International Journal of Mathematical Archive-9(3), 2018, Available online through ISSN

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1 Internatonal Journal of Matheatcal Archve-9(3), 208, Avalable onlne through ISSN CONSTRUCTION OF BALANCED INCOMPLETE BLOCK DESIGNS T. SHEKAR GOUD, JAGAN MOHAN RAO M AND N.CH. BHATRA CHARYULU* Departent of Statstcs, Unversty College of Scence, Osana Unversty, Hyderabad-7, Inda. (Receved On: 7-0-8; Revsed & Accepted On: ) ABSTRACT Bose and Nar (939) ntroduced a class of bnary, equreplcate and proper desgns, called Partally Balanced Incoplete Block Desgns. The Partally Balanced Incoplete Block Desgns are avalable wth saller nuber of replcatons for any ore nubers of treatents. Ths paper provdes a new seres of Partally Balanced Incoplete Block Desgns. Keywords: Partally Balanced Incoplete Block Desgns.. INTRODUCTION Balanced Incoplete Block Desgns are not always sutable for varetal trals, snce they requre large nuber of replcatons and also that sutable desgns are not avalable for all nuber of treatents. Bose and Nar (939) ntroduced a class of bnary, equreplcate and proper desgns, called Partally Balanced Incoplete Block Desgns. The Partally Balanced Incoplete Block Desgns are avalable wth saller nuber of replcatons for any ore nubers of treatents. The nubers of replcatons of par of treatents s not constant and are defned n general for types of replcatons of dfferent pars of treatents where s an nteger. The arrangeent of v treatents n b blocks each of sze k ( < v ) and each treatent occurs n r blocks such that n pars of treatents occurs n λ, n 2 pars of treatents occurs n λ 2 tes, so on and n pars of treatents occurs n λ tes then the ncoplete block desgn s sad to be Partally Balanced Incoplete Block Desgn (PBIBD) wth -assocate classes The nubers v, b, r, k, λ, λ 2,, λ, n, n 2,, n, P (, j, k =, 2, ) are called the jk paraeters of Partally Balanced Incoplete Block Desgn. Thus, there are 2+4 paraeters. The paraeters of -assocate class Partally Balanced Incoplete Block Desgn satsfy the followng relatons: vr = bk, n + n n = v-, n λ + n 2 λ 2 + +n λ = r(k-), P = n f j and jk j P = n f = j; jk j k P jk = Pkj n Pjk = n jpjk = n kp, j. When nuber of assocatons s 2 then called 2- assocate class Partally Balanced Incoplete Block Desgn. 2. CONSTRUCTION OF NEW SERIES OF PBIBD S Two new seres for the constructon of two assocate and - assocate class Partally Balanced Incoplete Block Desgns. Method 2.: Let N be the ncdence atrx of a Balanced Incoplete Block Desgn wth paraeters v, b, r, k, λ such that v+λ 2r and J be the atrx of untes. The cobnatoral arrangeent of the ncdence atrx N s N J N' = J N The resultng s a two assocate class Partally Balanced Incoplete Block Desgn wth ncdence atrx as Nwth paraeters v = 2v, b =2b, r = b+r, k =v+k, λ = v+ λ, λ 2 = 2r. Correspondng Author: N.Ch. Bhatra Charyulu*, Departent of Statstcs, Unversty College of Scence, Osana Unversty, Hyderabad-7, Inda. Internatonal Journal of Matheatcal Archve- 9(3), March k= k=

2 Constructon of Balanced Incoplete Block Desgns / IJMA- 9(3), March-208. Theore 2.: A Partally Balanced Incoplete Block Desgn wth paraeters v= 2v, b =2b, r = b+r, k =v+k, λ = v+ λ, λ 2 = 2r can be constructed usng the ncdence atrx of a Partally Balanced Incoplete Block Desgn wth paraeters v, b, r, k, λ such that v+λ 2r wth an arrangeent of N and J as N J N' = J N Proof: Let N vxb be the ncdence atrx of a Syetrc Balanced Incoplete Block Desgn wth paraeters v, b, r, k, λ such that v+λ 2r. Let J be the atrx of untes of order vxb. It can be observed drectly fro the arrangeent of N and J n N wll provdes an ncdence atrx of a desgn wth each treatent replcated b+r tes and the block sze s v+k. The 2v treatents can be parttoned nto two groups each consstng of v treatents such that () any par of treatents belongng to the sae group occur together n b+λ blocks, () any par of treatents belongng to dfferent groups occur n 2r blocks. Therefore the resultng ncdence atrx of a Partally Balanced Incoplete Block Desgn wth paraeters v = 2v, b = 2b, r = b+r, k = v+k, λ = v+ λ, λ 2 = 2r. The ethod s llustrated n the exaple 2. Exaple 2.: Let N 4x6 be the ncdence atrx of Balanced Incoplete Block Desgn wth paraeters v = 4, b = 6, r = 3, k = 2, λ = N = ; The arrange the ncdence atrces N and J n N as N J N' = = J N The resultng desgn N s the ncdence atrx of a SBIBD wth paraeters v = 8, b = 2, r = 9, k = 6, λ = 7, λ 2 = 6. Note: If b+λ=2r then N represents the ncdence atrx of BIBD wth paraeters of BIBD wth paraeters v = 2v = b, r = b+r = k λ = v+ λ. Method 2.2: Let N be the ncdence atrx of a Balanced Incoplete Block Desgn wth paraeters v, b, r, k, λ. Let N be the dual desgn of N. Arrange the ncdence atrx and ts dual n the for N = N N The resultng desgn s the ncdence atrx of three assocate class Partally Balanced Incoplete Block Desgn wth paraeters v = 2v, b = 2b, r = b, k = v, λ = b-2r+2λ, λ 2 = 2r-2λ, λ 3 = 0, n = v-, n 2 = v-, n 3 =. Theore 2.2: A three assocate class Partally Balanced Incoplete Block Desgn wth paraeters = v 2v, b = 2b, r = b, k = v, λ = b-2r+2λ, λ 2 = 2r-2λ, λ 3 = 0, n = v-, n 2 = v-, n 3 = can be constructed usng the cobnatoral arrangeent of N and ts dual n N, where N s the ncdence atrx of Balanced Incoplete Block Desgn. N =. 208, IJMA. All Rghts Reserved 2

3 Constructon of Balanced Incoplete Block Desgns / IJMA- 9(3), March-208. Proof: let N vxb be the ncdence atrx of a Balanced Incoplete Block Desgn wth paraeters v, b, r, k, λ,.then the ncdence atrx contans: the no of s n the each row r, the no of s n the each colun k, the nuber of 0 s n the each row b-k, the nuber of s n the each colun -k. Let N vxb be the copleentary atrx of N contans, no of s n each row are b-r, no of s n each colun are b-k, no of 0 s n each row are r, no of 0 s n each colun are k. In the ncdence atrx N of Balanced Incoplete Block Desgn, the nuber of pars (0, 0), (0, ), (, 0), (, ) are occurrng n any two rows are b-2r+λ, r-λ, r-λ, λ respectvely. Then N contans the no. of pars are λ, r-λ, r-λ, b-2r+λ. The arrangeent ncdence atrx N and ts copleent atrx N n the for N = 2vx2b As a results t wll contans 2v treatents, 2b blocks, each block sze s v and each treatent replcated b tes and λ = b-2r+2λ, λ 2 = 2r-2λ, λ 3 = 0. The ethod s llustrated n the exaple 2.2. Exaple 2.2: Consder a Balanced Incoplete Block Desgn wth paraeters v = 5, b = 0, r = 6, k = 3, λ = 3 whose ncdence atrx s N. where N and N are N = and N = The arrangeent of ncdence atrces results to N = The resultng desgn s the ncdence atrx of a Partally Balanced Incoplete Block Desgn wth paraeters v = 0, b = 20, r = 0, k = 5, λ = 4, λ 2 = 6, λ 3 = 0, n = 4, n 2 = 4, n 3 =. 4. EFFICIENCY AND OPTIMALITY CRIREIA OF PBIBD 4. Effcency of PBIBD: The pattern of NN atrx s, all ts dagonal eleents equal to r and off-dagonal eleents are ether λ s. Let B 0 = I v. Let us defne the assocaton atrces B = ((b () () jk )) =, 2,, where b jk = f j th and k th treatents are th assocates and = 0 otherwse. We know that B J v, = n J v, for =, 2, and B = JJ and B 0, B, B 2, B are lnearly ndependent. Snce B j B k can be nterpreted as the nuber of treatents coon to the j th assocates of α and k th assocate of β. Therefore NN can be expressed as NN = rb 0 + λ B + λ 2 B λ B. = 0 208, IJMA. All Rghts Reserved 22

4 Constructon of Balanced Incoplete Block Desgns / IJMA- 9(3), March-208. Then the deternant of NN = rk (r-θ ) α (r-θ ) α, t= α t = v-. In partcular when =2, NN = rb 0 + λ B + λ 2 B 2. NN = rk ( r - θ ) α. ( r - θ 2 ) α2, (4.3.) The effcency factors for PBIBD are E = - E 2 = r λ rk 2 n λ 2{λ + ( )λ 2} rk{λ + (n )λ } f th and j th are st assocates and 2 f th and j th are 2 nd assocates If λ < λ 2 then the par of treatents (α, α j ) are second assocates occur ore nuber of tes than the par of treatents (α, α ) whch are frst assocates then E 2 > E otherwse E 2 < E. The effcency of a partcular group of Group Dvsble Partally Balanced Incoplete Block Desgns effcences are evaluated and presented n Table 3.. Table-3.: The Effcences of PBIBDs v r k b n λ λ 2 E E 2 E , IJMA. All Rghts Reserved 23

5 Constructon of Balanced Incoplete Block Desgns / IJMA- 9(3), March ACKNOWLEDGMENTS The Frst author s grateful to UGC for provdng fnancal assstance to carry out ths work under BSR RFSMS. REFERENCES. Bose R.C., Clatworthy W.H. and Shrkande S.S. (954): Tables of Partally Balanced Desgns wth two Assocate Classes, North Carolna Agrcultural Experent Staton Tech. Bull, Vol. 07. pp 2. Bose R.C. and Nar K.R. (939): Partally Balanced Incoplete Block Desgns, Sankhya-B, Vol. 4, pp Bose R.C. And Mesner D.M. (959): on Lnear Assocatve Algebras Correspondng to Assocaton Schees of Partally Balanced Desgns, Analyss. Matheatcs Statstcs, Vol. 30, pp Bose R.C. and Shaoto T. (952): Classfcaton and Analyss of Partally Balanced Incoplete Block Desgns wth two Assocate Classes, Journal. Aerca. Statstcs. Assocaton. Vol.47, pp Bose R. C. and Shrkhande S. S. (960): On the Coposton of Balanced Incoplete Block Desgns, Canadan Journal of Matheatcs, Vol. 2, pp Source of support: Nl, Conflct of nterest: None Declared. [Copy rght 208. Ths s an Open Access artcle dstrbuted under the ters of the Internatonal Journal of Matheatcal Archve (IJMA), whch perts unrestrcted use, dstrbuton, and reproducton n any edu, provded the orgnal work s properly cted.] 208, IJMA. All Rghts Reserved 24