new coordinates is given by an incomplete response surface model. Although there exist no explicit solutions of the optimal design problem for incomplete response surface models so far, the corresponding design problem in the new coordinates is substantially more transparent, such that explicit or numerical solutions can be determined more easily. The designs for the original problem can finally be found by an inverse transformation of the optimal designs determined for the response surface model. We illustrate the method determining explicit solutions for the D-optimal design and for the optimal design problem for estimating the individual coefficients in a non-competitive enzyme inhibition kinetic model.     

Optimal design in average for inference in generalized linear models

This paper considers the problem of optimal design for inference in Generalized Linear Models, when prior information about the parameters is available. The general theory of optimum design usually requires knowledge of the parameter values. These are usually unknown and optimal design can, therefore, not be used in practice. However, one way to circumvent this problem is through so-called “optimal design in average”, or shortly, “ave optimal”. The ave optimal design is chosen to minimize the expected value of some criterion function over a prior distribution. We focus our interest on the aveD _A-optimality, including aveD- and avec-optimality and show the appropriate equivalence theorems for these optimality criterions, which give necessary conditions for an optimal design. Ave optimal designs are of interest when e.g. a factorial experiment with a binary or a Poisson response in to be conducted. The results are applied to factorial experiments, including a control group experiment and a 2×2 experiment.     

Some new lower bounds to various discrepancies on combined designs

The foldover is a useful technique in construction of two-level factorial designs. A foldover design is the follow-up experiment generated by reversing the sign(s) of one or more factors in the initial design. The full design obtained by joining the runs in the foldover design to those of the initial design is called the combined design. In this article, some new lower bounds of various discrepancies of combined designs, such as the centered, symmetric, and wrap-around L₂-discrepancies, are obtained, which can be used as a better benchmark for searching optimal foldover plans. Our results provide a theoretical justification for optimal foldover plans in terms of uniformity criterion.     

Locally optimal designs for some dose–response models with continuous endpoints

We consider the problem of constructing static (or non sequential), approximate optimal designs for a class of dose–response models with continuous outcomes. We obtain conditions for a design being D-optimal or c-optimal. The designs are locally optimal in that they depend on the model parameters. The efficiency studies show that these designs have high efficiency when the mis-specification of the initial values of model parameters is not severe. A case study indicates that using an optimal design may result in a significant saving of resources.     

Rational Regression Models with Continuous Chebyshev Design

In rational regression models, the G-optimal designs are very difficult to derive in general. Even when an G-optimal design can be found, it has, from the point of view of modern nonparametric regression, certain drawbacks because the optimal design crucially depends on the model. Hence, it can be used only when the model is given in advance. This leads to the problem of finding designs which would be nearly optimal for a broad class of rational regression models. In this article, we will show that the so-called continuous Chebyshev Design is a practical solution to this problem.     

Construction of optimal designs in random coefficient regression models

We consider the problem of optimal experimental design in random coefficient regression models with respect to a quadratic loss function. By application of WHITTLE'S general equivalence theorem we obtain the structure of optimal designs. An alogrithm is given which allows, under certain assumptions, the construction of the information matrix of an optimal design. Moreover, we give conditions on the equivalence of optimal designs with respect to optimality criteria which are analogous to usual A-D- and _E/-optimality.     

Conditions for D A -Maximin Marginal Designs for Generalized Linear Mixed Models to be Uniform

Optimal design theory deals with the assessment of the optimal joint distribution of all independent variables prior to data collection. In many practical situations, however, covariates are involved for which the distribution is not previously determined. The optimal design problem may then be reformulated in terms of finding the optimal marginal distribution for a specific set of variables. In general, the optimal solution may depend on the unknown (conditional) distribution of the covariates. This article discusses the D _A -maximin procedure to account for the uncertain distribution of the covariates. Sufficient conditions will be given under which the uniform design of a subset of independent discrete variables is D _A -maximin. The sufficient conditions are formulated for Generalized Linear Mixed Models with an arbitrary number of quantitative and qualitative independent variables and random effects.     

Optimal allocation of time points for the random effects model

In this article the problem of the optimal selection and allocation of time points in repeated measures experiments is considered. D‐ optimal designs for linear regression models with a random intercept and first order auto‐regressive serial correlations are computed numerically and compared with designs having equally spaced time points. When the order of the polynomial is known and the serial correlations are not too small, the comparison shows that for any fixed number of repeated measures, a design with equally spaced time points is almost as efficient as the D‐ optimal design. When, however, there is no prior knowledge about the order of the underlying polynomial, the best choice in terms of efficiency is a D‐ optimal design for the highest possible relevant order of the polynomial. A design with equally‐spaced time points is the second best choice     

A lower bound for the centred L 2-discrepancy on combined designs under the asymmetric factorials

The foldover is a useful technique in the construction of two-level factorial designs for follow-up experiments. To search an optimal foldover plans is an important issue. In this paper, for a set of asymmetric fractional factorials such as the original designs, a lower bound for centred L ₂-discrepancy of combined designs under a general foldover plan is obtained, which can be used as a benchmark for searching optimal foldover plans. All of our results are the extended ones of Ou et al. [Lower bounds of various discrepancies on combined designs, Metrika 74 (2011), pp. 109–119] for symmetric designs to asymmetric designs. Moreover, it also provides a theoretical justification for optimal foldover plans in terms of uniformity criterion.     

Bayesian D-optimal Accelerated Life Test plans for series systems with competing exponential causes of failure

This paper provides methods of obtaining Bayesian D-optimal Accelerated Life Test (ALT) plans for series systems with independent exponential component lives under the Type-I censoring scheme. Two different Bayesian D-optimality design criteria are considered. For both the criteria, first optimal designs for a given number of experimental points are found by solving a finite-dimensional constrained optimization problem. Next, the global optimality of such an ALT plan is ensured by applying the General Equivalence Theorem. A detailed sensitivity analysis is also carried out to investigate the effect of different planning inputs on the resulting optimal ALT plans. Furthermore, these Bayesian optimal plans are also compared with the corresponding (frequentist) locally D-optimal ALT plans.     

SkSP-R sampling plan based on process capability index

In this paper, we present an optimal designing methodology for the skip-lot sampling plan (SkSP) of type SkSP-R based on the most widely used process capability index (PCI) C_pk. The SkSP-R plan is one of the SkSPs that incorporate the provision of the reinspection concept. In order to design the optimal parameters, we consider both symmetric and asymmetric fraction non conforming cases. Tables are also constructed to determine the optimal parameters by formulating an optimization problem. The advantages of the proposed plan over the existing plan are also discussed. Application of the plan is explained with a real-life example.     

Computing A-optimal Designs for Weighted Polynomial Regression by Taylor Expansion

This article is concerned with the problem of constructing A-optimal design for polynomial regression with analytic weight function on the interval [m ? a, m + a], m, a > 0. It is shown that the structure of the optimal design depends on a and weight function only, as a close to 0. Moreover, if the weight function is an analytic function a, then a scaled version of optimal support points, and weights are analytic functions of a at a = 0. We make use of a Taylor expansion to provide a recursive procedure for calculating the A-optimal designs. Examples are presented to illustrate the procedures for computing the optimal designs.     

Optimal designs for the logit and probit models for binary data

We consider optimal designs for a class of symmetric models for binary data which includes the common probit and logit models. We show that for a large group of optimality criteria which includes the main ones in the literature (e.g. A-, D-, E-, F- and G-optimality) the optimal design for our class of models is a two-point design with support points symmetrically placed about the ED50 but with possibly unequal weighting. We demonstrate how one can further reduce the problem to a one-variable optimization by characterizing various of the common criteria. We also use the results to demonstrate major qualitative differences between the F - and c-optimal designs, two design criteria which have similar motivation.     

Optimal design for classification of functional data

We study the design problem for the optimal classification of functional data. The goal is to select sampling time points so that functional data observed at these time points can be classified accurately. We propose optimal designs that are applicable to either dense or sparse functional data. Using linear discriminant analysis, we formulate our design objectives as explicit functions of the sampling points. We study the theoretical properties of the proposed design objectives and provide a practical implementation. The performance of the proposed design is evaluated through simulations and real data applications. The Canadian Journal of Statistics 48: 285–307; 2020 © 2019 Statistical Society of Canada     

An evolutionary algorithm for A-optimal incomplete block designs

Evolutionary algorithms are heuristic stochastic search and optimization techniques with principles taken from natural genetics. They are procedures mimicking the evolution process of an initial population through genetic transformations. This paper is concerned with the problem of finding A-optimal incomplete block designs for multiple treatment comparisons represented by a matrix of contrasts. An evolutionary algorithm for searching optimal, or nearly optimal, incomplete block designs is described in detail. Various examples regarding the application of the algorithm to some well-known problems illustrate the good performance of the algorithm     

Bayesian sequential analysis for multiple-arm clinical trials

Use of full Bayesian decision-theoretic approaches to obtain optimal stopping rules for clinical trial designs typically requires the use of Backward Induction. However, the implementation of Backward Induction, apart from simple trial designs, is generally impossible due to analytical and computational difficulties. In this paper we present a numerical approximation of Backward Induction in a multiple-arm clinical trial design comparing k experimental treatments with a standard treatment where patient response is binary. We propose a novel stopping rule, denoted by τ ^p , as an approximation of the optimal stopping rule, using the optimal stopping rule of a single-arm clinical trial obtained by Backward Induction. We then present an example of a double-arm (k=2) clinical trial where we use a simulation-based algorithm together with τ ^p to estimate the expected utility of continuing and compare our estimates with exact values obtained by an implementation of Backward Induction. For trials with more than two treatment arms, we evaluate τ ^p by studying its operating characteristics in a three-arm trial example. Results from these examples show that our approximate trial design has attractive properties and hence offers a relevant solution to the problem posed by Backward Induction.     

Using orthogonal array for constructing three-level search designs

We consider the problem of constructing search designs for 3^m factorial designs. By using projection properties of some three-level orthogonal arrays, some search designs are obtained for 3 ? m ? 11. The new obtained orthogonal search designs are capable of searching and identifying up to four two-factor interactions and estimating them along with the general mean and main effects. The resulted designs have very high searching probabilities; it means that besides the well-known orthogonal structure, they have high ability in searching the true effects.     

Robust sequential designs for approximately linear models

We consider the problem of the sequential choice of design points in an approximately linear model. It is assumed that the fitted linear model is only approximately correct, in that the true response function contains a nonrandom, unknown term orthogonal to the fitted response. We also assume that the parameters are estimated by M-estimation. The goal is to choose the next design point in such a way as to minimize the resulting integrated squared bias of the estimated response, to order n^-1. Explicit applications to analysis of variance and regression are given. In a simulation study the sequential designs compare favourably with some fixed-sample-size designs which are optimal for the true response to which the sequential designs must adapt.     

Optimal Design of Acceptance Sampling Plans by Variables for Nonconforming Proportions When the Standard Deviation Is Unknown

This article presents an optimization-based approach for the design of acceptance sampling plans by variables for controlling nonconforming proportions when the standard deviation is unknown. The variables are described by rigorous noncentral Student’s t-distributions. Single and double acceptance sampling (AS) plans are addressed. The optimal design results from minimizing the average sampling number (ASN), subject to conditions holding at producer’s and consumer’s required quality levels. The problem is then solved employing a nonlinear programming solver. The results obtained are in close agreement with previous sampling plans found in the literature, outperforming them regarding the feasibility.