Partially Balanced Incomplete Block Research Articles

PBIB-Designs Arising From BI-Connected Dominating Sets of Cubic Graphs of Order at Most 12

A $(v, b, \gamma_{bc}, r, \lambda_{i})$-design over regular graph $G = (V, E)$ of degree $k$ is an ordered pair $D = (V, B)$, where $|V| = v$ and $B$ is the set of bi-connected dominating sets of $G$ called blocks such that two vertices $\alpha$ and $\beta$ which are $i^{th}$ associates occur together in $ \lambda_{i}$ blocks, the numbers $\lambda_{i}$ being independent of the choice of the pair $\alpha$ and $\beta$. In this paper, we obtain Partially Balanced Incomplete Block (PBIB)-designs arising from bi-connected dominating sets in cubic graphs. Also, we give a complete list of PBIB-designs with respect to the bi-connected dominating sets for cubic graphs of order at most $12$. The discussion of non-existence of some designs corresponding to bi-connected dominating sets from certain graphs concludes the article.

Generalized extended triangular designs: Construction and online generation

Method of constructing new and useful series of m-associate class partially balanced incomplete block (PBIB) designs, named as Generalized Extended Triangular (GET) designs, in m replications has been proposed along with an outline of analytical procedure. Triangular designs, one of the most popular and widely used class of PBIB(2) designs, and the extended triangular PBIB(3) designs form other two important particular cases of this generalized class of designs. To avoid the complexity involved in the construction methodology and to save time, an R package ‘‘GETdesigns’’ has been developed for the generation of the proposed designs. Further, these designs have been effectively used for obtaining mating designs and Latin hypercube designs.

Incomplete Block Designs for Comparing Test Treatments with Multiple Controls

Background: This article provides some new construction methods of partially balanced bipartite block (PBBB) designs for comparing test treatments with more than one control. Methods: Partially balanced incomplete block (PBIB) designs based on some association schemes such as group divisible association, and cyclic association are used for developing these methods of construction. Result: A catalogue of efficient PBBB designs is included for parameter values n1 (number of test treatments) ≤10, n2 (number of control treatments) ≤10 and r2 (replications of test treatments) and (replications of control treatments) ≤15 along with computed variances using SAS code.

Partially balanced bipartite block designs

This article provides some new construction methods of partially balanced bipartite block (PBBB) designs for comparing test treatments with more than one control. Partially balanced incomplete block (PBIB) designs based on some association schemes such as triangular association, Latin-square association, group divisible association, and cyclic association are used for developing these methods of construction. A catalog of efficient PBBB designs is included for parameter values v 1 (number of test treatments) ≤ 10 , v 2 (number of control treatments) = 2 , b (number of blocks) ≤ 8 , r 1 (replications of test treatments) ≤ 10 and k (block size) ≤ 6 along with computed variances.

T-designs involving treatment combinations for animal experimentation

Small and marginal farmers in India prefer an Integrated Farming System (IFS) approach i.e. a mixture of crop and livestock that enables them to maintain a stable income throughout the year. Identifying the best location-specific t-component crop-livestock combination to generate maximum profit is the major concern. t-designs can play a vital role in bringing out those best combinations. However, in many situations, combinations derived from specific components may not be practically feasible in a certain locality. Partially balanced t-designs with few selected combinations may prove to be useful in such situations. t-designs are special type of partially balanced incomplete block (PBIB) designs with some additional properties. Here, two series of partially balanced t-designs with some interesting characterization properties have been obtained by using the triangular association scheme. These designs find high application potential in crop and animal experimentation, especially in IFS research involving both crop and livestock components.

Partially balanced incomplete block (PBIB)-designs arising from diametral paths in some strongly regular graphs

The partially balanced incomplete block (PBIB)-designs and association schemes arising from different graph parameters is a well-studied concept. In this paper, we define and construct PBIB-designs with association schemes in strongly regular graphs through the nodes belonging to the diametral paths as blocks. A [Formula: see text]-design, called a diametral-design (in short) over a strongly regular graph [Formula: see text] of degree [Formula: see text], diameter [Formula: see text], is an ordered pair [Formula: see text], where [Formula: see text] and [Formula: see text] is the set of all diametral paths of [Formula: see text], called blocks, containing the nodes belonging to the diametral paths, satisfying the following conditions: (i) If [Formula: see text] and [Formula: see text], then there are exactly [Formula: see text] blocks containing [Formula: see text]; (ii) If [Formula: see text], [Formula: see text] and [Formula: see text], then there are exactly [Formula: see text]-blocks containing [Formula: see text]. We construct PBIB-designs with two-association schemes through diametral paths corresponding to the strongly regular graphs such as the square lattice graphs [Formula: see text], the triangular graphs [Formula: see text], the three Chang graphs, the Petersen graph, the Shrikhande graph, the Clebsch graph, the complement of Clebsch graph, the complement of Schläfli graph, complete bipartite graphs, the Hoffman–Singleton graph, etc. and in each case we give the composition of the diametral paths. These serve as extensions for the collection of near impossible class of strongly regular graphs with given parameters having two-association schemes.

Partially balanced incomplete block (PBIB)-designs associated with geodetic sets in graphs

A [Formula: see text]-design over regular graph [Formula: see text] of degree [Formula: see text] is an ordered pair [Formula: see text], where [Formula: see text] and [Formula: see text] is the set of geodetic sets in [Formula: see text] called blocks such that two vertices [Formula: see text] and [Formula: see text] which are [Formula: see text] associates occur together in [Formula: see text] blocks, the numbers [Formula: see text] being independent of the choice of the pair [Formula: see text] and [Formula: see text]. In this paper, we obtain Partially Balanced Incomplete Block (PBIB)-designs arising from geodetic sets in some classes of graphs. Also, we give a complete list of PBIB-designs with respect to the geodetic sets for cubic graphs of order at most [Formula: see text]. The discussion of non-existence of some designs corresponding to geodetic sets from certain graphs concludes the paper.

Higher associate class partially balanced incomplete block designs using graphs for agricultural experiments

Some new association schemes and construction methods of 3- and 4-associate class partially balanced incomplete block (PBIB) designs based on these schemes using chosen graphs have been given. A catalogue is included of efficient PBIB designs with number of treatments (v) ≤ 100.

Constant block-sum Youden-m square and PBIB designs using Galois field

In the present paper, we have discussed the construction of constant block-sum designs using Galois field. In this direction, the construction of constant block-sum Youden-m square designs and the constant block-sum Partially Balanced Incomplete Block (PBIB) designs has been discussed. In addition to this, a generalized m-associate class association scheme, on which the PBIB designs constructed in this paper are based, is also discussed.

Construction of three associate PBIB designs using K - maps

A new series of s (s ≥ 3) associate class Partially Balanced Incomplete Block (PBIB) designs have been constructed by applying hamming distance on Karnaugh maps. In this series, every character is expressed in binary set and calculate the weight, then hamming distance for each literal of k – map. We considersꞌ different factors (s≥3) simultaneously and each factor has two levels, it means each factor is introduced in primal as well as dual form.

Three-associate class partially balanced incomplete block designs through kronecker product

Kronecker product of matrices can be advantageously used to obtain three-associate class partially balanced incomplete block (PBIB) designs from two-associate class PBIB designs, for a larger number of treatments. This method has been described here and two such series of PBIB designs are obtained. If the experimenter is constrained of resources, theses designs can be used as an alternative to balanced incomplete block designs or 2-associate class partially balanced incomplete block designs.

PBIB-designs arising from diametral paths in cubic graphs

The Partially balanced incomplete block (PBIB)-designs and association schemes arising from different graph parameters is a well studied concept. In this paper, we determine the PBIB-designs with some association schemes through diametral paths in cubic graphs of order at most ten. The discussion of non-existence of some designs corresponding to diametral paths from certain graphs concludes the article.

A Necessary Condition for the Existence of a Certain Resolvable Pairwise Balanced Designa

A mathematical topic using the property of resolvability and affine resolvability was introduced in 1850 and the designs having such concept have been statistically discussed since 1939. Their combinatorial structure on existence has been discussed richly since 1942. This concept was generalized to<i>α</i>-resolvability and affine <i>α</i>-resolvability in 1963. When <i>α</i> = 1, they are simply called a resolvable or an affine resolvable design, respectively. In literature these combinatorial arguments are mostly done for a class of block designs with property of balanced incomplete block (BIB) designs and <i>α</i>-resolvability. Due to these backgrounds, Kadowaki and Kageyama have tried to clarify the existence of affine <i>α</i>-resolvable partially balanced incomplete block (PBIB) designs having association schemes of two associate classes. The known 2-associate PBIB designs have been mainly classified into the following types depending on association schemes, i.e., group divisible (GD), triangular, Latin-square (L<SUB>2</SUB>), cyclic. First, it could be proved that an affine <i>α</i>-resolvable cyclic 2-associate PBIB design does not exist for any <i>α</i> ≥ 1. Also, Kageyama proved the non-existence of an affine <i>α</i>-resolvable triangular design for 1 ≤ <i>α</i> ≤ 10 in 2008. Furthermore, the existence of affine resolvable GD designs and affine resolvable L<SUB>2</SUB> designs with parameters <i>υ</i> ≤ 100 and <i>r</i>, <i>k</i> ≤ 20 was mostly clarified by Kadowaki and Kageyama in 2009 and 2012. As a result, only three designs (i.e., two semi-regular GD designs, only one L<SUB>2</SUB> design) are left unknown on existence within the practical range of parameters. In the present paper, a necessary condition for the existence of a certain resolvable pairwise balanced (PB) design (i.e., some block sizes are not equal) is newly provided. Existence problems on PB designs are far from the complete solution. By use of the necessary condition derived here, we can also show a non-existence result of the affine resolvable L<SUB>2</SUB> design which is left as only one unknown among L<SUB>2</SUB> designs.

Construction of incomplete Sudoku square and partially balanced incomplete block designs

The present article deals with the study of association among the elements of a Sudoku square. In this direction, we have defined an association scheme and constructed incomplete Sudoku square designs which are capable of studying four explanatory variables and also happen to be the designs for two-way elimination of heterogeneity. Some series of Partially Balanced Incomplete Block (PBIB) designs have also been obtained.

PBIB-designs and association schemes arising from minimum bi-connected dominating sets of some special classes of graphs

A dominating set D of a connected graph $$G = (V, E)$$ is said to be bi-connected dominating set if the induced subgraphs of both $$\langle D \rangle $$ and $$\langle V-D \rangle $$ are connected. The bi-connected domination number $$\gamma _{bc}(G)$$ is the minimum cardinality of a bi-connected dominating set. A $$\gamma _{bc}$$ -set is a minimum bi-connected dominating set of G. In this paper, we obtain the Partially Balanced Incomplete Block (PBIB)-designs with m = 1, 2, 3, 4 and $$\lfloor \frac{p}{2}\rfloor $$ association schemes arising from $$\gamma _{bc}$$ -sets of some special classes of graphs.

Constructions of Partially Balanced Crossover Designs Based on Two and Higher Order Association Schemes

Partially balanced crossover designs (PBCODs) are needed in case a balanced design is not possible due to some reasons. Blaisdell and Raghavarao (1980) and Raghavarao and Blaisdell (1985) introduced the concept of PBCODs and gave certain classes of such designs, along with their efficiency factors. The work on the construction of PBCODs was further considered by Aggarwal and Jha (2006), wherein they gave a number of classes of such designs, based on certain partially balanced incomplete block (PBIB) (2) designs and rectangular association scheme. The present work provides some new classes of connected PBCODs based on 2-, 3-, 4-, and general m-associate class association schemes.

Augmented partial diallel cross plans involving two sets of parental lines

Augmented Partial Diallel Cross (APDC) plans are suitable for situations in which resources are limited where a complete diallel is not feasible, but some lines are believed to be superior and are therefore of prime interest. It is desirable to have these primary lines represent a high proportion of crosses and hence they are crossed with every other lines, but a partial diallel system is used for lines of secondary interest. Here, some classes of APDC plans have been obtained using association schemes of Partially Balanced Incomplete Block (PBIB) designs. Variances pertaining to different groups of interline comparisons have been computed and a list consisting of parameters of plans for useful range has been prepared.

Augmented partial diallel cross plans involving two sets of parental lines

Augmented Partial Diallel Cross (APDC) plans are suitable for situations in which resources are limited where a complete diallel is not feasible, but some lines are believed to be superior and are therefore of prime interest. It is desirable to have these primary lines represent a high proportion of crosses and hence they are crossed with every other lines, but a partial diallel system is used for lines of secondary interest. Here, some classes of APDC plans have been obtained using association schemes of Partially Balanced Incomplete Block (PBIB) designs. Variances pertaining to different groups of interline comparisons have been computed and a list consisting of parameters of plans for useful range has been prepared.

English

In this paper, we have constructed partially balanced incomplete block (PBIB) designs with two and three associate classes by establishing a link between PBIB designs and graphs through the chosen lines and number of triangles. We have considered six configurations for this purpose and constructed four two associate class PBIB designs and two three class PBIB designs.

NEW CONSTRUCTION METHOD OF RECTANGULAR PARTIALLY BALANCED INCOMPLETE BLOCK DESIGNS AND SINGULAR GROUP DIVISIBLE DESIGNS

In this study, we propose a new method of construction of rectangular Partially Balanced Incomplete Block (PBIB) designs of size k = 2s and 2≤s < l and singular Group Divisible (GD) designs of size k = 2l, l is the number of columns in a rectangular matrix containing v = n × l treatments. We give the parameters expression of the obtained designs. Furthermore, we provide two algorithms of calculation for practical use.