Abstract

This paper deals with general construction methods of nearly µ - resolvable balanced incomplete block designs with illustration using nearly one resolvable balanced incomplete block (BIB) designs and some known group divisible (GD) designs.

1. Introduction

An incomplete block design (IBD) is a pair (V, D) where V is a v-set of symbols and D is a collection of k – subsets of V called blocks where k < v. A balanced incomplete block design (BIBD) is an IBD (V, D) such that each pair of V is contained in exactly λ blocks. We denote such design as a (v, k, λ) – BIBD.

A near parallel class of an IBD (V, D), with respect to a symbol s, is a set of blocks that partitions the set V–{S} into (v – 1) / k blocks of that design. We call s the missing symbol / hole of this parallel class.

A design is (near) µ-resolvable if it has a (near) resolution such that any (v – 1) treatments occur µ-times in each of the resolution sets. When µ = 1, it leads to a near resolvable design in the sense of Abel and Furino ². The constructions and existence results of nearly resolvable designs can be seen in literatures as Morales et al. ¹³, Haanpaa and Kaski ⁸ and Abel and Funiro ².

The existence of group divisible (GD) designs has been of interest over the years, going back to at least the work of Bose and Shimamoto ⁵, who began classifying such designs. A group divisible design is a 2-associates partially balanced incomplete block design based on v = mn treatments (being m groups of n treatments each), consisting of b blocks of size k (< v), such that each treatment appears in r blocks, and any two treatments (called the first associates) belonging to the same group occur together in λ₁ blocks whereas any two treatments (called the second associates) belonging to different groups occur together in λ₂ blocks. Furthermore, group divisible (GD) designs are classified into three types: (i) Singular (S) design for which r - λ₁=0; (ii) Semi regular (SR) design for which r - λ₁ > 0 and rk - vλ₂ = 0; (iii) Regular (R) design for which r - λ₁ > 0 and rk - vλ₂ > 0. For notations of parameters in a GD association scheme, we refer the reader to Raghavarao ¹⁷.

Bose and Nair ⁴ first introduced the concept of “dualisation” in the field of design of experiments. They derived a new class of block designs by interchanging the role of treatments and blocks in a block design. Shrikhande ¹⁸ applied the concept of dualisation to an asymmetric BIB design with λ = 1 or 2 to produce 2-associate PBIB designs. Dualisation of an incomplete block design with respect to unordered pairs of treatments could be found in Vanstone ¹⁹, who constructed a BIB design through symmetric BIB designs with λ ≥ 2. Using similar technique Mohan and Kageyama ¹⁴ constructed 2-associate group divisible designs. The generalization of the concept due to Vanstone ¹⁹ i.e. dualisation with respect to s-tuples could be seen in Kageyama and Mohan ¹⁰ in constructing PBIB designs for any s ≥ 1. Kageyama et al. ¹² and Philip et al. ¹⁶ used the concept of “restricted dualisation” to construct some nested BIB designs and PBIB designs.

In the present paper, we have proposed some construction methods of nearly μ – resolvable BIB designs. In one section, we have proposed construction methods of getting nearly μ – resolvable balanced block designs having same parameters using two different techniques, firstly by rearrangements of blocks and secondly by using some restrictions in dualisation with respect to treatments of the blocks. In the next section we have discussed two elegant methods of getting nearly μ – resolvable balanced block designs using some known group divisible designs.

2. Method of Construction I

Consider a BIB design D with parameters (v, b, r, k, λ). Considering the r blocks containing any treatment Rearranging the r–blocks corresponding to any i-th treatment as

Now eliminate the treatment and consider the remaining structure of the r - blocks as D_i, where as

(2.1.1)

This denotes the nearly λ–resolvable block design with parameters

(2.1.2)

Theorem 2.1: The existence of BIB design with parameters v, b, r, k and λ implies the existence of nearly λ – resolvable block designs with parameters:

Proof: Let denotes the v treatments of the chosen design D. Now rearrange the r blocks of D corresponding to any i-th treatment (i = 1, 2,3, ..., v) and then eliminate . Doing the same for all the v treatments, we will get the required nearly λ–resolvable design. The parameters are obvious by construction. As in the original BIB design any pair of treatments say occurs in λ blocks, the method produces this pair will occur in (k–2) sets of D_i ; and hence the λ blocks, where occurs in the original BIB design, produce . It is now obvious to note that in , there is a natural partition of v resolution sets of r blocks each and every (v – 1) treatments occur λ times in i-th resolution set with a hole i.e the i-th resolution set do not contain . This completes the proof.

Corollary 2.2: The existence of SBIB design with parameters (v, k, λ) implies the existence of nearly λ – resolvable block designs with parameters:

Corollary 2.3: If λ = 1, then the structure define in (2.1.2) leads to a near resolvable design.

Example 2.4: Consider a SBIB design with parameters, and. Then the theorem 2.1 yields a nearly 2 – resolvable block design with parameters and ; whose blocks are given as

Example 2.5: Consider a SBIB design with parameters, and. Then the theorem 2.1 yields a nearly 2 – resolvable block designs with parameters , whose blocks are given as

We can also obtained the nearly λ – resolvable block design defined in (2.1.1) with same parameters given in (2.1.2) using another technique of restricted dualisation with respect to the treatments of the blocks.

3. Method of Construction II

Consider a symmetric balanced incomplete block design D (v, k, λ) having v treatments (with replication number r) and b number of blocks (of size k). Choose any one block from it and call this chosen block as a fixed block (say B_j). Now ignore this fixed block and dualise D with respect to the treatments of this fixed block B_j. Then write the block numbers corresponding to any treatment of this fixed block. A new design so formed is called D₁. Doing this dualisation for all the v treatments, we get a required structure as follows

(3.1.1)

This designs denotes the nearly λ – resolvable designs with parameters same as defined in (2.1.2).

Theorem 3.1: The existence of SBIB design with parameters v = b, r = k and λ implies the existence of nearly λ – resolvable block designs with parameters:

Which are same as parameters given in (2.1.2).

Remark: The restricted dualisation used in Method II when made for Non-Symmetrical BIB designs; it produces 2-association PBIB designs.

Example 3.2: Consider a SBIB design with parameters, and. Then the theorem 3.1 yields a nearly 3 – resolvable block designs with parameters and This gives six multiple solution of (11, 5, 2) design with no repeated blocks. The structure is as follows

In the next sections, we have discussed two elegant methods of construction of nearly μ – resolvable balanced incomplete block designs.

Preposition 1: Malcolm ¹⁴ proved that the two multiple of a SBIB design v = b = 4t+3, r = k = 2t+1 and λ = t; where 4t+3 is a prime or a prime power is always near one resolvable balanced incomplete block design with parameters v’ = 4t+3, b’ = 2(4t+3), r’ = 2(2t+1), k’ = 2t+1 and λ’ = 2t respectively.

4. Method of Construction III

Consider a two multiple of a SBIB (4t+3, 2t+1, t) design. This design D’ is always near one resolvable balanced incomplete block design since one treatment is missing from each resolution set. Also D’ gives column-wise BIB design D_c with parameters V_c = 4t+3, B_c = 2(4t+3), R_c = 2(2t+1), K_c = 2t+1, λ_c = 2t and row-wise BIB design D_r with parameters V_r = 4t+3, B_r = (2t+1) (4t+3), R_r = 2(2t+1), K_r = 2, λ_r = 1 respectively.

If we consider rows of D_r as an association scheme, then there are (4t+3) such association schemes with (2t+1) groups of size two which meets with the association scheme of chosen GD design with parameters and The association scheme for each resolution set is given as

(4.1.1)

Considering all the (4t+3) resolution sets, we obtained the resultant design as given below

(4.1.2)

This gives nearly – resolvable balanced incomplete block design with parameters

Theorem 4.1: The existence of a row-wise BIB design and group divisible (GD) design with parameters V_r = 4t+3, B_r = (2t+1) (4t+3), R_r = 2 (2t+1), K_r = 2, λ_r = 1 and respectively; implies the existence of nearly – resolvable balanced incomplete block design with parameters

Proof: Let us consider a row-wise BIB design D_r having parameters V_r = 4t+3, B_r = (2t+1) (4t+3), R_r = 2(2t+1), K_r = 2, λ_r = 1 and considering rows of D_r as an association scheme which meets with the association scheme of chosen GD design having parameters (since one treatment is missing from each resolution set in D’), . In each resolution set, there are treatments arranged in groups of size.

There are (4t+3) resolution sets in all. For each resolution set, using the association scheme defined in (4.1.1), we construct group divisible designs D_i (i = 1,2,...,v). Then their juxtaposition gives the resultant design as defined in (4.1.2).

In ; and are obvious from the construction. Under the present method of construction, since each D_i (i = 1,2,...,v) contains blocks and there are (4t+3) resolution sets in . Thus in resultant design there are blocks in all.

Furthermore any treatment, say appears in blocks of D_i (i = 1,2,...,v). These blocks contribute 2(2t+1) times in, since one treatment is missing in each D_i (i = 1,2,...,v). Thus .

Also any pair of treatment, say, , which is first associate in D_i , will contributes λ₁^* times in one resolution set and in the remaining 4t resolution sets, it will contributes λ₂^* times. Hence .

It can be noted that the resultant design is nearly – resolvable balanced incomplete block design as there is a natural partition of v resolution sets of r blocks each and every (v – 1) treatments occur λ times in i-th resolution set with a hole i.e the i-th resolution set do not contain . This completes the proof.

Example 4.2: Consider a SBIB design with parameters, and. A two multiple of this design yields a nearly one resolvable balanced incomplete block design D’ as given below

Here the rows of D’ forms a BIB design D_r with parameters V_r = 7, B_r = 21, R_r = 6, K_r = 2, λ_r = 1 and considering rows of this design D_r as an association scheme as follows

Each resolution set meets with the GD design having parameters and

Then the theorem 4.1 yields a nearly 3 – resolvable balanced incomplete block design with parameters and ; whose blocks are given as

5. Method of Construction IV

Consider a two multiple of a SBIB (4t+3, 2t+1, t) design. This design D’ is always near one resolvable balanced incomplete block design since one treatment is missing from each resolution set. Also D’ gives column-wise BIB design D_c with parameters V_c = 4t+3, B_c = 2(4t+3), R_c = 2(2t+1), K_c = 2t+1, λ_c = 2t and row-wise BIB design D_r with parameters V_r = 4t+3, B_r = (2t+1) (4t+3), R_r = 2(2t+1), K_r = 2, λ_r = 1 respectively.

If we consider columns of D_c as an association scheme, then there are (4t+3) such association schemes with two groups of size (2t+1) which meets with the association scheme of chosen GD design with parameters and The association scheme for each resolution set is given as

(5.1.1)

Considering all the (4t+3) resolution sets, we obtained the resultant design is given as

(5.1.2)

This gives nearly – resolvable balanced incomplete block design with parameters

Theorem 5.1: The existence of a column-wise BIB design and group divisible (GD) design with parameters V_c = 4t+3, B_c = 2(4t+3), R_c = 2(2t+1), K_c = 2t+1, λ_c = 2t and respectively, implies the existence of nearly – resolvable balanced incomplete block design with parameters

Proof: Let us consider a column-wise BIB design D_c having parameters V_c = 4t+3, B_c = 2(4t+3), R_c = 2(2t+1), K_c = 2t+1, λ_c = 2t and considering columns of D_c as an association scheme which meets with the association scheme of chosen GD design having parameters [since one treatment is missing from each resolution set in D’], . In each resolution set, there are treatments arranged in groups of size.

There are (4t+3) resolution sets in all. For each resolution set, using the association scheme defined in (5.1.1), we construct group divisible designs D_i (i = 1,2,...,v). Then their juxtaposition gives the resultant design as defined in (5.1.2).

In ; and are obvious from the construction. Under the present method of construction, there are (4t+3) resolution sets in the resultant design and each resolution set contains blocks. Thus in resultant design there are blocks in all.

Furthermore any treatment, say appears in blocks of D_i (i = 1,2,...,v). These blocks contribute 2(2t+1) times in, since one treatment is missing in each D_i (i= 1,2,...,v). Thus .

Also any pair of treatment, say , which is first associate in D_i , will contributes λ₁^* times in 2t resolution set and in the remaining (2t+1) resolution sets, it will contributes λ₂^* times. Hence

It can be noted that the resultant design is nearly – resolvable balanced incomplete block design as there is a natural partition of v resolution sets of r blocks each and every (v – 1) treatments occur λ times in i-th resolution set with a hole i.e the i-th resolution set do not contain . This completes the proof.

Example 5.2: Consider a SBIB design with parameters , and . A two multiple of this design yields a nearly one resolvable balanced incomplete block design D’ as given below

Here the columns of D’ forms a BIB design D_c with parameters V_c = 11, B_c = 22, R_c = 10, K_c = 5, λ_c = 4 and considering columns of this design D_c as an association scheme as follows

Each resolution set meets with the GD design having parameters and

Then the theorem 5.1 yields a nearly 10 – resolvable balanced incomplete block design with parameters; whose blocks are given as

6. Results and Discussion

The following Table 6.1 provides the list of nearly µ – resolvable incomplete block designs for Methods I, II which are obtained by using certain known SBIB designs with k ≤15.

The following Table 6.2 provides the list of nearly µ – resolvable incomplete block designs for Method III which are obtained by using nearly one-resolvable balanced incomplete block design and certain known GD designs from Clatworthy table ⁶.

References