Problem YOUDEN RECTANGLE'S PROBLEM BIBD- Balanced Incomplete Block Design . There is a matrix m x n. This matrix can be 26 x 4 but we can change this dimension if 26 x 4 does not admit a feasible solution. • m> n + 1 • The rows are called "blocks" and the columns are called "forms" • Every form is composed by the same number of blocks • • Every block is composed by the same number of forms The matrix is full and every single spot should be filled with a treatment Every treatment is is replicated r times in the Youden rectangle • Every treatment appears in every form maximum one time • Every couple of treatment appear together in every form 1 times Every treatment appear in all single blocks the same number of times v = number of treatments (1,2,3,4,.....,n) n = blocks (columns) m = forms (rows) r = number of times every treatment is replicated λ = number of times two treatment appear together in the same form Constraints: 1) nxm=vxr - 2) 1 (v 1) = r (k-1) 3) m > v then r > n Goal The matrix m x n should be fully filled with treatments respecting the above mentioned constraints. The filling of the matrix should be done using balanced Incomplete Block Design. Software The Problem should be solved using r sosftware. The r package "ibd" and the function "bibd" didn't solved the problem. The r package "agricolae" neither. Example that doesn work [,1] [,2] [,3] [,4] [1,] "C" "E" "B" "D" [2,] "D" "B" "A" "E" [3,] "B" "D" "F" "A" [4,] "A" "B" "C" "D" [5,] "F" "E" "C" "D" [6,] "F" "A" "B" "E" [7,] "F" "A" "C" "B" [8,] "D" "A" "C" "E" [9,] "B" "E" "D" "F" [10,] "A" "F" "C" "D" [11,]"E" "F" "C" "A" [12,] "C" "B" "D" "F" [13,] "A" "B" "E" "C" [14] "B" "F" "C" "E" [15,] "F" "A" "E" "D" 4 3 1 2 In this example treatment "F" is not replicated the same number of times in all blocks (4 in first block, 3 in second, 1 in third, 2 in fourth)
Problem YOUDEN RECTANGLE'S PROBLEM BIBD- Balanced Incomplete Block Design . There is a matrix m x n. This matrix can be 26 x 4 but we can change this dimension if 26 x 4 does not admit a feasible solution. • m> n + 1 • The rows are called "blocks" and the columns are called "forms" • Every form is composed by the same number of blocks • • Every block is composed by the same number of forms The matrix is full and every single spot should be filled with a treatment Every treatment is is replicated r times in the Youden rectangle • Every treatment appears in every form maximum one time • Every couple of treatment appear together in every form 1 times Every treatment appear in all single blocks the same number of times v = number of treatments (1,2,3,4,.....,n) n = blocks (columns) m = forms (rows) r = number of times every treatment is replicated λ = number of times two treatment appear together in the same form Constraints: 1) nxm=vxr - 2) 1 (v 1) = r (k-1) 3) m > v then r > n Goal The matrix m x n should be fully filled with treatments respecting the above mentioned constraints. The filling of the matrix should be done using balanced Incomplete Block Design. Software The Problem should be solved using r sosftware. The r package "ibd" and the function "bibd" didn't solved the problem. The r package "agricolae" neither. Example that doesn work [,1] [,2] [,3] [,4] [1,] "C" "E" "B" "D" [2,] "D" "B" "A" "E" [3,] "B" "D" "F" "A" [4,] "A" "B" "C" "D" [5,] "F" "E" "C" "D" [6,] "F" "A" "B" "E" [7,] "F" "A" "C" "B" [8,] "D" "A" "C" "E" [9,] "B" "E" "D" "F" [10,] "A" "F" "C" "D" [11,]"E" "F" "C" "A" [12,] "C" "B" "D" "F" [13,] "A" "B" "E" "C" [14] "B" "F" "C" "E" [15,] "F" "A" "E" "D" 4 3 1 2 In this example treatment "F" is not replicated the same number of times in all blocks (4 in first block, 3 in second, 1 in third, 2 in fourth)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Transcribed Image Text:Problem
YOUDEN RECTANGLE'S PROBLEM
BIBD- Balanced Incomplete Block Design
.
There is a matrix m x n. This matrix can be 26 x 4 but we can change this dimension
if 26 x 4 does not admit a feasible solution.
•
m> n + 1
•
The rows are called "blocks" and the columns are called "forms"
•
Every form is composed by the same number of blocks
•
•
Every block is composed by the same number of forms
The matrix is full and every single spot should be filled with a treatment
Every treatment is is replicated r times in the Youden rectangle
• Every treatment appears in every form maximum one time
•
Every couple of treatment appear together in every form 1 times
Every treatment appear in all single blocks the same number of times
v = number of treatments (1,2,3,4,.....,n)
n = blocks (columns)
m = forms (rows)
r = number of times every treatment is replicated
λ = number of times two treatment appear together in the same form
Constraints:
1) nxm=vxr
-
2) 1 (v 1) = r (k-1)
3) m > v then r > n
Goal
The matrix m x n should be fully filled with treatments respecting the above mentioned
constraints. The filling of the matrix should be done using balanced Incomplete Block
Design.
Software
The Problem should be solved using r sosftware.
The r package "ibd" and the function "bibd" didn't solved the problem. The r package
"agricolae" neither.
![Example that doesn work
[,1] [,2] [,3] [,4]
[1,] "C" "E" "B" "D"
[2,] "D" "B" "A" "E"
[3,] "B" "D" "F" "A"
[4,] "A" "B" "C" "D"
[5,] "F" "E" "C" "D"
[6,] "F" "A" "B" "E"
[7,] "F" "A" "C" "B"
[8,] "D" "A" "C" "E"
[9,] "B" "E" "D" "F"
[10,] "A" "F" "C" "D"
[11,]"E" "F" "C" "A"
[12,] "C" "B" "D" "F"
[13,] "A" "B" "E" "C"
[14] "B" "F" "C" "E"
[15,] "F" "A" "E" "D"
4 3 1 2
In this example treatment "F" is not replicated the same number of times in all blocks (4 in
first block, 3 in second, 1 in third, 2 in fourth)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf7381b5-894e-49ad-ac17-f5e999ebfcd8%2Feae13965-8228-4f85-84a0-2602e3c7db0d%2F4op91c7u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Example that doesn work
[,1] [,2] [,3] [,4]
[1,] "C" "E" "B" "D"
[2,] "D" "B" "A" "E"
[3,] "B" "D" "F" "A"
[4,] "A" "B" "C" "D"
[5,] "F" "E" "C" "D"
[6,] "F" "A" "B" "E"
[7,] "F" "A" "C" "B"
[8,] "D" "A" "C" "E"
[9,] "B" "E" "D" "F"
[10,] "A" "F" "C" "D"
[11,]"E" "F" "C" "A"
[12,] "C" "B" "D" "F"
[13,] "A" "B" "E" "C"
[14] "B" "F" "C" "E"
[15,] "F" "A" "E" "D"
4 3 1 2
In this example treatment "F" is not replicated the same number of times in all blocks (4 in
first block, 3 in second, 1 in third, 2 in fourth)
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