Some Construction Methods of A-Optimum Chemical Balance Weighing Designs

Some new construction methods of the optimum chemical balance weighing designs and pairwise efficiency and variance balanced designs are proposed, which are based on the incidence matrices of the known symmetric balanced incomplete block designs. Also the conditions under which the constructed chemical balance weighing designs become A-optimal are also been given.


Introduction
Sir R. A. Fisher, a founder of modern concept of experimental designs gave the new ideas of designing in his first book Design of Experiment in the year 1935.Fisher's work was continued by others; see [1]- [4].The necessary and sufficient condition for a general block design to be variance balanced and efficiency balanced was given in the literature [5]- [8].The concept of repeated blocks was introduced by Van Lint; see [9].Further some potential applications of the balanced incomplete block designs with repeated blocks were presented in the literature [10]- [13].
Another important concept which we discuss in this paper is weighing designs.The concept of weighing design was originally given by Yates and formulated as a weighing problem by Hotelling and the condition of attaining the lower bound by each of the variance of the estimated weights was given by him; see [14] [15].In the latter developments, attention has been made in the direction of obtaining optimum weighing designs.Prominent work has been done by many researchers in this field; see [16]- [20].In recent years, the new methods of constructing the optimum chemical balance weighing designs and a lower bound for the variance of each of the es-timated weights from this chemical balance weighing design were obtained and a necessary and sufficient condition for this lower bound to be attained was proposed in the literature; see [21]- [24].The constructions were based on the incidence matrices of balanced incomplete block designs, balanced bipartite block designs, ternary balanced block designs and group divisible designs.
Awad et al. [25] [26] gave the construction methods of obtaining optimum chemical balance weighing designs using the incidence matrices of symmetric balanced incomplete block designs and some pairwise balanced designs were also been obtained which were efficiency as well as variance balanced.In that series we now propose another new construction methods of obtaining optimum chemical balance weighing designs using the incidence matrices of symmetric balanced incomplete block designs and some more pairwise efficiency as well as variance balanced designs are proposed.Also we present the conditions under which the chemical balance weighing designs constructed by new construction methods leading to the A-optimal designs.
Let us consider ν treatments arranged in b blocks, such that the th j block contains j k experimental units and the th i treatment appears i r times in the entire design, 1, 2, , i  .For any block design there exist a incidence matrix , where ij n denotes the number of experiment units in the th j block getting the th i treatment.When 1 ij n = or 0 i ∀ and j , the design is said to be binary.Otherwise it is said to be nonbinary.In this paper we consider binary block designs only.The following additional notations are used [ ] An equi-replicate, equi-block sized, incomplete design, which is also balanced in the sense given above is called balanced incomplete block design, which is an arrangement of ν symbols (treatments) into b sets (blocks) each containing k ( ) distinct symbols, such that any pair of distinct symbols occurs in exactly λ sets.Then it is easy to see that each treatment occurs in r ( ) λ > sets.ν , b , r , k , λ are called pa- rameters of the BIBD and the parameters satisfies the relations r bk ν = , ( ) ( ) A BIB design is said to be symmetric if b ν = and r k = .In this case incidence matrix is a square matrix i.e.N N ′ = .In case of symmetric balanced incomplete block design any two blocks have λ treatments in common.
Though there have been balanced designs in various senses (see [6] [27]).We will consider a balanced design of the following type.
A block design is called variance balanced if and only if 1) It permits the estimation of all normalized treatment contrasts with the same variance (see [7]).
2) If the information matrix for treatment effects where µ is the unique nonzero eigen value of the matrix C with the multiplicity ( ) A block design is called efficiency balanced if 1) Every contrast of treatment effects is estimated through the design with the same efficiency factor. 2) ; see [2], and since , where ψ is the unique non zero eigen value of o M with multiplicity ( ) ; see [28].
A block design is said to be pairwise balanced if ≠ and a pair- wise balanced block design is said to be binary if 0 ij n = or 1 only, for all i , j and it has parameters ν , b , r , k , Λ ( λ = , say) [in this case, when Weighing designs consists of n groupings of the p objects and suppose we want to determine the individual weights of p objects.We can fit the results into the general linear model where Y is an 1 n × random column vector of the observed weights, w is the 1 p × column vector representing the unknown weights of objects and e is an 1 n × random column vector of errors such that The normal equations estimating w are of the form where ŵ is the vector of the weights estimated by the least squares method.The matrix X is called the design matrix.A weighing design is said to be singular or nonsingular, depend- ing on whether the matrix X X ′ is singular or nonsingular, respectively.It is obvious that the matrix X X ′ is nonsingular if and only if the matrix X is of full column ( ) ′ is nonsingular, the least squares estimate of w is given by ( ) and the variance-covariance matrix of ŵ is When the objects are placed on two pans in a chemical balance, we shall call the weighings two pan weighing and the design is known as two pan design or chemical balance weighing design.In chemical balance weighing design, the elements of design matrix takes the values as +1 if the th j object is placed in the left pan in the th i weighing, 1 − if the th j object is placed in the right pan in the th i weighing and 0 if the th j object is not weighted in the th i weighing.Hotelling has shown that if n weighing operations are to determine the weights of p n = objects, the minimum attainable variance for each of the estimated weights in this case is 2 n σ and proved the theorem that each of the variance of the estimated weights attains the minimum if and only if p X X nI ′ = (see [14]).

Variance Limit of Estimated Weights
Let X be an n p × matrix of rank p of a chemical balance weighing design and let j m be the number of times in which th j object is weighed, 1, 2, , j p =  (i.e. the j m be the number of elements equal to −1 and 1 in th j column of matrix X ).Then Ceranka et al. (see [21]) proved the following theorem: Theorem 2.1.For any n p × matrix X , of a nonsingular chemical balance weighing design, in which maximum number of elements equal to 1 − and 1 in columns is equal to m, where Then each of the variances of the estimated weights attains the minimum if and only if Also a nonsingular chemical balance weighing design is said to be optimal for the estimating individual weights of objects if the variances of their estimators attain the lower bound given by, In SBIB design ( ) , , D v r λ ; the block intersection between any two blocks is constant i.e. λ .Using this concept Banerjee (see [29]) proved the following results; Proposition 2.2.Existence of SBIB design ( )

Construction of Design Matrix: Method I
In SBIB design D with the parameters v b = , r k = , λ ; fix the th j block ( ) to the th j fixed block, give negative sign to all the λ common treatments of remaining ( ) Then combining the incidence matrix N of SBIB design repeated s-times with 1 N * we get the matrix X of a chemical balance weighing design as Under the present construction scheme, we have ments equal to zero.Clearly such a design implies that each object is weighted ( ) weighing operations.
Lemma 3.1.A design given by X of the form ( 8) is non singular if and only if ( ) ( )( ) For the design matrix X given by ( 8), we have and the determinant ( 10) is equal to zero if and only if ) The non-singular chemical balance weighing design with matrix X given by ( 8) is optimal if and only if ( ) ( ) Proof.From the conditions (5) and ( 9) it follows that a chemical balance weighing design is optimal if and only if the condition (11) holds.Hence the theorem.
If the chemical balance weighing design given by matrix X of the form (8) is optimal then ( ) ( Clearly such a design implies that each object is weighted 32 m = times in 56 n = weighing operations and . Then the structure -times form a pairwise VB and EB design D *1 with parameters

Construction of Design Matrix: Method II
In Then combining the incidence matrix N of SBIB design repeated s -times with 2 N * we get the matrix X of a chemical balance weighing design as Under the present construction scheme, we have 2 Thus the each column of X will contain 1 3 3 --n ρ ρ ele- ments equal to zero.Clearly such a design implies that each object is weighted Lemma 4.1.A design given by X of the form ( 14) is non singular if and only if Proof.For the design matrix X given by ( 14), we have and the determinant ( 16) is equal to zero if and only if is positive and then ( ) . So the lemma is proved.□ Theorem 4.2.The non-singular chemical balance weighing design with matrix X given by ( 8) is optimal if and only if Proof.From the conditions ( 5) and ( 15) it follows that a chemical balance weighing design is optimal if and only if the condition (17) holds.Hence the theorem.
If the chemical balance weighing design given by matrix X of the form ( 14) is optimal then ( ) . Theorem 4.2 yields a design matrix X of optimum chemical balance weighing design as Clearly such a design implies that each object is weighted 12 m = times in 28 n = weighing operations and λ ≤ ; then the design matrix X so formed using above method II is optimum chemical balance weighing design.
form a pairwise VB and EB design 2

A-Optimality of Chemical Balance Weighing Design
Some problems related to the optimality of chemical balance weighing designs were considered in the literature; see [17] [30] [31].Wong and Masaro [32] [33] gave the lower bound for ( ) and some construction methods of the A-optimal chemical balance weighing designs.
Let X be a n p × design matrix of a chemical balance weighing design.Then the following results from Ceranka et al. [34] give the lower bound for ( ) Theorem 5.1.For any nonsingular chemical balance weighing design with the design matrix where ( )  .The case when q p = ; we get the inequality given in Wong and Masaro [32].Definition 5.2.Any nonsingular chemical balance weighing design with the design matrix Comparing these two equalities we get Comparing these two equalities we get ( ) ( ) 24) is satisfied then we get the condition (25) from the last equation.Hence the theorem.

Discussion
The following Table 1 and Table 2 provide the list of pairwise variance and efficiency balanced block designs for Methods I and II respectively, which can be obtained by using certain known SBIB designs.

Conclusion
It is well known that pairwise balanced designs are not always efficiency as well as variance balanced.But in this research we have significantly shown that the proposed pairwise balanced designs are efficiency as well as variance balanced.Further there is a scope to propose different methods of construction to obtain the optimum chemical balance weighing designs and pairwise variance and efficiency balanced block designs, which will fulfill the optimality criteria by means of efficiency.In this research paper we also gave the conditions under which  ( ) MH α denote the reference number α in Raghavrao [30] and Marshal Halls [35] list.
the constructed chemical balance weighing designs lead to A-optimal designs.The only limitation of this research is that the obtained pairwise balanced designs all have large number of replications.[35] Hall Jr., M. (1986) Combinatorial Theory.John Wiley, New York.

4 . 1 NCorollary 3 . 5 . 1 D * ; 1 − is replaced by zero then the new design 1 D
If the SBIB design exists with parameters b N * so formed using above method is optimum chemical balance weighing design.If in the design * * so formed is a BIB design with parameters

2 N * of the new design 2 D 1 −
SBIB design D with the parameters b ν = , r k = , λ ; consider the λ blocks containing any pair of treatments say ( ) , θ ψ .Now rearranging the λ -blocks corresponding to the pair ( ) , θ ψ and giving the nega- tive sign to the treatments θ and ψ both; the matrix 1 N of design 1 D is obtained.Now doing the same procedure for all the 2 * so formed is the matrix having the elements 1, and

4 .
If the SBIB design exists with block size 6 r ≤ and 5

Corollary 4 . 5 . 4 . 6 . 2 D * ; 1 − is replaced by zero then the new design 2 D
If the SBIB design exists with parameters ( ) the design matrix X given by (14) is optimum chemical balance weighing design if and only if 7 v ≤ .Corollary If in the design * * so formed is a BIB design with parameters

Table 1 .
For method I.

Table 2 .
For method II.