Some Construction Methods of Optimum Chemical Balance Weighing Designs I
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1 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(6): Scholarlin Research Institute Journals, 3 (ISS: 4-76) jeteas.scholarlinresearch.org Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(6): (ISS: 4-76) Some Construction Methods of Optimum Chemical Balance Weighing Designs I Rashmi Awad and Shati Banerjee School of Statistics, Devi Ahilya University, Indore-45, M.P., India. Corresponding Author: Rashmi Awad Abstract Some construction methods of the optimum chemical balance weighing design with repeated blocs are proposed which are based on the incidence matrices of the nown symmetric balanced incomplete bloc designs and some pairwise balanced designs are also obtained which are efficiency as well as variance balanced Keywords: Balance incomplete bloc design; symmetric balanced incomplete bloc design; variance balanced design; efficiency balanced design; chemical balance weighing design. ITRODUCTIO The modern concept of experimental designs is primarily due to Sir R.A. Fisher, who formulated and developed the basic ideas of statistical designing in the period Later different methods of construction of balanced incomplete bloc designs have been given in literature, lie, Agrawal et al. (98), Calińsi (97), Hanani (975), Shrihande et al. (963) etc. Out of the two main concepts of balancing in incomplete bloc designs, Rao (958) gives a necessary and sufficient condition for a general bloc design to be variance balanced. The concept of efficiency balanced was introduced by Jones (959) and the nomenclature Efficiency Balanced is due to Puri et al. (975) and Williams (975). In many of the applications, BIB design would contain repeated blocs. Indeed, the statistical optimality of BIB designs is unaffected by the presence of repeated blocs. Van Lint (973) gave the concept of repeated blocs. Foody et al. (977), Hedayat et al. (97, 984), Ghosh et al. (), Cerana et al. (7, 9) presented some potential applications of the balanced incomplete bloc designs with repeated blocs. Another important concept is weighing design which was originally given by Yates (935). Later this illustration was formulated as a weighing problem by Hotelling (944). Over the years the problem has attained a distinctive growth and has acquired the status of a problem in the design of experiments. In the latter developments, attention has been made in the direction of obtaining "optimum weighing designs. Hotelling (944) gave the condition under which each of the variance of the estimated weights attained the lower bound. Prominent wor has been done by Dey (969, 97), Benerjee (948, 975), Raghavrao (959, 96) and many others in this area of research. In recent years Cerana et al. (,, 4, ) provided the way to deal with the 778 problem of estimating individual weights of objects, using a chemical balance weighing design under the restriction on the number in which each object is weighed. The new methods of constructing the optimum chemical balance weighing designs and a lower bound for the variance of each of the estimated weights from this chemical balance weighing design were obtained and a necessary and sufficient condition for this lower bound to be attained was proposed by Cerana et al. (). The constructions were based on the incidence matrices of balanced incomplete bloc designs, balanced bipartite bloc designs, ternary balanced bloc designs and group divisible designs. In this paper we proposed a construction method of obtaining optimum chemical balance weighing design using the incidence matrix of symmetric balanced incomplete bloc design. Let us consider ν treatments arranged in b blocs, such that the j th bloc contains j experimental units and the i th treatment appears r i times in the entire design, i =,,.,ν; j =,,,b. For any bloc design there exist a incidence matrix = [n ij ] of order ν x b, where n ij denotes the number of experiment units in the j th bloc getting the i th treatment. When n ij = or i and j, the design is said to be binary. Otherwise it is said to be nonbinary. In this paper we consider binary bloc designs only. The following additional notations are used = [.. b ] is the column vector of bloc sizes, r = [r r..r v ] is the column vector of treatment replication, K bxb = diag [.. b ], R vxv = diag [r r..r v ], Σr i = Σ j =n is the total number of experimental units, with this b = r and v =, Where a is the a x vector of ones. A bloc design with parameters (ν, b, r,, λ) consists of a of ν treatments in b blocs, each of size ( < ν) is
2 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(6): (ISS: 4-76) nown as BIBD if (i) In a bloc design each treatment is contained in precisely r blocs and (ii) each pair of points is contained in precisely λ blocs; where the parameters satisfy b = νr, r(-) = λ (ν-) and b v (Fisher s Inequality). A BIB design is said to be symmetric if b=ν and r=. In this case incidence matrix is a square matrix i.e. =. In case of symmetric balanced incomplete bloc design any two blocs have λ treatments in common. The information matrix for treatment effects C defined below as C = R K - () Though there have been balanced designs in various senses (see Puri and igam (975), Calińsi (977)). We will consider a balanced design of the following type. A bloc design is said to be balanced if every elementary contrast of treatment is estimated with the same variance (see Rao (958)). In this sense this design is also called a variance balance design. It is well nown that bloc design is a variance balanced if and only if it has C = μ [ I ν (/ ν ) ν ν ] () where μ is the unique nonzero eigenvalue of the matrix C with the multiplicity (v ), I v is the v v identity matrix. (3) A bloc design is called efficiency balanced if every contrast of treatment effects is estimated through the design with the same efficiency factor. Let us consider the matrix M o given by Calińsi (97) M o = R - K - (/n) ν r (4) and since M o S = ψ S, where ψ is the unique non zero eigen value of M o with multiplicity (ν-) and M o is given as (4). Calińsi (97) showed that for such designs every treatment contrast is estimated with the same efficiency (- ψ) and is a EB bloc design if and only if M o = ψ (I ν (/n) ν r) (5) Kageyama (974) proved that for the EB bloc design, eq n (5) is fulfilled if and only if C = (- ψ) (R (/n) r r) A bloc design is said to be pairwise balanced if b n j i j n i j (a constant) for all i, i, i i and a pairwise balanced bloc design is said to be binary if n ij = or only, for all i, j and it has parameters ν, b, r,, Λ (= λ, say) [in this case, when r = r v and = b, it is a BIB design with parameters ν, b, r,, λ]. 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 y X w e (6) n n p p n The elements of matrix X ( xij ), i,..., n, j,... p, tae the values as xij if thejth object isplacedin theleft pan in thei if thejth object isplacedin theright pan in thei if thejth object isnot weight edin thei where y is an n random column vector of the observed weights, w is the p column vector representing the unnown weights of objects and e is an n random column vector of errors such that E(e) = n and E(ee ) =. Where n is the n column vector with zero elements everywhere; in the nn identity matrix E stands for expectation and e is used for transpose of e. The normal equations estimating w are of the form X X wˆ X y, (7) where ŵ is the vector of the weights estimated by the least squares method. When the objects are placed on two pans in a chemical balance, we shall call the weighings two pan weighing and the design is nown as two pan design or chemical balance weighing design. In chemical balance weighing design, the elements of design matrix X = {x ij } taes the values as + if the j th object is placed in the left pan in the i th weighing, - if the j th object is placed in the right pan in the i th weighing and if the j th object is not weighted in the i th weighing. A chemical balance weighing design is said to be singular or nonsingular, depending 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 ran (= p). ow, if X is of full ran, that is, when X X is nonsingular, the least squares estimate of w is given by wˆ ( X X ) X y (8) and the variance - covariance matrix of ŵ is Var( wˆ) ( X X ) (9) Hotelling (944) 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 and proved the theorem that each of the variance of the estimated weights attains the minimum if and only if X X=nI p. VARIACE LIMIT OF ESTIMATED WEIGHTS Let X be an n p matrix of ran p of a chemical balance weighing design and let m j be the number of times in which j th object is weighed, j=,,., p (i.e. the m j be the number of elements equal to -and in j th column of matrix X). Then Cerana et al. () proved the following theorem: th th th weighing, weighing weighing 779
3 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(6): (ISS: 4-76) Theorem : For any n x p matrix X, of a nonsingular chemical balance weighing design, in which maximum number of elements equal to - and in columns is equal to m, where m=max {m,m,,m p }. Then each of the variances of the estimated weights attains the minimum if and only if X X () m I p 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, Var ( wˆ), j,,... p () m METHOD OF COSTRUCTIO In SBIB design D with the parameters v = b, r =, λ ; considering the r-blocs containing the element, =,,.,v. Rearranging the r-blocs corresponding to any element and doing the same for all the v treatments, then the vr blocs so formed by these r-blocs, is a BIB design D with incidence matrix. Theorem : The existence of a SBIB Design D with the parameters v = b, r =, λ ; implies the existence of a BIB design D with parameters v = v, b = vr, r =r, =, λ = λ. Proof : In SBIB design D with the parameters v = b, r =, λ ; the vr blocs can be obtained. Under the present method of construction, the design D yields the parameters v = v, b = vr, r = r + r(-) = r, =, which are obvious. Since in the original SBIB design any ( ) pair occurs in λ blocs. Due to new construction method these λ blocs contribute to λ in the following three ways: (i) When is repeated in λ blocs. (ii) When is repeated in λ blocs. (iii) When each of the (-) treatments [other than ( )] in each of the λ blocs is repeated. So the frequency of ( ) pair in the new design can be calculated as λ = λ + λ + λ ( ) = λ (iii)] This completes the proof. [For (i)] [For (ii)] [For COSTRUCTIO OF DESIG MATRIX Let be the incidence matrix of the SBIB design D with the parameters v = b, r =, λ. Consider the r- blocs containing the treatment; =,,., v. ow rearranging the r-blocs corresponding to any treatment ( =,,.,v ) and putting = -; the matrix of design D is obtained. Then doing the same for all the v treatments, the new incidence matrix * of new design D * so formed is the matrix having the elements,- and ; given as follows () Then combining the incidence matrix of SBIB design repeated s-times with * we get the matrix X of a chemical balance weighing design as stimes X (3) Under the present construction scheme, we have n = vr + sb and p = v. thus the each column of X will contain r( ) sr elements equal to, r elements equal to - and n ρ ρ elements equal to zero. Clearly such a design implies that each object is weighted m = ρ + ρ = r + sr times in n = vr + sb weighing operations. Lemma : A design given by X of the form (3) is non-singular if and only if r ( s) {( 4) s }. Proof : For the design matrix X given by (3), we have ( rsr) ( 4) s I ( 4 s J X X ) (4) r( s) ( 4) s I ( 4 s J ) and ( rsr) ( ) ( 4) s ( rsr) ( 4) s X X (5) the determinant in (5) is equal to zero if and only if r sr ( 4) s r ( s) {( 4) s } r sr ( ) ( 4) s r sr) ( ) ( 4) s or but ( is positive and then det (X X) = if and only if r ( s) {( 4) s }. So the lemma is proved. Theorem : The non-singular chemical balance weighing design with matrix X given by (3) is optimal if and only if ( 4) s. (6) Proof : From the conditions () and (4) it follows that a chemical balance weighing design is optimal if and only if the condition (6) holds. Hence the theorem. 78
4 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(6): (ISS: 4-76) 78 If the chemical balance weighing design given by matrix X of the form (3) is optimal then p j sr r w Var j.,,,... ; ˆ Example : Consider a SBIB design with parameters v = b =7, r = = 3, λ=; whose blocs are given by (,, 4), (, 3, 5), (3, 4, 6), (4, 5, 7), (, 5, 6), (, 6, 7), (, 3, 7). Theorem yields a design matrix X of optimum chemical balance weighing design as X Clearly such a design implies that each object is weighted m = times in n = 8 weighing operations and ˆ w j Var for each j =,,, 7. Corollary : If the SBIB design exists with parameters ν = b =, r = = ( ± d)/, λ = ( ± d +)/4; then the design matrix * so formed using above method is optimum chemical balance weighing design. Corollary : If the SBIB design exists with parameters (ν, ν-, ν-); then the design matrix X given by (3) is optimum chemical balance weighing design if and only if < v 5 s. Corollary 3: If in the design D * ; - is replaced by zero then the new design D ~ so formed is a BIB design with parameters V = ν, B = νr, R = r(-), K = -, Λ = λ (-). Then the structure times s ~ (7) (7) form a pairwise VB and EB design D* with parameters ν* = V, b* = B + sb, r* = R + sr,,, λ*= Λ + sλ, μ* s and ψ* s r. RESULT AD DISCUSSIO The following table provides a list of pairwise variance and efficiency balanced bloc designs which can be obtained by using certain nown SBIB designs. S. o. ν* b* r* λ* μ* ψ* Reference o. ** R(), MH() R(9), MH(5) R(3) R(43), MH(6) R(44) R(47) R(58) R(75) R(87), MH(5) ** The symbols R(α) and MH(α) denote the reference number α in Raghavrao (97) and Marshal Hall s (986) list. COCLUSIO It is well nown 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
5 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(6): (ISS: 4-76) 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 bloc designs, which will fulfill the optimality criteria by means of "efficiency. The only limitation of this research is that the obtained pairwise balanced designs are all have large number of replications. ACKOWLEDGEMET We are grateful to the anonymous reviewer for his constructive comments and valuable suggestions. REFERECES Agrawal, H.L. and Prasad, J. (98). Some methods of construction of balanced incomplete bloc designs with nested rows and columns. Biometria, 69, pp: Banerjee, K. S. (948). Weighing designs and balanced incomplete blocs. Ann. Math. Stat., 9, pp: Banerjee, K. S. (975). Weighing Designs for Chemistry, Medicine, Economics, Operations Research, Statistics. Marcel Deer Inc., ew Yor. Banerjee, S., Kageyama, S. and Rudra, S. (5). Constructions of nested pairwise efficiency and variance balanced designs. Commun. Statist.: Theory and Methods, 37, 7, pp: Calińsi, T. (97). On some desirable patterns in bloc designs. Biometrics, 7, pp: Calińsi, T. (977). On the notation of balance bloc designs. Recent Developments in Statistics, Amsterdam, orth-holland Publishing Company, pp: Cerana, B. and Graczy, M. (). Optimum chemical balance weighing designs under the restriction on the number in which each object is weighed. Discussiones Mathematicae Probability and Statistics,, pp: 3-. Cerana, B. and Graczy, M. (). Optimum chemical balance weighing designs based on balanced incomplete bloc designs and balanced bipartite bloc designs. Mathematica,, pp: 9-7. Cerana, B. and Graczy, M. (4). Ternary balanced bloc designs leading to chemical balance weighing designs for v + objects. Biometrica, 34, pp: Cerana, B. and Graczy, M. (7). Variance Balanced Bloc Designs with repeated blocs. Applied Mathematical Sciences, Hiari Ltd., Vol., o. 55, pp: Cerana, B. and Graczy, M. (9). Some notes about Efficiency Balanced Bloc Designs with repeated blocs. Metodološi Zvezi, Vol. 6, o., pp: Cerana, B. and Graczy, M. (). Some construction of optimum chemical balance weighing designs. Acta Universitatis Lodziensis, Folia economic, pp: Dey, A. (969). A note on weighing designs. Ann. Inst. Stat. Math.,, pp: Dey, A. (97). On some chemical balance weighing designs. Aust. J. Stat., 3 (3), pp: Dey, A. (986). Theory of Bloc Designs. Wiley Eastern Limited. Foody, W. and Hedayat, A. (977). On Theory and Applications of balanced incomplete bloc designs with repeated blocs. The annals of statistics, Vol. 5, o. 5, pp: Ghosh, D.K. and Shrivastava, S.B. (). A class of balanced incomplete bloc designs with repeated blocs. Journal of Applied Statistics, Vol. 8, o. 7, pp: Hall, M. Jr. (986). Combinatorial Theory. John Wiley, ew Yor. Hanani, H. (975): Balanced Incomplete Bloc Designs and related designs. Discrete Math., pp: Hedayat, A. and Federer, W.T. (97). Pairwise and variance balanced incomplete bloc designs. Ann. Inst. Statist. Math., 6, pp: Hedayat, A. and Hwang, H. L. (984). BIB(8, 56,, 3, 6) and BIB(, 3, 9, 3, ) designs with repeated blocs. J. Comb. Th., A, 36, pp: Hotelling, H. (944). Some improvements in weighing and other experimental techniques. Ann. Math. Stat., 5, pp: Jones, R.M. (959). On a property of incomplete blocs. J. Roy. Statist. Soc. B,, pp: Kageyama, S. (974). On properties of efficiency balanced designs. Commun. Statist. Theor. Math., A, 9, pp: Puri, P.D. and igam, A.K. (975). On patterns of efficiency balanced designs. J. Roy. Statist. Soc., B, 37, pp: Raghavarao, D. (959). Some optimum weighing designs. Ann. Math. Stat., 3, pp:
6 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(6): (ISS: 4-76) Raghavarao, D. (96). Some aspects of weighing designs. Ann. Math. Stat., 3, pp: Raghavarao, D. (97). Constructions and Combinatorial Problems in Designs of Experiments. John Wiley, ew Yor. Rao, V.R. (958). A note on balanced designs. Ann. Math. Statist, 9, pp: Shrihande, S.S. and Raghavarao, D. (963). A method of construction of incomplete bloc designs. Sanhya, A, 5, pp: Van Lint, J.H. (973). Bloc Designs with repeated blocs and (b, r, λ) =. Journal of Combi. Theory, Series A, 5, pp: Williams, E.R. (975). Efficiency Balanced Designs. Biometria, 6, pp: Yates, F. (935). Complex experiments. J. Roy. Stat. Soc. Suppl.,, pp:
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