New Results Regarding the Construction Method for D‑optimal Chemical Balance Weighing Designs

Bronisław Ceranka,Małgorzata Graczyk

doi:10.18778/0208-6018.349.08

Bronisław Ceranka, Małgorzata Graczyk

Open Access

https://doi.org/10.18778/0208-6018.349.08

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Abstract

We study an experiment in which we determine unknown measurements of p objects in n weighing operations according to the model of the chemical balance weighing design. We determine a design which is D‑optimal. For the construction of the D‑optimal design, we use the incidence matrices of balance incomplete block designs, balanced bipartite weighing designs and ternary balanced block designs. We give some optimality conditions determining the relationships between the parameters of a D‑optimal design and we present a series of parameters of such designs. Based on these parameters, we will be able to set down D‑optimal designs in classes in which it was impossible so far.

Highlights

In this paper, we consider the linear model y = Xw + e, where: y is an n × 1 random vector of observations, XXÖ∈Î Φnp (–1,0,1), the class of n × p matrices X = of known elements where xij equals –1, 0 or 1, w is a p × 1 vvector of unknown measurements of objects, e is an n × 1 random vector of errors.We assume that E(e) = 0n and Var(e) = σ2In, where 0n is the n × 1 vector with zero elements everywhere, In denotes the identity matrix of rank n
The aim of this paper is an investigation of a new construction method of a D‐optimal chemical balance weighing design
We will be able to set down D‐optimal designs in classes in which it was impossi‐ ble so far

Summary

Introduction

Any chemical balance weighing design XXÖ∈Î Φnp (–1,0,1),with the variance matrix of errors σ2In is regular D‐optimal if and only if X'X = mI p , where m is the maximal number of elements different from zero in the j‐th column, where j = 1, ..., p. The aim of this paper is an investigation of a new construction method of a D‐optimal chemical balance weighing design. Based on this method, we will be able to set down D‐optimal designs in classes in which it was impossi‐ ble so far. New matrix construction methods will allow us to determine the D‐optimal chemical balance weighing de‐ sign for new combinations of the number of objects and the number of measure‐ ments which are not known in the literature.

Balanced block design

Examples

Discussion