Some Construction Methods of Optimum Chemical Balance Weighing Designs II

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1 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 5(): Scholarlin Research Institute Journals, 4 (ISS: 4-76) jeteas.scholarlinresearch.org Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 5():39-44 (ISS: 4-76) Some Construction Meods of Optimum Chemical Balance Weighing Designs II Rashmi Awad and Shati Banerjee School of Statistics, Devi Ahilya University, Indore-45, M.P., India. Corresponding Auor: Rashmi Awad Abstract Some construction meods of e optimum chemical balance weighing designs and pairwise efficiency and variance balanced designs are proposed, which are based on e incidence matrices of e nown symmetric balanced incomplete bloc designs. Keywords: balance incomplete bloc design; symmetric balanced incomplete bloc design; variance balanced design; efficiency balanced design; weighing design; chemical balance weighing design; optimum chemical balance weighing design ITRODUCTIO The modern concept of experimental designs is primarily due to Sir R.A. Fisher, who formulated and developed e basic ideas of statistical designing in e period Later different meods 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 e 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 e nomenclature Efficiency Balanced is due to Puri et al. (975) and Williams (975). In many of e applications, BIB design would contain repeated blocs. Indeed, e statistical optimality of BIB designs is unaffected by e presence of repeated blocs. Van Lint (973) gave e concept of repeated blocs. Hedayat et al. (97), Ghosh et al. (), Cerana et al. (7, 9) presented some potential applications of e balanced incomplete bloc designs wi repeated blocs. Anoer important concept is weighing design which was originally given by Yates (935). Later is illustration was formulated as a weighing problem by Hotelling (944). Over e years e problem has attained a distinctive grow and has acquired e status of a problem in e design of experiments. In e latter developments, attention has been made in e direction of obtaining "optimum weighing designs. Hotelling (944) gave e condition under which each of e variance of e estimated weights attained e lower bound. Prominent wor has been done by Dey (969, 97), Benerjee (948, 975), Raghavrao (959) and many oers in is area of research. In recent years Cerana et al. (,, 4, ) provided e way to deal wi e problem of estimating individual weights of objects, using a chemical balance weighing design under e restriction on e number in which each object is weighed. The new meods of constructing e optimum chemical balance weighing designs and a lower bound for e variance of each of e estimated weights from is chemical balance weighing design were obtained and a necessary and sufficient condition for is lower bound to be attained was proposed by Cerana et al. (). The constructions were based on e incidence matrices of balanced incomplete bloc designs, balanced bipartite bloc designs, ternary balanced bloc designs and group divisible designs. Awad et al. (3) gave e construction meods of obtaining optimum chemical balance weighing designs using e incidence matrices of symmetric balanced incomplete bloc designs and some pairwise balanced designs were also been obtained which were efficiency as well as variance balanced. In at series we now propose anoer new construction meod of obtaining optimum chemical balance weighing designs using e incidence matrices of symmetric balanced incomplete bloc designs and some more pairwise efficiency as well as variance balanced designs are also propose. Let us consider ν treatments arranged in b blocs, such at e j bloc contains j experimental units and e i treatment appears r i times in e entire design, i =,,.,ν; j =,,,b. For any bloc design ere exist a incidence matrix = [n ij ] of order ν x b, where n ij denotes e number of experiment units in e j bloc getting e i treatment. When n ij = or i and j, e design is said to be binary. Oerwise it is said to be nonbinary. 39

2 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 5():39-44 (ISS: 4-76) In is paper we consider binary bloc designs only. The following additional notations are used = [.. b ] is e column vector of bloc sizes, r = [r r..r v ] is e column vector of treatment replication, K bxb = diag [.. b ], R vxv = diag [r r..r v ], Σr i = Σ j =n is e total number of experimental units, wi is b = r and v =, Where a is e a x vector of ones. A bloc design wi parameters (ν, b, r,, λ) consists of a of ν treatments in b blocs, each of size ( < ν) is 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 e 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 is 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 ere have been balanced designs in various senses (see Puri and igam (975), Calińsi (977)). We will consider a balanced design of e following type. A bloc design is said to be balanced if every elementary contrast of treatment is estimated wi e same variance (see Rao (958)). In is sense is design is also called a variance balance design. It is well nown at bloc design is a variance balanced if and only if it has C = μ [ I ν (/ ν ) ν ν ] () where μ is e unique nonzero eigen value of e matrix C wi e multiplicity (v ), I v is e v v identity matrix. A bloc design is called efficiency balanced if every contrast of treatment effects is estimated rough e design wi e same efficiency factor. Let us consider e matrix M o given by Calińsi (97) M o = R - K - (/n) ν r if e j object is placed in e right pan in e (3) i and since M o S = ψ S, where ψ is e unique non zero eigen value of M o wi multiplicity (ν-) and M o is given as (3). Calińsi (97) showed at for such designs every treatment contrast is estimated wi e same efficiency (- ψ) and is a EB bloc design if and only if M o = ψ (I ν (/n) ν r) (4) Kageyama (974) proved at for e EB bloc design, eq n (4) is fulfilled if and only if C = (- ψ) (R (/n) r r) (5) 4 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 is case, when r = r v and = b, it is a BIB design wi parameters ν, b, r,, λ]. Weighing designs consists of n groupings of e p objects and suppose we want to determine e individual weights of p objects. We can fit e results into e general linear model y w e (6) n n p p n The elements of matrix ( xij ), i,..., n, j,... p, tae e values as if ej object is placedin eleft pan in ei weighing, xij if ej object is placedin eright pan in ei weighing if ej object is not weight edin ei weighing where y is an n random column vector of e observed weights, w is e p column vector representing e unnown weights of objects and e is an n random column vector of errors such at E(e) = n and E(ee ) = I n. Where n is e n column vector wi zero elements everywhere; in e nn identity matrix E stands for expectation and e is used for transpose of e. The normal equations estimating w are of e form wˆ y, (7) where ŵ is e vector of e weights estimated by e least squares meod. When e objects are placed on two pans in a chemical balance, we shall call e weighings two pan weighing and e design is nown as two pan design or chemical balance weighing design. In chemical balance weighing design, e elements of design matrix = {x ij } taes e values as + if e j object is placed in e left pan in e i weighing, - weighing and if e j object is not weighted in e i weighing. A chemical balance weighing design is said to be singular or nonsingular, depending on wheer e matrix is singular or nonsingular, respectively. It is obvious at e matrix is nonsingular if and only if e matrix is of full column ran (= p). ow, if is of full ran, at is, when is nonsingular, e least squares estimate of w is given by wˆ ( ) y (8) and e variance - covariance matrix of ŵ is Var( wˆ) ( ) Hotelling (944) has shown at if n weighing operations are to determine e weights of

3 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 5():39-44 (ISS: 4-76) p = n objects, e minimum attainable variance for each of n e estimated weights in is case is and proved e eorem at each of e variance of e estimated weights attains e minimum if and only if =ni p. VARIACE LIMIT OF ESTIMATED WEIGHTS Let be an n p matrix of ran p of a chemical balance weighing design and let m j be e number of times in which j object is weighed, j=,,., p (i.e. e m j be e number of elements equal to - and in j column of matrix ). Then Cerana et al. () proved e following eorem: Theorem: For any n x p matrix, 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 e variances of e estimated weights attains e minimum if and only if () m I p Also a nonsingular chemical balance weighing design is said to be optimal for e estimating individual weights of objects if e variances of eir estimators attain e lower bound given by, Var ( wˆ), j,,... p () m In SBIB design D (v, r, λ); e bloc intersection between any two blocs is constant i.e. λ. Using is concept Banerjee (985) proved e following result; Proposition : In SBIB Design D (v, r, λ); each pair of treatments intersect in blocs. Then e blocs formed by ese λ common treatments considering all possible pairs of blocs, is a BIB design D wi parameters v = v, b =, r = r ( ), =, λ =. METHOD OF COSTRUCTIO OF DESIG MATRI In SBIB design D wi e parameters v = b, r =, λ; each pair of treatments occur togeer in λ blocs. Consider any pair of treatments, say, (θ, φ) from is design. Repeating all e λ blocs which contain e pair (θ, φ); so we obtain λ blocs in all, corresponding to at (θ, φ) pair. ow ) Corresponding to e pair (θ, φ), give e negative sign to θ while e treatment φ and e oer (-) remaining treatments of e same bloc remain as it is. ) Then corresponding to e same pair (θ, φ), give e negative sign to φ in e repeated bloc while e treatment θ and e oer (-) remaining treatments of e repeated bloc remain as it is. Thus matrix of design D is obtained. ow doing e same procedure for all possible pairs of treatments, e incidence matrix * of e new design D * so formed is e matrix having e elements,- and ; given as follows * () Then combining e incidence matrix of SBIB design repeated s-times wi * we get e matrix of a chemical balance weighing design as stimes (3) Under e present construction scheme, we have n= + sb and p = v. Thus e each column of will contain ( ) ( ) sr elements equal to, ( ) elements equal to - and [ + s b] ρ ρ elements equal to zero. Clearly such a design implies at each object is weighted m = ρ + ρ = λ (v ) + s r times in n = + s b weighing operations. Lemma : A design given by of e form (3) is non singular if and only if λ (v ) ( ) + s r = λ [( ) ( 4) + s]. Proof: For e design matrix given by (3), we have { ( )( ) sr} ( )( 4) s I ( )( 4 s J { ( )( ) sr} ( )( 4) s I ( )( 4 s J ) ) and { ( )( ) sr} ( ) ( )( { ( )( ) sr} ( ) ( 4) s (4) 4) s (5) e determinant (5) is equal to zero if and only if ( )( ) sr ( ) ( 4) s ( )( ) sr ( ) ( 4) s or ( )( ) sr ( ) ( ) ( 4) s but { ( )( ) sr} ( ) ( ) ( 4) s is positive and en det ( ) = if and only if λ (v ) ( ) + s r = λ [( ) ( 4) + s]. So e lemma is proved. Theorem : The non-singular chemical balance weighing design wi matrix given by (3) is optimal if and only if λ {( ) ( 4) + s} =. (6) 4

4 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 5():39-44 (ISS: 4-76) 4 Proof: From e conditions () and (4) it follows at a chemical balance weighing design is optimal if and only if e condition (6) holds. Hence e eorem. If e chemical balance weighing design given by matrix of e form (3) is optimal en p j sr w Var j.,,,... ; ) ( ˆ Example : Consider a SBIB design wi parameters v = b =4, r = = 3, λ=; whose blocs are given by (,, 3), (,, 4), (, 3, 4), (, 3, 4). Theorem yields a design matrix of optimum chemical balance weighing design as Clearly such a design implies at each object is weighted m = 4 times in n = 3 weighing operations and 4 ˆ w j Var for each j =,, 3, 4. Corollary : If e SBIB design exists wi bloc size = 4; en e design matrix * so formed using above meod is optimum chemical balance weighing design. Corollary : If e SBIB design exists wi parameters (ν, ν -, ν-); en e design matrix given by (3) is optimum chemical balance weighing design if and only if 3 v 4. Corollary 3: If e SBIB design exists wi parameters (ν, r, λ); en e design matrix given by (3) is optimum chemical balance weighing design if and only if ν 7 and = 3. Corollary 4: If in e design D * ; - is replaced by zero en e new design D ** so formed is a BIB design wi parameters V = ν, B =, R = λ (- ) (v-), K = -, Λ =. Then e structure times s (7) form a pairwise VB and EB design D* wi parameters ν* = V, b* = B + sb, r* = R + sr, *, *, λ*= Λ + sλ, s ) ( and 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() R(8) R(), MH() R(9), MH(5) R(58) ** The symbols R(α) and MH(α) denote e reference number α in Raghavrao (97) and Marshal Hall s (986) list.

5 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 5():39-44 (ISS: 4-76) COCLUSIO It is well nown at pairwise balanced designs are not always efficiency as well as variance balanced. But in is research we have significantly shown at e proposed pairwise balanced designs are efficiency as well as variance balanced. Furer ere is a scope to propose different meods of construction to obtain e optimum chemical balance weighing designs and pairwise variance and efficiency balanced bloc designs, which will fulfill e optimality criteria by means of "efficiency. The only limitation of is research is at e obtained pairwise balanced designs are all have large number of replications. ACKOWLEDGEMET We are grateful to e anonymous reviewer for his constructive comments and valuable suggestions. REFERECES Agrawal, H.L. and Prasad, J. (98). Some meods of construction of balanced incomplete bloc designs wi nested rows and columns. Biometria, 69, pp: Awad, R. and Banerjee, S. (3). Some Construction Meods of Optimum Chemical Balance Weighing Designs I. Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS), 4(6), pp: Banerjee, S. (985). Some Combinatorial Problems in Incomplete Bloc Designs. Thesis, Devi Ahilya University. Indore. Banerjee, K. S. (948). Weighing designs and balanced incomplete blocs. Ann. Ma. 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 Meods, 37, 7, pp: Calińsi, T. (97). On some desirable patterns in bloc designs. Biometrics, 7, pp: Calińsi, T. (977). On e notation of balance bloc designs. Recent Developments in Statistics, Amsterdam, or-holland Publishing Company, pp: Cerana, B. and Graczy, M. (). Optimum chemical balance weighing designs under e restriction on e number in which each object is weighed. Discussiones Maematicae 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. Maematica,, 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 wi repeated blocs. Applied Maematical Sciences, Hiari Ltd., Vol., o. 55, pp: Cerana, B. and Graczy, M. (9). Some notes about Efficiency Balanced Bloc Designs wi 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. Ma.,, 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. Ghosh, D.K. and Shrivastava, S.B. (). A class of balanced incomplete bloc designs wi 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 Ma., pp: Hedayat, A. and Federer, W.T. (97). Pairwise and variance balanced incomplete bloc designs. Ann. Inst. Statist. Ma., 6, pp: Hotelling, H. (944). Some improvements in weighing and oer experimental techniques. Ann. Ma. 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. Ma., A, 9, pp:

6 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 5():39-44 (ISS: 4-76) 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. Ma. 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. Ma. Statist., 9, pp: Shrihande, S.S. and Raghavarao, D. (963). A meod of construction of incomplete bloc designs. Sanhya, A, 5, pp: Van Lint, J.H. (973). Bloc Designs wi 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:

Some Construction Methods of Optimum Chemical Balance Weighing Designs I

Some Construction Methods of Optimum Chemical Balance Weighing Designs I Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(6): 778-783 Scholarlin Research Institute Journals, 3 (ISS: 4-76) jeteas.scholarlinresearch.org Journal of Emerging Trends in Engineering