﻿ U-Type Designs via New Generalized Partially Balanced Incomplete Block Designs with m = 4, 5 and 7 Associated Classes

Applied Mathematics
Vol.06 No.02(2015), Article ID:53791,22 pages
10.4236/am.2015.62024

U-Type Designs via New Generalized Partially Balanced Incomplete Block Designs with m = 4, 5 and 7 Associated Classes

Imane Rezgui1, Zebida Gheribi-Aoulmi1, Hervé Monod2

1Department of Mathematics, University of Constantine 1, Constantine, Algeria

2INRA, UR MaIAGE, Jouy-en-Josas, Paris, France   Received 10 January 2015; accepted 28 January 2015; published 4 February 2015

ABSTRACT

The traditional combinatorial designs can be used as basic designs for constructing designs of computer experiments which have been used successfully till now in various domains such as engineering, pharmaceutical industry, etc. In this paper, a new series of generalized partially balanced incomplete blocks PBIB designs with m associated classes (m = 4, 5 and 7) based on new generalized association schemes with number of treatments v arranged in w arrays of n rows and l columns (w ≥ 2, n ≥ 2, l ≥ 2) is defined. Some construction methods of these new PBIB are given and their parameters are specified using the Combinatory Method (s). For n or l even and s divisor of n or l, the obtained PBIB designs are resolvable PBIB designs. So the Fang RBIBD method is applied to obtain a series of particular U-type designs U (wnl; ) (r is the repetition number of each treatment in our resolvable PBIB design).

Keywords:

Association Scheme, Combinatory Method (s), Resolvable Partially Balanced Incomplete Block Design, U-Type Design 1. Introduction

Designs of computer experiments drew a wide attention in the previous two decades and were still being used successfully till now in various domains. Among the various construction methods of these designs, the traditional combinatorial designs can be used as basic designs (example:   ) and particularly the PBIB designs. The association schemes of two or three associated classes have been widely studied, while it is not the case of those over three associated classes. However, some association schemes with five associated classes have been studied (example:  ). Besides, a method to obtain new association schemes by crossing or nesting other association schemes was given by Bailey  using the character tables and strata of the initial association schemes to find the parameters of the obtained association schemes.

In this paper, new association schemes with 4, 5 and 7 associated classes are described starting by a geometric representation. The parameter expressions of these association schemes are given. Moreover, some methods to construct the PBIB designs based on these association schemes are explained using an accessible construction method called the Combinatory Method (s)  , which allows obtaining a series of PBIB designs by only using these association schemes. The parameters expressions of these new designs are given. In addition, for n or l even

and s divisor of n or l, the obtained PBIB are resolvable PBIB. Then, a series of U-type designs U (r is the repetition number of each treatment in the resolvable PBIB design) is obtained by applying the RBIBD method  on these designs.

The paper is organized as follows. In Section 2, we give new definitions of generalized association schemes with m (= 4, 5 and 7) associated classes, starting by geometric representation and we give their parameters as properties. Section 3 describes a series of construction method using the Combinatory Method (s) for obtaining the PBIB designs associated to our generalized association schemes. We give the series of the U-type designs associated to our constructed PBIB designs in Section 4. We achieve our paper with a Conclusion.

Recall some definitions:

Definition 1. An m-association scheme (m ≥ 2) of v treatments  is a relation satisfying the following conditions:

1) Any two treatments are either 1st, 2nd, ∙∙∙, or mth associates. The relation of association is symmetric, i.e., if the treatment is an ith associate of β, then β is an ith associate of .

2) Each treatment has ni ith associates, the number ni being independent of .

3) If any two treatments and β are ith associates, then the number of treatments that are jth associates of and kth associates of β is and is independent of the pair of ith associates and β .

The numbers v, ni and  are called the parameters of the association scheme.

Definition 2. A PBIB design  , based on an m-association scheme (m ≥ 2), with parameters v, b, r, k, λi, , is a block design with v treatments and b blocks of size k each such that every treatment occurs in blocks and any two distinct treatments being ith associate occur together in exactly λi blocks. The number λi is independent of the pair of ith associates.

A parallel class of PBIB is a collection of disjoint blocks from the b blocks whose union is V. A partition of the b blocks into parallel classes is called a resolution, and PBIB design is resolvable if it has at least one resolution and it denoted by RPBIB design.

The Combinatory Method (s)  :

Let an array of n rows and l columns as follows:

Consider s different elements of the same row i, and associate with them s other elements of a row , respecting the correspondence between the elements aij and. Bringing together the 2s elements in the same block and making all possible combinations, we obtain a partially balanced incomplete block design of size.

Definition 3. Let U denote a design of v runs and r factors with respective levels. This design corresponds to an v × r matrix such that the ith column xi takes values from a set of qi elements, say, equally often. The set of all such designs, called U-type designs in the statistical literature, is denoted by U. Obviously, v must be a multiple of qi. When all the qi are equal to q, we denote it by U. Note that the rows and columns of X are identified with the runs and factors respectively  .

2. Generalized Rectangular Right Angular Association Schemes (m) (m = 4, 5 and 7 Associated Classes)

2.1. Generalized Rectangular Right Angular Association Scheme (4)

Let V be a set of v = wnl treatments, (w ≥ 2, n ≥ 2, l ≥ 2), to which we associate a geometrical representation in the following way:

Each treatment of V is associated with a unique triplet of the set where:

Let be a treatment of coordinates

corresponds to the treatments 1st associated to

corresponds to the treatments 2nd associated to.

corresponds to the treatments 3rd associated to.

corresponds to the treatments 4th associated to.

This geometric representation describes a new association scheme, we call it for convenience, generalized rectangular right angular association scheme (4) with four associated classes, to which we give the following equivalent definition:

Definition 4. A generalized rectangular right angular association scheme (4) is an arrangement of v = wnl (w ≥ 2, n ≥ 2, l ≥ 2) treatments in w arrays of n rows and l columns such that, with respect to each treatment:

1) The first associates of are the other treatments of the same row in the same array.

2) The second associates of are the other treatments of the same column in the same array.

3) The third associates of are the remaining treatments in the same array.

4) The fourth associates of are the other treatments of the other arrays.

Property 1. The parameters of generalized rectangular right angular association schemes (4) are:

, , , ,

Definition 5. A PBIB design based on a generalized rectangular right angular association scheme (4) is called generalized rectangular right angular GPBIB4 design.

2.2. Generalized Rectangular Right Angular Association Scheme (5)

Let V be a set of v = wnl treatments, (w ≥ 2, n ≥ 2, l ≥ 2), to which we associate a geometrical representation in the following way:

Each treatment of V is associated with a unique triplet (of coordinates) of the set where:

Let be a treatment of coordinates

corresponds to the treatments 1st associated to

corresponds to the treatments 2nd associated to.

corresponds to the treatments 3rd associated to.

corresponds to the treatments 4th associated to.

corresponds to the treatments 5th associated to.

This geometric representation describes a new association scheme, we call it for convenience, generalized rectangular right angular association scheme (5) with five associated classes, to which we give the following equivalent definition:

Definition 6. A generalized rectangular right angular association scheme (5) is an arrangement of v = wnl (w ≥ 2, n ≥ 2, l ≥ 2) treatments in w (n × l) rectangular arrays such that, with respect to each treatment:

1) The first associates of are the other treatments of the same row in the same array.

2) The second associates of are the other treatments of the same column in the same array.

3) The third associates of are the remaining treatments in the same array.

4) The fourth associates of are the treatments of same row as, of the other arrays.

5) The fifth associates of are the remaining treatments in the other arrays.

Property 2. The parameters of the generalized rectangular right angular association schemes (5) are:

, , , , ,

Definition 7. A PBIB design based on a generalized rectangular right angular association scheme (5) is called generalized rectangular right angular GPBIB5 design.

2.3. Generalized Rectangular Right Angular Association Scheme (7)

Let V be a set of v = wnl treatments, (w ≥ 2, n ≥ 2, l ≥ 2), to which we associate a geometrical representation in the following way:

Each treatment of V is associated with a unique triplet of the set where:

Let be a treatment of coordinates

corresponds to the treatments 1st associated to

corresponds to the treatments 2nd associated to.

corresponds to the treatments 3th associated to.

corresponds to the treatments 4th associated to.

corresponds to the treatments 5th associated to.

corresponds to the treatments 6th associated to.

corresponds to the treatments 7th associated to.

This geometric representation describes a new association scheme, we call it for convenience, generalized rectangular right angular association scheme (7) with seven associated classes, to which we give the following equivalent definition:

Definition 8. A generalized rectangular right angular association scheme (7) is an arrangement of v = wnl (w ≥ 2, n ≥ 2, l ≥ 2) treatments in w arrays of n rows and l columns such that, with respect to each treatment:

1) The first associates of are the other treatments of the same row in the same array.

2) The second associates of are the other treatments of the same column in the same array.

3) The third associates of are the remaining treatments in the same array.

4) The fourth associates of are the treatments in the same row and the same column as, of the other arrays.

5) The fifth associates of are the treatments of the same row as in the other arrays, that are different from the fourth associates of.

6) The sixth associates of are the treatments of the same column as in the other arrays, that are different from the fourth associates of.

7) The seventh associates of are the remaining treatments in the other arrays.

Property 3. the parameters of generalized rectangular right angular association schemes (7) are:

, , , , ,

, ,

Definition 9. A PBIB design based on a generalized rectangular right angular association scheme (7) is called generalized rectangular right angular GPBIB7 design.

3. Construction Method of GPBIBm Designs (m = 4, 5 and 7 Associated Classes)

Let v = wnl (w ≥ 2, n ≥ 2, l ≥ 2) treatments be arranged in w arrays of n rows and l columns, ..., and written as follows:

3.1. Construction Method of GPBIB4 Designs

3.1.1. First Construction Method of GPBIB4 Designs

Applying the Combinatory Method (s) on each of the w arrays, with chosen, then we obtain w rectangular PBIB designs. The set of all blocks gives a PBIB design with 4 associated classes.

Theorem 1. The partially balanced incomplete block designs with the parameters:

, , , , , , ,

are generalized rectangular right angular GPBIB4 designs.

Proof. For each array of the w arrays, we obtain a rectangular design with parameters:, , , , , ,. (see  ).

For the w arrays we obtain a generalized rectangular right angular GPBIB4 design with parameters:, , , , , ,.

: Two treatments and from the arrays and respectively they never appear together in the same block thus.

Lemma 1. For the special case s = l, the previous method can also be used for the construction of nested group divisible designs, with parameters:

, , , , , ,

Remark 1.

・ For w = 1, the GPBIB4 design of Theorem 1 is a rectangular design with parameters as in the Theorem 1 of  .

・ For w = 2, the GPBIB4 design of Theorem 1 is a rectangular right angular PBIB4 design with parameters as in Proposition 1 of  .

Proposition 2. Let GPBIB4 be a design with parameters:

, , , , , , ,.

For n or l even and s divisor of l or s, the GPBIB4 design is a resolvable PBIB designs (RGPBIB4) with r parallel classes where each parallel classes contain blocks.

Proof.

n or l is even and s is divisor of n or l, then.

Example 1. Let v = 3 × 4 × 4 treatments be arranged in the three following arrays:

The construction method for (s = 2), give the following resolvable generalized rectangular right angular GPBIB4 design, with the parameters:

3.1.2. Second Construction Method of GPBIB4 Designs with

Let, and,.

Applying the Combinatory Method with chosen on each array of the form for, and, by only considering the combinations of s treatments that always contain a component of the vector, the set of all the obtained blocks provides a PBIPwith 4 associated classes.

Theorem 3. The partially balanced incomplete block designs with the parameters:

, , , , ,

, ,

are generalized rectangular right angular GPBIB4 designs.

Proof.

・ The v and k values are obvious.

・ r: For each treatment of the array , we have:

○ On an array , applying the procedure with the l other elements of the same row. There is possibilities, each one being repeated times, with n permutations, then we have repetitions. Therefore, we have arrays of the form ; so we have repetitions of the treatment.

○ On an array , and, applying the procedure with the

other elements of the same row. There is possibilities each one being repeated times, with n permutations, then we have appearances repeated themselves n times. Therefore, for an array we have repetitions of. Considering all the arrays for and; so we have repetitions of.

Thus:.

: Consider two treatments and from the array (and), they appear together times with the other elements of the same row i of the arrays , with n permutations, we obtain times in which the two treatments appear together. Considering all the arrays for and; then we have times where the two treatments and appear together.

: Consider two treatments and from the array, (and), we have:

○ In an array : the two treatments appear together times, with the n permutations they appear together times. Considering all the arrays of the form for and, they appear together times.

○ In the array: both treatments appear together times, with the n permutations

they appear times. Considering all the arrays of the form for; they appear together times.

In total.

: Consider two treatments and from the array (,and), they appear together times for each array of the form , with the n permutations they appear together times. Considering all the arrays for and, then the two treatments and appear together times.

: Consider two treatments and from the arrays and respectively (g,):

○ If, for the array and h = 1 the two treatments appear together

times. For the array and h = 1 they also appear together time, so we

have times. On the other hand, for, the two treatments appear together times for the array and times for the array, so we have for one value of h. Taking all the values of, we obtain times where the two treatments appear together.

In total.

○ If, then the two treatments appear together times for the array of the form and appear together times for the array of the form for, so we have times. For, among the permutations of the vector, and for a given value of, the treatment takes the same row as then the two treatments appear together times, for the remaining values the two treatments appear together times. For, among the permutations of the vector and for a given value of, the treatment takes the same row as then the two treatments appear together times, for the remaining values the two treatments appear together times.

In total.

: Using the above construction method on each array of the form:, we obtain blocks. So for the l arrays of the form we have blocks, with the n permutations we obtain blocks. Considering all the arrays for, then in total we have blocks. Considering all the arrays for, then in total we have blocks.

Remark 2. For w = 2, the GPBIB4 design of Theorem 3 is a rectangular right angular PBIB4 design with parameters as in Proposition 2 of  .

Proposition 4 Let GPBIB4 be a design with parameters:

, , , , ,

, ,

For n or l even and s divisor of l or s, the GPBIB4 design is a resolvable PBIB designs (RGPBIB4) with r parallel classes where each parallel classes contain blocks.

3.2. Construction Method of GPBIB5 Designs

Let be the jth column of the gth array. Applying the Combinatory Me-

thod (s) with s chosen, on each array in the form for and by only considering the combinations of s treatments that contain a component of the vector, then the set of all the blocks obtained, gives a PBIB design with 5 associated classes.

Theorem 5. The incomplete block designs with parameters:

, , , , ,

, , ,

are generalized rectangular right angular GPBIB5 designs.

Remark 3. For w = 2, the GPBIB5 design of Theorem 5 is a rectangular right angular PBIB5 design with parameters as in Proposition 3 of  .

Proposition 6. Let GPBIB5 be a design with parameters:

, , , , ,

, , ,

For n or l even and s divisor of l or s, the GPBIB5 design is a resolvable PBIB designs (RGPBIB5) with r parallel classes where each parallel classes contain blocks.

Example 2. Let v = 3 × 2 × 3 treatments be arranged in the two following arrays:

The construction method for (s = 3), give the following generalized rectangular right angular GPBIB5 design, with parameters:

3.3. Construction Method of GPBIB7 Designs

3.3.1. First Construction Method of GPBIB7 Designs

Applying the Combinatory Method (s) on each of the w arrays, with chosen and fixed, then we obtain w rectangular PBIB designs. The juxtaposition of the blocks of the w rectangular PBIB designs, such that the blocks containing treatment and are put side by side, gives a PBIB design with 7 associated classes.

Theorem 7. The incomplete block designs with parameters:

, , , ,

, ,

are generalized rectangular right angular GPBIB7 designs.

Proof. The design parameters are deduced from the construction method.

Remark 4.

・ For w = 1, the GPBIB7 design of Theorem 7 is a rectangular design with parameters as in Theorem 1 of  .

・ For w = 2, the GPBIB7 design of Theorem 7 is a rectangular right angular PBIB7 design with parameters as in Proposition 4 of  .

Proposition 8. Let GPBIB7 be a design with parameters:

, , , ,

, ,

For n or l even and s divisor of l or s, the GPBIB7 design is a resolvable PBIB designs (RGPBIB7) with r parallel classes where each parallel classes contain blocks.

Example 3. Let v = 3 × 4 × 4 treatments be arranged in the three following arrays:

The construction method for (s = 2), give the following resolvable generalized rectangular right angular GPBIB7 design, with the parameters:

3.3.2. Second Construction Method of GPBIB7 Designs with

Let be the jth column of the gth array and let be the ith row

of the gth array. Then applying the Combinatory Method (s) with chosen on each

array of the form for and g, (respectively

for and g,), by only considering the combinations of s

treatments that always contain a component of the column (respectively the row), the set of all the obtained blocks provides a PBIP with 7 associated classes.

Theorem 9. The partially balanced incomplete block designs with the parameters:

, , , ,

, , , ,

, ,

are generalized rectangular right angular GPBIB7 designs.

Proof. The design parameters are deduced from the construction method.

Remark 5. For w = 2, the GPBIB7 design of Theorem 9 is a rectangular right angular PBIB7 design with parameters as in Proposition 5 of  .

Proposition 10. Let GPBIB7 be a design with parameters:

, , , ,

, , , ,

, ,

For n or l even and s divisor of l or s, the GPBIB7 design is a resolvable PBIB designs (RGPBIB7) with r parallel classes where each parallel classes contain blocks.

Example 4. Let v = 3 × 3 × 3 treatments be arranged in the three following arrays:

The construction method for (s = 3), give a generalized rectangular right angular GPBIB7 design with parameters:

To illustrate the method, we applying the construction method for the columns and rows of the first array, where each column represents a block:

4. Construction of the U-Type Designs Based on Resolvable GPBIBm Designs (m = 4, 5 and 7)

In this section we apply the RBIBDmethod (see  ) on our resolvable rectangular right angular RGPBIBm designs (m = 4, 5 and 7) to obtain a series of U-type designs.

Let a resolvable GPBIBm designs (m = 4, 5 and 7) with r parallel classes where each j-th class contains blocks (1 ≤ j ≤ r). Then we can construct a U-type design from resolvable GPBIBm designs (m = 4, 5 and 7) as follows:

Algorithm RGPBIBm − UD

・ Step 1. Give a natural order to the q blocks in each parallel class PCj,.

・ Step 2. For each PCj, construct a q-level column as follows: Set, if treatment is contained in the u-th block of PCj,

・ Step 3. The r q-level columns constructed from PCj, form a.

Proposition 11. For v = wnl runs (w ≥ 2, n ≥ 2, l ≥ 2), a series of U-type designs exist:

・ U, n or l even and s divisor of n or l.

・ U, n or l even and s divisor of n or l.

・ U, n or l even and s divisor of n or l.

・ U, n or l even and s divisor of n or l.

・ U, n or l even and s divisor of n or l.

Proof. applying the RGPBIBm − UD Algorithm on each resolvable rectangular right angular GPBIBm (m = 4, 5 and 7) of the Proposition 1, 5.

Example 5. Applying the RGPBIBm − UD Algorithm on the resolvable rectangular right angular GPBIB7 of Example 1, we obtain the following U-type U (48, 49) with 48 runs and nine 4-level factors.

5. Conclusions

New association schemes with m = 4, 5 and 7 associated classes called generalized rectangular right angular association schemes for v = wnl treatments arranged in w ≥ 2 (n × l) arrays were described and their parameters expressions were given exactly and directly. Some construction methods of PBIB designs based on these association schemes accommodated by accessible method called the Combinatory Method (s) which facilitates the

construction application were explained. Moreover, a series of U-type designs U, by applying the

Fang RBIBD method on resolvable generalized rectangular right angular GPBIBm designs (m = 4, 5 and 7) was constructed.

We note that all the construction methods described in this article were programmed with the R-package “CombinS”  (the ameliorated version).

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