CONSTRUCTION METHOD OF A-OPTIMAL CHEMICAL BALANCE WEIGHING DESIGNS

Here, we study the design in that we determine unknown measurements of p objects by used of n measurement operations. For that reason we consider the chemical balance weighing design under assumption that the measurement errors are uncorrelated and they have the same variances. We give new construction method of the A-optimal chemical balance weighing design based on the incidence matrices of the balanced bipartite weighing designs and the ternary balanced block designs. The consequence of the proposed method is widening of possible classes in that A-optimal design exists

Optimum Chemical Balance Weighing Design under Certain Condition

2000 Mathematics Subject Classification: 62K05, 05B05.The problem of estimation of unknown weights of p objects is considered. The experiment is carried out according to the standard Gauss-Markoff model of the chemical balance weighing design. Existence conditions of the optimum design are given. New construction method of the optimum design based on the set of the incidence matrices of the ternary balanced block designs is presented

Nested Balanced Incomplete Block Designs

If the blocks of a balanced incomplete block design (BIBD) with v treatments and with parameters (v; b1;r;k1) are each partitioned into sub-blocks of size k2, and the b2 =b1k1=k2 sub-blocks themselves constitute a BIBD with parameters (v; b2;r;k2), then the system of blocks, sub-blocks and treatments is, by de4nition, a nested BIBD (NBIBD). Whist tournaments are special types of NBIBD with k1 =2k2= 4. Although NBIBDs were introduced in the statistical literature in 1967 and have subsequently received occasional attention there, they are almost unknown in the combinatorial literature, except in the literature of tournaments, and detailed combinatorial studies of them have been lacking. The present paper therefore reviews and extends mathematical knowledge of NBIBDs. Isomorphism and automorphisms are defined for NBIBDs, and methods of construction are outlined. Some special types of NBIBD are de4ned and illustrated. A first-ever detailed table of NBIBDs with v⩽16, r⩽30 is provided; this table contains many newly discovered NBIBDs. © 2001 Elsevier Science B.V. All rights reserved

Balanced bipartite weighing designs leading to regular D-optimal spring balance weighing designs

W pracy przedstawiono nową metodę konstrukcji regularnego D-optymalnego sprężynowego układu z tarowaniem przy założeniu, że wariancje błędów pomiarów nie są jednorodne. Do konstrukcji macierzy układu wykorzystano macierze incydencji dwudzielnych układów bloków

Balanced bipartite weighing designs leading to regular D-optimal spring balance weighing designs

W pracy przedstawiono nową metodę konstrukcji regularnego D-optymalnego sprężynowego układu z tarowaniem przy założeniu, że wariancje błędów pomiarów nie są jednorodne. Do konstrukcji macierzy układu wykorzystano macierze incydencji dwudzielnych układów bloków

Optimum chemical balance weighing design with diagonal matrix of errors based on balanced block designs

The paper is studying the estimation problem of individual weights of objects using the chemical balance weighing design under the restriction on the number of times in which each object is weighed. Additionally, we assume that the errors are uncorrelated and they have different variances. There are given the lower bound of the variance of each of the estimated weights and the sufficient and necessary conditions under which this lower bound is attained. For the new construction method for optimum chemical balance weighing designs, we use the incidence matrices of the balanced incomplete block designs and the balanced bipartite weighing designs

Optimum chemical balance weighing designs with diagonal variance-covariance matrix of errors

In this paper we study the estimation problem of individual measurements (weights) of objects in a model of chemical balance weighing design with diagonal variance - covariance matrix of errors under the restriction k₁ + k₂ < p, where k₁ and k₂ represent the number of objects placed on the right and left pans, respectively. We want all variances of estimated measurments to be equal and attaining their lower bound. We give a necessary and sufficient condition under which this lower bound is attained by the variance of each of the estimated measurements. To construct the design matrix X of the considered optimum chemical balance weighing design we use the incidence matrices of balanced bipartite weighing designs

Regularny D‑optymalny układ wagowy z ujemnie skorelowanymi błędami

The issues concerning optimal estimation of unknown parameters in the model of chemical balance weighing designs with negative correlated errors are considered. The necessary and sufficient conditions determining the regular D‑optimal design and some new construction methods are presented. They are based on the incidence matrices of balanced incomplete block designs and balanced bipartite weighing designs. W artykule rozważa się problematykę dotyczącą istnienia regularnego D‑optymalnego chemicznego układu wagowego przy założeniu, że błędy pomiarów są ujemnie skorelowane i mają takie same wariancje. Przedstawiono warunki konieczne i dostateczne, wyznaczające układ regularnie D‑optymalny oraz podano nowe metody konstrukcji. Są one oparte na macierzach incydencji układów zrównoważonych o blokach niekompletnych oraz dwudzielnych układów bloków