Some Construction of Regular A-Optimal Spring Balance Weighing Designs for Even Number of Objects

[103]

Maágorzata Graczyk*

SOME CONSTRUCTION OF REGULAR A–OPTIMAL

SPRING BALANCE WEIGHING DESIGNS

FOR EVEN NUMBER OF OBJECTS

ABSTRACT. In the paper, the problem of construction of the spring balance weighing de-signs satisfying the criterion of A-optimality is discussed. The incidence matrices of the partially incomplete block designs are used for constructing the regular A-optimal spring balance weighing design.

Key words: A–optimal design, partially balanced incomplete block design, spring balance weighing design

I. INTRODUCTION

We study the experiment in which using

n

measurement operations we de-termine unknown measurements of p objects. The results of experiment can be written as y Xwe, where y is an nu1 random vector of the observations,

 

0,1

p nu

ĭ

X , where ĭ_nup

 

0,1 denotes the class of nu matrices p X

 

x_ij ,

n

i 1,2,..., , j 1,2,...,p, having entries x_ij 1 or 0 depending upon whether the

j

th object is included or excluded on the ith weighing operation,





'

2 1,w ,...,wp

w

w is a vector representing unknown measurements of objects and

e

is the nu1 random vector of errors. We assume that there are not system-atic errors and the errors are uncorrelated and have constant variance V2, i.e.

 

e 0n

E and Var

 

e V2I_n, where 0_n is nu1 vector of zeros, I_n is the

n

u

n

identity matrix. If the design matrix X is of full column rank, then all w_j are estimable and the variance matrix of their best linear unbiased estimator is

 

' 1.

2 

X X

V The matrix

 

X'X1 is called the information matrix of .X

*

Ph.D., Department of Mathematical and Statistical Methods, PoznaĔ University of Life Sci-ences.

In many problems concerning weighing experiments the A–optimal designs are considered. There are designs for that the trace of

 

X'X1 is minimal. The concept of A–optimality was considered, for instance, in the books of Raghava-rao (1971), Banerjee (1975) and in the paper of Jacroux and Notz (1983). More-over, the design for which the sum of variances of estimated measurements at-tains the lower bound is called the regular A–optimal design.

The main purpose of this paper is to obtain a new construction method, which gives the regular A–optimal spring balance weighing designs.

II. A–OPTIMAL DESIGN

For even ,p let us consider the design matrix X₁ĭ_hup

 

0,1 satisfying the condition given by Jacroux and Notz (1983)













' 1 ' 1 1 4 2 1 4 p p p p p h p hp 1 1 I X X     (1) where



1



4 p hp ,







1



4 2   p p h

are some integers, 1_p is the pu1 vector of ones. Moreover, for h let n

1 X X , (2) for h n1 let » ¼ º « ¬ ª ' 1 x X X , (3)

where x is the pu1 vector of elements equal to 0 or 1. The theorem below is from Graczyk (2010).

Theorem 1. For even p, any nonsingular spring balance weighing design

 

0,1

p nu

ĭ

X of the form (3) is regular A–optimal if condition (1) is fulfilled and 2 ' p p 1 x .

Some method of construction of the regular A–optimal spring balance weighing design for matrix Xĭ_nup

 

0,1 in the form (2) based on the inci-dence matrices of the balanced incomplete block designs is given in by Jacroux

and Notz (1983). Graczyk (2010) gave the conditions determining the regular A–optimal spring balance weighing design X in the form (3).

In this paper, for even ,p we propose new construction method of the regu-lar A–optimal spring balance weighing design Xĭ_nup

 

0,1 in the form (2) which widest the class of the design matrices given by Jacroux and Notz (1983) and in the form (3). This method is based on the incidence matrices of two group divisible designs with the same association scheme.

III. CONSTRUCTION OF THE DESIGN MATRIX

Now, we recall the definition of the partially balanced incomplete block de-sign with two associate classes given, for instance in Raghavarao and Padgett (2005).

A partially balanced incomplete block design with two associate classes is an arrangement of v treatments in b blocks, each of size k such that every treatment occurs at most once in a block and occurs in r blocks. Each treatment has exactly

n

_h treatments which are its hth associates, h 1,2. Two treatments which are hth associate occur together in exactly O_h blocks. The numbers

2 1, , , , , b r k O O

v are the parameters of the partially balanced incomplete block design. This design is usually identified by the associate scheme of treatments.

A group divisible design is a partially balanced incomplete block design with two associate classes for which the

v

ms

treatments may be divided into

m

groups of

s

distinct treatments each, such that treatments belonging to the same group are first associates and two treatments belonging to different groups are second associates, n1 s1,

n

2

m

s





1



,



s1



O1s



m1



O2 r



k 1



.

Now, based on the incidence matrices of two group divisible designs with the same association scheme, we construct the regular A–optimal spring balance weighing design.

For p let us consider the design matrix with v, X₁

>

N₁ N₂

@

', where N_t

is the incidence matrix of the group divisible design with the same association scheme with the parameters v, bt, rt, kt, O1t, O2t, t 1,2 and let

O O O O

O₁₁ _{12 21} ₂₂ . (4)

Theorem 2. Let p be even. If there exist the incidence matrices N1 and N2

of the group divisible design with the same association scheme with the parame-ters v, b_t, r_t, k_t, O_1t, O_2t, t 1,2, and if the conditions

(ii) 4O



v1

 

v2



b₁b₂



are fulfilled simultaneously and

(a) if n b1b2 then Xĭnup

 

0,1 in the form (2),

(b) if n b₁b₂1 then Xĭ_nup

 

0,1 in the form (3) for

2

' p

p

1

x ,

with X₁

>

N₁ N₂

@

' is the regular A–optimal spring balance weighing design. Proof. Let p and v h b1b2. From the condition (1) we have







2





1



2







' 1 ' 2 2 ' 1 1 1 ' 1 1 4 2 1 4 v v v v v b b v v b b 1 1 I N N N N X X        . (5)

On the other hand, N₁N₁' N₂N'₂



r₁r₂O



I_vO1_v1_v'. Thus (5) is satis-fied if and only if











1 4 2 2 1    v v b b

O , thus we are given in the condition (ii).

Considering Theorem 1 and the equality











 O  2 1 2 1 1 4 v r r v b b we obtain the condition (i). Hence the result.

Based on the book of Clatworthy (1973) we formulate next theorems giving the parameters of group divisible design having appropriate design numbers.

Theorem 3. Let v 4. Let N₁ and N₂ be the incidence matrices of the group divisible design with the same association with the parameters

(i) b₁ 8, r₁ 4, k₁ 2, O₁₁ 2, O₂₁ 1 (R1) and 1 , 2 , 5 , 10 _{2 2 12} 2 r k O b , O₂₂ 2 (R3), (ii) b₁ 8, r₁ 4, k₁ 2, O₁₁ 2, O₂₁ 1 (R1) and 2 , 2 , 8 , 16 _{2 2 12} 2 r k O b , O₂₂ 3 (R10), (iii) b₁ 8, r₁ 4, k₁ 2, O₁₁ 2, O₂₁ 1 (R1) and 1 , 0 , 2 , 2 , 4 _{2 2 12 22} 2 r k O O b (SR1), (iv) b₁ 10, r₁ 5, k₁ 2, O₁₁ 3, O₂₁ 1 (R2) and 3 , 1 , 2 , 7 , 14 _{2 2 12 22} 2 r k O O b (R7), (v) b₁ 10, r₁ 5, k₁ 2, O₁₁ 3, O₂₁ 1 (R2) and 2 , 2 , 10 , 20 _{2 2 12} 2 r k O b , O₂₂ 4 (R17), (vi) b₁ 10, r₁ 5, k₁ 2, O₁₁ 3, O₂₁ 1 (R2) and 2 , 0 , 2 , 4 , 8 _{2 2 12 22} 2 r k O O b (SR2), (vii) b₁ 10, r₁ 5, k₁ 2, O₁₁ 1, O₂₁ 2 (R3) and

b

₂

14

,

r

₂

7

,

k

₂

2

,

O

₁₂

3

, O₂₂ 2 (R6), (viii) b₁ 10, r₁ 5, k₁ 2, O₁₁ 1, O₂₁ 2 (R3) and b₂ 20, r₂ 10, k₂ 2, O₁₂ 4,

O

₂₂

3

(R16), (ix)

b

₁

12

,

r

₁

6

,

k

₁

2

,

O

₁₁

4

,

O

₂₁

1

(R4) and 0 , 2 , 6 , 12 _{2 2 12} 2 r k O b , O₂₂ 3 (SR3), (x) b₁ 14, r₁ 7, k₁ 2, O₁₁ 5, O₂₁ 1 (R5) and 0 , 2 , 8 , 16 _{2 2 12} 2 r k O b ,

O

₂₂

4

(SR4), (xi) b₁ 14, r₁ 7, k₁ 2, O₁₁ 3, O₂₁ 2 (R6) and 2 , 2 , 8 , 16 _{2 2 12} 2 r k O b , O₂₂ 3 (R10), (xii) b₁ 14, r₁ 7, k₁ 2, O₁₁ 3, O₂₁ 2 and (R6) 1 , 0 , 2 , 2 , 4 _{2 2 12 22} 2 r k O O b (SR1), (xiii) b₁ 14, r₁ 7, k₁ 2, O₁₁ 1, O₂₁ 3 (R7) and 4 , 2 , 8 , 16 _{2 2 12} 2 r k O b , O₂₂ 2 (R9), (xiv)

b

₁

16

,

r

₁

8

,

k

₁

2

,

O

₁₁

6

,

O

₂₁

1

(R8) and 0 , 2 , 10 , 20 _{2 2 12} 2 r k O b , O₂₂ 5 (SR5), (xv) b₁ 16, r₁ 8, k₁ 2, O₁₁ 4, O₂₁ 2 (R9) and 2 , 2 , 10 , 20 _{2 2 12} 2 r k O b , O₂₂ 4 (R17), (xvi) b₁ 16, r₁ 8, k₁ 2, O₁₁ 4, O₂₁ 2 (R9) and 0 , 2 , 4 , 8 _{2 2 12} 2 r k O b , O₂₂ 2 (SR2), (xvii) b₁ 16, r₁ 8, k₁ 2, O₁₁ 2, O₂₁ 3 (R10) and 4 , 2 , 10 , 20 _{2 2 12} 2 r k O b , O₂₂ 3 (R16), (xviii)

b

₁

18

,

r

₁

9

,

k

₁

2

,

O

₁₁

5

,

O

₂₁

2

(R12) and 1 , 2 , 9 , 18 _{2 2 12} 2 r k O b , O₂₂ 4 (R13), (xix)

b

₁

18

,

r

₁

9

,

k

₁

2

,

O

₁₁

5

,

O

₂₁

2

(R12) and 0 , 2 , 6 , 12 _{2 2 12} 2 r k O b , O₂₂ 3 (SR3), (xx)

b

₁

20

,

r

₁

10

,

k

₁

2

,

O

₁₁

6

,

O

₂₁

2

(R15) and 0 , 2 , 8 , 16 _{2 2 12} 2 r k O b , O₂₂ 4 (SR4), (xxi) b₁ 20, r₁ 10, k₁ 2, O₁₁ 4, O₂₁ 3 (R16) and

0

,

2

,

2

,

4

_{2 2 12} 2

r

k

O

b

,

1

,

0

,

2

,

2

,

4

_{2 2 12 22} 2

r

k

O

O

b

(SR1) and let X₁

>

N₁ N₂

@

'.

(b) if n b₁b₂ 1 then Xĭ_nup

 

0,1 in the form (3) for 2 ' v p 1 x

is the regular A–optimal spring balance weighing design.

Proof. This is proved by checking that the parameters given in (i) – (xxi) sat-isfy conditions (i) and (ii) of Theorem 2.

Theorem 4. Let v 6. Let

N

₁ and

N

₂ be the incidence matrices of the group divisible design with the same association with the parameters

(i) b₁ 12, r₁ 6, k₁ 3, O₁₁ 3, O₂₁ 2 (R43) and 3 , 3 , 9 , 18 _{2 2 12} 2 r k O b , O₂₂ 4 (R52), (ii)

b

₁

4

,

r

₁

2

,

k

₁

3

,

O

₁₁

0

,

O

₂₁

1

(SR18) and 2 , 3 , 3 , 6 _{2 2 12} 2 r k O b , b₂ 6, r₂ 3, k₂ 3, O₁₂ 2, O₂₂ 1 (R42), (iii) b₁ 4, r₁ 2, k₁ 3, O₁₁ 0, O₂₁ 1 (SR18) and 4 , 3 , 8 , 16 _{2 2 12} 2 r k O b , O₂₂ 3 (R48), (iv) b₁ 8, r₁ 4, k₁ 3, O₁₁ 0, O₂₁ 2 (SR19) and 4 , 3 , 6 , 12 _{2 2 12} 2 r k O b , b₂ 12, r₂ 6, k₂ 3, O₁₂ 4, O₂₂ 2 (R44), (v) b₁ 12, r₁ 6, k₁ 3, O₁₁ 0, O₂₁ 3 (SR20) and 6 , 3 , 9 , 18 _{2 2 12} 2 r k O b , O₂₂ 3 (R50), (vi) b₁ 6, r₁ 3, k₁ 3, O₁₁ 2, O₂₁ 1 (R42) and 2 , 3 , 7 , 14 _{2 2 12} 2 r k O b , O₂₂ 3 (R46), (vii)

b

₁

12

,

r

₁

6

,

k

₁

3

,

O

₁₁

4

,

O

₂₁

2

(R44) and 2 , 3 , 9 , 18 _{2 2 12} 2 r k O b , O₂₂ 4 (R51), (viii) b₁ 14, r₁ 7, k₁ 3, O₁₁ 2, O₂₁ 3 (R46) and 4 , 3 , 8 , 16 _{2 2 12} 2 r k O b , b₂ 16, r₂ 8, k₂ 3, O₁₂ 4, O₂₂ 3 (R48), and let X₁

>

N₁ N₂

@

'.

(a) If n b₁b₂ then Xĭ_nuv

 

0,1 in the form (2),

(b) if n b₁b₂1 then Xĭ_nuv

 

0,1 in the form (3) for

2

' v

v

1

x is the regular A–optimal spring balance weighing design.

Proof. It is easily seen that the parameters given in (i) – (viii) satisfy conditions (i) and (ii) of Theorem 2.

Theorem 5. Let v 8. Let

N

₁ and

N

₂ be the incidence matrices of the group divisible design with the same association with the parameters

(i) b₁ 12, r₁ 6, k₁ 4, O₁₁ 2, O₂₁ 3 (SR38) and 4 , 4 , 8 , 16 _{2 2 12} 2 r k O b , b₂ 16, r₂ 8, k₂ 4, O₁₂ 4, O₂₂ 3 (R98), (ii) b₁ 6, r₁ 3, k₁ 4, O₁₁ 3, O₂₁ 1 (S6) and 2 , 0 , 4 , 4 , 8 _{2 2 12 22} 2 r k O O b (SR36), (iii) b₁ 12, r₁ 6, k₁ 4, O₁₁ 6, O₂₁ 2 (S7) and 0 , 4 , 8 , 16 _{2 2 12} 2 r k O b , O₂₂ 4 (SR39), (iv) b₁ 8, r₁ 4, k₁ 4, O₁₁ 0, O₂₁ 2 (SR36) and 6 , 4 , 10 , 20 _{2 2 12} 2 r k O b , O₂₂ 4 (R103), (v)

b

₁

12

,

r

₁

6

,

k

₁

4

,

O

₁₁

0

,

O

₂₁

3

(SR37) and

6

,

4

,

8

,

16

_{2 2 12} 2

r

k

O

b

,

O

₂₂

3

(R99), (vi) b₁ 10, r₁ 5, k₁ 4, O₁₁ 3, O₂₁ 2 (R97) and 3 , 4 , 9 , 18 _{2 2 12} 2 r k O b , O₂₂ 4 (R101), and let X₁

>

N₁ N₂

@

'.

(a) If n b₁b₂ then Xĭ_nuv

 

0,1 in the form (2),

(b)if n b₁b₂1 then Xĭ_nuv

 

0,1 in the form (3) for

2

' v

v

1

x is the regular A–optimal spring balance weighing design.

Proof. An easy computation shows that the parameters given in (i) – (vi) sat-isfy conditions (i) and (ii) of Theorem 2.

Theorem 6. Let

N

₁ and

N

₂ be the incidence matrices of the group divisi-ble design with the same association with the parameters

(i) v 10, b₁ 8, r₁ 4, k₁ 5, O₁₁ 0, O₂₁ 2 (SR52) and 5 , 10 , 10 b₂ r₂ v , k₂ 5, O₁₂ 4, O₂₂ 2 (R139), (ii) v 10, b₁ 16, r₁ 8, k₁ 5, O₁₁ 0, O₂₁ 4 (SR54) and 10 , 20 , 10 b₂ r₂ v , k₂ 5 O₁₂ 8, O₂₂ 4 (R142), (iii) v 12, b₁ 10, r₁ 5, k₁ 6, O₁₁ 5, O₂₁ 2 (S28) and 6 , 12 , 12 b₂ r₂ v , k₂ 6, O₁₂ 0, O₂₂ 3 (SR67), (iv) v 14, b₁ 12, r₁ 6, k₁ 7, O₁₁ 0, O₂₁ 3 (SR81) and 7 , 14 , 14 b₂ r₂ v , k₂ 7, O₁₂ 6, O₂₂ 3 (R177),

(v) v 16, b₁ 14, r₁ 7, k₁ 8, O₁₁ 7, O₂₁ 3 (S63) and 8 , 8 , 16 , 16 b₂ r₂ k₂ v , O₁₂ 0, O₂₂ 4 (SR92), (vi) v 18, b₁ 16, r₁ 8, k₁ 9, O₁₁ 0, O₂₁ 4 (SR100) and 9 , 18 , 18 b₂ r₂ v , k₂ 9, O₁₂ 8, O₂₂ 4 (R197), (vii) v 20, b₁ 18, r₁ 9, k₁ 10, O₁₁ 9, O₂₁ 4 (S109) and 10 , 20 , 20 b₂ r₂ v , k₂ 10, O₁₂ 0, O₂₂ 5 (SR108), and let X₁

>

N₁ N₂

@

'.

(a) If n b₁b₂ then Xĭ_nuv

 

0,1 in the form (2),

(b)if n b₁b₂1 then Xĭ_nuv

 

0,1 in the form (3) for

2

' v

v

1

x is the regular A–optimal spring balance weighing design.

Proof. The main idea of proof is to show that the parameters given in (i) – (vii) satisfy conditions (i) and (ii) of Theorem 2.

IV. EXAMPLE

Let us consider the class ĭ_11u6

 

0,1 . Let note that n 11 and p 6. For 11

h , according to (1)





1 4 p

hp

is not integer. Hence Xĭ_11u6

 

0,1 in the form (2) doesn’t exist. On the other hand, for h 10, according to (1)





1 4 p

hp

is integer. Thus we consider Xĭ_11u6

 

0,1 in the form (3) for b₁b₂ 10 n1. Based on Theorem 4 (ii) there exist the group divisible block design with the association scheme with the parameters v 6, b₁ 4, r₁ 2, k₁ 3, O₁₁ 0, O₂₁ 1 (SR18) and

2 , 3 , 3 , 6 , 6 b₂ r₂ k₂ O₁₂ v , v 6, b₂ 6, r₂ 3, k₂ 3, O₁₂ 2, O₂₂ 1

(R42) given by the incidence matrices

» » » » » » » » ¼ º « « « « « « « « ¬ ª 0 1 1 0 1 0 1 0 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 1 1 N and » » » » » » » » ¼ º « « « « « « « « ¬ ª 1 1 0 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 1 0 0 1 1 1 0 1 0 0 1 2

scheme 6 3 5 2 4 1 . Hence » » » » » » » » » » » » » » » ¼ º « « « « « « « « « « « « « « « ¬ ª 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 0 1 0 0 1 1 0 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 0 1 1 0 0 0 1 0 0 0 1 1 1 X ĭ_11u6

 

0,1 for

>

1 10010

@

'

x is the regular A–optimal spring balance weighing design.

V. CONCLUSIONS

The problem of estimation of unknown measurements of objects in the model of spring balance weighing design is presented. Of particular interest is new construction method of design matrix X which allows to determine optimal design in the class of matrices ĭ_nup

 

0,1 in the cases not considered in

litera-ture.

REFERENCES

Banerjee K.S. (1950), How balanced incomplete block designs may be made to furnish orthogonal estimates in weighing designs, Biometrica, 37, 50 – 58.

Clatworthy W.H. (1973), Tables of Two-Associate-Class Partially Balanced Designs, NBS Ap-plied Mathematics Series 63.

Graczyk M. (2010), Some notes about spring balance weighing design (to appear.)

Jacroux M., Notz W. (1983), On the optimality of spring balance weighing designs, The Annals

of Statistics, 11, 970-978.

Raghavarao D. (1971), Constructions and Combinatorial Problems in Designs of Experiments, John Wiley Inc., New York.

Raghavarao D., Padgett L.V. (2005), Block Designs, Analysis, Combinatorics and Applications, Series of Applied Mathematics, 17, Word Scientific Publishing Co. Pte. Ltd.

Maágorzata Graczyk

O PEWNEJ KONSTRUKCJI REGULARNEGO A–OPTYMALNEGO SPRĉĩYNOWEGO UKàADU WAGOWEGO

W pracy przedstawiono zagadnienie konstrukcji sprĊĪynowego ukáadu wagowego speániają-cego kryterium A–optymalnoĞci. Do konstrukcji macierzy ukáadu wykorzystano macierze incy-dencji czĊĞciowo zrównowaĪonych ukáadów bloków.