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).
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: [
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) [
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 [
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 [
1) Any two treatments are either 1st, 2nd, ∙∙∙, or mth associates. The relation of association is symmetric, i.e., if the treatment
2) Each treatment
3) If any two treatments
The numbers v, ni
Definition 2. A PBIB design [
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
The Combinatory Method (s) [
Let an array of n rows and l columns as follows:
Consider s different elements of the same row i
Definition 3. Let U
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
Let
・
・
・
・
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
2) The second associates of
3) The third associates of
4) The fourth associates of
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.
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
Let
・
・
・
・
・
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
2) The second associates of
3) The third associates of
4) The fourth associates of
5) The fifth associates of
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.
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
Let
・
・
・
・
・
・
・
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
2) The second associates of
3) The third associates of
4) The fourth associates of
5) The fifth associates of
6) The sixth associates of
7) The seventh associates of
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.
Let v = wnl (w ≥ 2, n ≥ 2, l ≥ 2) treatments be arranged in w arrays of n rows and l columns
Applying the Combinatory Method (s) on each of the w arrays, with chosen
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:
For the w arrays we obtain a generalized rectangular right angular GPBIB4 design with parameters:
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
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:
Let
Applying the Combinatory Method
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
○ On an array
○ On an array
other elements of the same row. There is
Thus:
・
・
○ In an array
○ In the array
In total
・
・
○ If
times. For the array
have
In total
○ If
In total
・
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
Let
thod (s) with s chosen
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
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:
Applying the Combinatory Method (s) on each of the w arrays, with chosen and fixed
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
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:
Let
of the gth array
array of the form
treatments that always contain a component of the column
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
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:
In this section we apply the RBIBDmethod (see [
Let a resolvable GPBIBm designs (m = 4, 5 and 7) with r parallel classes
・ Step 1. Give a natural order
・ Step 2. For each PCj, construct a q-level column
・ Step 3. The r q-level columns constructed from PCj,
Proposition 11. For v = wnl runs (w ≥ 2, n ≥ 2, l ≥ 2), a series of U-type
・ U
・ U
・ U
・ U
・ U
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.
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
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” [