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Methods of constructing the optimum chemical balance weighing designs from symmetric balanced incomplete block designs are proposed with illustration. As a by-product pairwise efficiency and variance balanced designs are also obtained.

Originally Yates (see [

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. [

Let us consider a block design in which

matrix N are denoted by

perimental units and the

A balanced incomplete block design is an arrangement of

A balanced incomplete block design is said to be symmetric if

Balancing of design in various senses has been given in the literature (see [

1) A block design is said to be variance balanced if and only if its C-matrix,

2) A block design is said to be efficiency balanced if

design N, the information matrix is given as

3) A block design is said to be pairwise balanced if

and

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 [

so that

The following additional notations are used

the

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

where

The normal equations for estimating

where

Singularity or non-singularity of a weighing design depends on whether the matrix

and the variance-covariance matrix of

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

Hotelling (see [

Ceranka et al. (see [

Theorem 2.1. For any

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 [

Proposition 2.2. Existence of BIB design D 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

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

Consider a SBIB design D with the parameters (

1. Give the negative sign to the treatment

2. Give the negative sign to the treatment

The matrix

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

Under the present construction scheme, we have

column of X will contain

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

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

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

Corollary 3.7. If the SBIB design exists with parameters (

Remark: In SBIB design with parameters (

Corollary 3.8. If in the design

design with parameters

form a pairwise VB and EB design

Consider a SBIB design D with the parameters (

1. Give the negative sign to the treatment

2. Give the negative sign to the treatment

3. Give the negative sign to both the treatments

The matrix

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

Under the present construction scheme, we have

will contain

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

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

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

Corollary 4.12. If the SBIB design exists with parameters (

Corollary 4.13. If in the design

design with parameters

form a pairwise VB and EB design

The following

S. No. | Reference No.^{**} | ||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 5 | 70 | 32 | 2 | 4 | 12 | 22.5 | 0.2968 | R (4) |

2 | 7 | 98 | 32 | 2 | 4 | 8 | 21 | 0.3437 | R (11) |

3 | 13 | 182 | 32 | 2 | 4 | 4 | 19.5 | 0.3906 | R (37), MH (3) |

S. No. | Reference No.^{**} | ||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 5 | 110 | 32 | 2 | 4 | 12 | 22.5 | 0.2968 | R (4) |

2 | 7 | 84 | 12 | 1 | 3 | 2 | 4.667 | 0.6111 | R (10), MH (1) |

3 | 7 | 154 | 32 | 2 | 4 | 8 | 21 | 0.3437 | R (11) |

4 | 13 | 286 | 32 | 2 | 4 | 4 | 19.5 | 0.3906 | R (37), MH (3) |

^{**}The symbols R(a) and MH(a) denote the reference number a in Raghavrao [

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