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

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 ma‐ trices of balance incomplete block designs, balanced bipartite weighing designs and ternary balanced block designs. We give some optimality conditions determining the relationships between the pa‐ rameters 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.


Introduction
In this paper, we consider the linear model , = y Xw + e where: y is an n × 1 random vector of observations, the class of n × p matrices X = (x ij ) of known elements where x ij 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) = 0 n and Var(e) = σ 2 I n , where 0 n is the n × 1 vector with zero elements everywhere, I n denotes the identity matrix of rank n. Such form of the matrix Var(e) indicates that errors are uncorrelated and have the same variance.
In order to estimate w, we use the least squares method and the normal equations of the form X Xw = X y . Any chemical balance weighing design is singular or non-singular, depending on whether the matrix ' X X is singular or non-singular, respectively. If X is of full column rank, the least squares estimator of w is equal to X X is called the information matrix for the design X. In the literature, basic problems of weighing designs are discussed. Jacroux, Wong and Masaro (1983), Sathe and Shenoy (1990) gave the introduction to different optimality criteria.
Here, we consider chemical balance weighing designs under the basic assumption that the design is D-optimal. The weighing design is stated by entering its matrix. The design X D is called D-optimal in the given class X ∈ Φ Φ ( ) -1, 0,1 , ). Moreover, if det(M) attains the upper bound, then the design is called regular D-optimal. For more theory, we refer the reader to the papers of Katulska and Smaga (2013), Ceranka and Graczyk (2016).
Based on the results given in Ceranka and Graczyk (2017), we have: with the variance matrix of errors σ 2 I n is regular D-optimal if and only if ' where m is the maximal number of elements different from zero in the j-th column, where j = 1, …, p.
The relations between the parameters of the D-optimal chemical balance weighing design imply that for any combination of numbers p and n, we are not able to determine a D-optimal design. In other words, in any class Φ Φ ( ) -1, 0,1 , n ṕ Î X Ö , a D-optimal chemical balance weighing design may not exist. Therefore, 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 impossible so far. Thus, we can determine estimators of unknown parameters having the smallest possible product of its variances.
We construct the design matrix of the D-optimal chemical balance weighing design by use of incidence matrices of known block designs. Here we take the incidence matrices of the balanced incomplete block design, the balanced bipartite weighing design and the ternary balanced block design. New matrix construction methods will allow us to determine the D-optimal chemical balance weighing design for new combinations of the number of objects and the number of measurements which are not known in the literature. The properties of mentioned designs are presented in Section 2, whereas Section 3 contains the methods of construction of the design matrix. Finally, some examples of experimental plans are given.

Balanced block design
In this section, we present the definition and properties of the balanced incomplete block design given in Raghavarao (1971), the balanced bipartite weighing design given in Huang (1976) and the ternary balanced block design given in Billington (1984).
A balanced incomplete block design (BIBD) with the parameters ν, b, r, k, λ is an arrangement of ν treatments into b blocks, each of size k. Each treatment occurs at most once in each block, occurs in exactly r blocks, and every pair of treatments occurs together in exactly λ blocks. Let N be the incidence matrix of a balanced incomplete block design. The parameters are related by the following identities vr bk A balanced bipartite weighing design (BBWD) with the parameters ν, b, r, k 1 , k 2 , λ 1 , λ 2 is an arrangement of ν treatments into b blocks. Each block containing k distinct treatments is divided into 2 subblocks containing k 1 and k 2 treatments, respectively, where k = k 1 + k 2 . Each treatment appears in r blocks. Every pair of treatments from different subblocks appears together in λ 1 blocks and every pair of treatments from the same subblocks appers together in λ 2 blocks. Let N * be the incidence matrix of such a design. The parameters are not independent and they are related by the following equalities A ternary balanced block design (TBBD) with the parameters ν, b, r, k, λ, ρ 1 , ρ 2 is an arrangement of ν treatments in b blocks each of size k. Each treatment appears 0, 1, 2 times in a given block, repeated r times. Each of the distinct pairs of treatments occurs λ times. Each element appears once in ρ 1 block and twice in ρ 2 blocks, where ρ 1 and ρ 2 are a known constant for the design. Let N be the incidence matrix of a ternary balanced block design. The following relations are satisfied

Construction
A large number of publications presenting construction methods of optimal chemical balance weighing designs can be found in the literature. Generally, the construction methods are based on the incidence matrices of known block designs, see Ceranka and Graczyk (2018), Graczyk and Janiszewska (2019). When we determine the design matrix of the D-optimal chemical balance weighing design, then we prepare a plan of an experiment in which we determine unknown measurements of p objects by using n measurement operations. Let N 1 be the incidence matrix of BIBD with the parameters ν, b 1 , r 1 , k 1 , λ 1 . Moreover, let 2 , N * be the incidence matrix of BBWD with the parameters ν, b 2 , r 2 , k 12 , k 22 , λ 12 , λ 22 . Based on the matrix 2 , N * we form the matrix N 2 by replacing k 12 elements equal to +1 in each column which corresponds to the elements belonging to the first subblock by -1. Consequently, each column of N 2 will contain k 12 elements equal to -1, k 22 elements equal to 1 and ν -k 12 -k 12 elements equal to 0. Furthermore, let N 3 be the incidence matrix of TBBD with the parameters ν, b 3 , From Graczyk and Janiszewska (2019) In particular, the equality (3.2) is true, when any combination of these parameters which in total gives zero is true. Based on the series of parameters given by Raghavaro (1971), Huang (1976, Billington (1984), and Ceranka and Graczyk (2004a;2004b) of the block designs presented in Section 2, we formulate the following corollaries. In the special case when s = t = u = 1, we obtain the Corollary 3.14 (Graczyk, Janiszewska, 2019). In the special case when s = t = u = 1, we obtain the Corollary 3.7 (Graczyk, Janiszewska, 2019), when s = 3, t = u = 1, we obtain the Corollary 3.31 (ii) (Graczyk, Janiszewska, 2019). In the special case when s = t = u = 1, we obtain the Corollary 3.31 (ii) (Graczyk, Janiszewska, 2019).

Examples
Let us consider an experiment in which we determine unknown measurements of p = 5 objects and n = 30 measurements. According to the Theorem 3.3, we consider the balanced incomplete block design with the parameters ν = 5, b 1 = 10, r 1 = 4, k 1 = 2, λ 1 = 1 and the incidence matrix N 1 , the balanced bipartite weighing design with the parameters ν = 5, b 2 = r 2 = 5, k 12 = 1, k 22 = 4, λ 12 = 2, λ 22 = 3 and the incidence matrix 2 , N * and also the ternary balanced block design with the parameters ν = 5, b 3 = 15, r 3 = 9, k 3 = 3, λ 3 = 4, ρ 13 = 7, ρ 23 = 1 and the incidence matrix Here, 1 h denotes the element belonging to the h-th subblock, h = 1, 2. Thus, the design matrix of the regular D-optimal chemical balance weighing design X ∈ Φ Φ ( ) -1, 0,1 , n ṕ Î X Ö is given in the form estimator. The design matrix is interpreted as a plan of an experiment and it sets the allocation of objects to particular weighing. From this point of view, the parameters presented in corollaries 3.1-3.8 allow us to construct the incidence matrices of block designs and simultaneously experimental plans with the required properties. Given this interpretation and for different optimality criteria, the application of chemical balance weighing designs in economic research is presented in Banerjee (1975) and Ceranka and Graczyk (2014). The applications of such designs are not limited to only one field of science. In addition, these types of experiments are used in agricultural experimental practice. A detailed description of the applications was given in Ceranka and Katulska (1987) and Graczyk (2013).
It is worth emphasising that other optimality criteria are also considered in the literature. For example, detailed research on A-optimal chemical balance weighing designs is given in Ceranka and Graczyk (2015).