Originally Yates (see [1] ) gave the concept of weighing design. Afterward his work was formulated by Hotelling (see [2] ) and he gave the condition of attaining the lower bound by each of the variance of the estimated weights. Many statisticians did prominent work in obtaining optimum weighing designs (see [3] -[7] ). In recent years, different methods of constructing the optimum chemical balance weighing designs; using the incidence matrices of known balanced incomplete block designs, balanced bipartite block designs, ternary balanced block designs and group divisible designs have been given in the literature (see [8] -[11] ).

Construction methods of obtaining optimum chemical balance weighing designs using the incidence matrices of symmetric balanced incomplete block designs have been given by Awad et al. [12] - [14] ; some pairwise balanced designs are also been obtained which are efficiency as well as variance balanced. In this paper; some other new construction methods of obtaining optimum chemical balance weighing designs using the incidence matrices of known symmetric balanced incomplete block designs are propose. Some more pairwise efficiency as well as variance balanced designs are also been proposed.

Let us consider a block design in which treatments arranged in b blocks and elements of the incidence

matrix N are denoted by, for all;, such that the block contains ex-

perimental units and the treatment appears times in the entire design. A balanced block design is said to be (a) binary when or 1,. Otherwise, it is said to be nonbinary (see [7] ); (b) ternary if or 2, , and it has parameters, b, , , r, k,; where, are the number of times 1, 2 occurs in the incidence matrix, respectively (see [15] ); (c) generalized binary if or x, and some positive integer, and it has parameters, b, r, k, (see [16] ).

A balanced incomplete block design is an arrangement of symbols (treatments) into b sets (blocks) such that (1) each block contains distinct treatments; (2) each treatment appears in different blocks and; (3) every pair of distinct treatments appears together in exactly blocks. Here, the parameters of balanced incomplete block design (, b, r, k,) are related by the following relations

A balanced incomplete block design is said to be symmetric if ( consequently,). In this case, incidence matrix N is a square matrix i.e.. In case of symmetric balanced incomplete block design any two sets have symbols in common.

Balancing of design in various senses has been given in the literature (see [17] [18] ). In this paper, we consider the balanced design of the following types: 1) variance balanced block designs; 2) efficiency balanced block designs and; 3) pairwise balanced block designs.

1) A block design is said to be variance balanced if and only if its C-matrix, , satisfies, for some constant (see [19] - [22] ); where is the unique nonzero eigen value of the matrix C with the multiplicity, is the identity matrix.

2) A block design is said to be efficiency balanced if, for some constant (see [20] - [22] ); where is the unique non zero eigen value with multiplicity. For the EB block

design N, the information matrix is given as; see [23]

3) A block design is said to be pairwise balanced if (a constant) for all i,; where

and. A pairwise balanced block design is said to be binary if or 1 only, for all i, j and it has parameters, b, , , (=, say) (in this case, when and, it is a BIB design with parameters, b, , ,).

A design is said to form a nested structure, when there are two sources of variability and one source is nested within other. Preece (see [24] ) introduced a class of nested BIB designs with treatments, each replicated r times, with two systems of blocks, such that (a) the second system nested within the first, with each block from the first system (called super blocks) containing exactly “m” blocks from the second system (called sub-blocks). (b) Ignoring the sub-blocks, leaves a BIB design with parameters, , r, ,. (c) Ignoring the super- blocks, leaves a BIB design with parameters, , r, ,. The, , , r, , , , and m are called the parameters of a nested BIB design. The parameters satisfy the following conditions:

so that

The following additional notations are used is the column vector of block sizes, is the column vector of treatment replication, , ,

is the total number of experimental units, with this and, Where is

the vector of ones. Furthermore represents the loss of information, i.e., represents an effi- ciency of the design, represents the variance of any normalized contrast in the intra block analysis (see [20] [23] [25] [26] ).

For given p objects to be weighted in groups in n weightings, a weighing designs consists of n groupings of the p objects and the least square estimates of the weight of the objects can be obtained by the usual methods when. Using matrix notations the general linear model can be written as:

where is column vector of the weights of the objects, is the column vector of the unknown weights of objects and is vector of error components in the different observations such that and. Also, is the matrix of known quantities called design matrix whose entries are +1, −1 or 0. Let matrix X takes the values as:

The normal equations for estimating are as follows:

where is the vector of the weights estimated by the least squares method.

Singularity or non-singularity of a weighing design depends on whether the matrix is singular or non- singular, respectively. When X is of full rank, then it is obvious that the matrix is non-singular. Then in this case the least squares estimate of is given by

and the variance-covariance matrix of is

A weighing design is said to be the chemical balance weighing design if the objects are placed on two pans in a chemical balance. In a chemical balance weighing design, the elements of design matrix takes the values as +1 if the object is placed in the left pan in the weighing, −1 if the object is placed in the right pan in the weighing and 0 if the object is not weighted in the weighing.

Hotelling (see [2] ) has shown that the precision of the estimates of the weight of the object increases further by placing in the other pan of the scale those objects not included in the weighing and thus using two pan chemical balance. He proved that if n weighing operations have been done to determine the weight of p = n objects, the minimum attainable variance for each of the estimated weights in this case is and he also shown that each of the variance of the estimated weights attains the minimum if and only if. A design satisfying this condition is called an optimum chemical balance weighing design.

Ceranka et al. (see [8] ) studied the problem of estimating individual weights of objects, using a chemical balance weighing design under the restriction on the number of times in which each object is weighed. A lower bound for the variance of each of the estimated weights from this chemical balance weighing design is obtained and a necessary and sufficient condition for this lower bound to be attained was given. Then Ceranka et al. (see [8] ) proved the following theorem:

Theorem 2.1. For any matrix X, of a nonsingular chemical balance weighing design, in which maxi- mum number of elements equal to −1 and 1 in columns is equal to m, where (where be the number of elements equal to −1 and 1 in column of matrix X). 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,

Preece [24] introduced a class of nested BIB designs. Using this concept Banerjee; (see [27] - [29] ) constructed the nested EB as well as VB designs, where the sub-blocks form ternary design D with parameters (, b, , , r, k,) while the super-blocks form generalized binary design D with parameters (, b, r, k,); where i = 1, 2.

Proposition 2.2. Existence of BIB design D with parameters implies the existence of a nested EB as well as VB design. The sub-blocks form an EBT as well as VBT design with parameters

, , , , , ,

, ,

while the super-blocks form a generalized binary EB as well as VB design with parameters

, , , , ,

,

Proposition 2.3. Existence of BIB design D with parameters implies the existence of a nested EB as well as VB design. The sub-blocks form an EBT as well as VBT design with parameters

, , , , , ,

, ,

while the super-blocks form a generalized binary EB as well as VB design with parameters

, , , , ,

,

Consider a SBIB design D with the parameters (, k,); each pair of treatments occurs together in blocks. Take any pair of treatments, say, (,) from this design. Then make two blocks from the design which contains the pair (,).

1. Give the negative sign to the treatment and eliminate the treatment while the other (k−2) remaining treatments of the same block remain as it is.

2. Give the negative sign to the treatment and eliminate the treatment while the other (k−2) remaining treatments of the same block remain as it is.

The matrix of design is obtained. Now doing the same procedure for all possible pairs of treatments, we obtain matrix;. Then the incidence matrix of the new design so

formed is the matrix having the elements 1, −1 and 0; given as follows by juxtaposition

Then combining the incidence matrix N of SBIB design repeated s-times with we get the matrix X of a chemical balance weighing design as

Under the present construction scheme, we have and. Thus the each

column of X will contain elements equal to 1, elements equal to −1 and elements equal to zero. Clearly such a design implies that each object is weighted

times in weighing operations.

Lemma 3.4. A design given by X of the form (8) is non singular if and only if

.

Proof. For the design matrix X given by (8), we have

and

the determinant (10) is equal to zero if and only if

or

but is positive and then if and only if

. So the lemma is proved.

Theorem 3.5. 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

Example 3.6. Consider a SBIB design with parameters, ,; whose blocks are given by (1,2,3), (1,2,4), (1,3,4), (2,3,4).

Theorem 3.5 yields a design matrix X of optimum chemical balance weighing design as

Clearly such a design implies that each object is weighted m = 18 times in n = 32 weighing operations and for each j = 1, 2, 3, 4.

Corollary 3.7. If the SBIB design exists with parameters (, k,); then the design matrix X so formed in (8) is optimum chemical balance weighing design iff and.

Remark: In SBIB design with parameters (, k,); if k = 0 then the design matrix given in (8) is perfectly optimum and s = 0 in this case.

Corollary 3.8. If in the design; −1 is replaced by zero then the new design so formed is a BIB

design with parameters, , , ,. Then the struc- ture

form a pairwise VB and EB design with parameters

, , , , , ,

and

.

Consider a SBIB design D with the parameters (, k,); each pair of treatments occurs together in blocks. Take any pair of treatments, say, (,) from this design. Then make three blocks from the design which contains the pair (,).

1. Give the negative sign to the treatment and eliminate the treatment while the other (k−2) remaining treatments of the same block remain as it is.

2. Give the negative sign to the treatment and eliminate the treatment while the other (k−2) remaining treatments of the same block remain as it is.

3. Give the negative sign to both the treatments and while the other remaining treatments of the same block remain as it is.

The matrix of design is obtained. Now doing the same procedure for all possible pairs of treatments, we obtain matrix;. Then the incidence matrix of the new design so

formed is the matrix having the elements 1, −1 and 0; given as follows juxtaposition:

Then combining the incidence matrix N of SBIB design repeated s-times with we get the matrix X of a chemical balance weighing design as:

Under the present construction scheme, we have and. Thus the each column of X

will contain elements equal to 1, elements equal to −1 and elements equal to zero. Clearly such a design implies that each object is weighted

times in weighing operations.

Lemma 4.9. 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

or

but is positive and then if and only if. So the lemma is proved.

Theorem 4.10. The non-singular chemical balance weighing design with matrix X given by (14) 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

Example 4.11. Consider a SBIB design with parameters, ,; whose blocks are given by (1,2,3), (1,2,4), (1,3,4), (2,3,4).

Theorem 4.10 yields a design matrix X of optimum chemical balance weighing design as

X=

Clearly such a design implies that each object is weighted m = 30 times in n = 48 weighing operations and

for each j = 1, 2, 3, 4.

Corollary 4.12. If the SBIB design exists with parameters (, k,); then the design matrix X of the form (14) is optimum chemical balance weighing design iff and.

Corollary 4.13. If in the design; −1 is replaced by zero then the new design so formed is a BIB

design with parameters, , , ,. Then the structure

form a pairwise VB and EB design with parameters

, , , , , ,

and

.

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.

^**The symbols R(a) and MH(a) denote the reference number a in Raghavrao [30] and Marshal Halls [31] list.

In this research, we have significantly shown that the obtained designs are pairwise balanced as well as effi- ciency balanced. The only limitation of this research is that the obtained pairwise balanced designs are all have large number of replications.

We are grateful to the anonymous referees for their constructive comments and valuable suggestions.

RashmiAwad,ShaktiBanerjee, (2016) Some Construction Methods of Optimum Chemical Balance Weighing Designs III. Open Journal of Statistics,06,37-48. doi: 10.4236/ojs.2016.61006