# rcd: Program for Constructing Non-Resolvable Row-column Designs

## Introduction

A (non-resolvable) row-column design (RCD) is an arrangement of v treatments set out in kxb array such that each treatment is replicated r times (vr=kb). We say this RCD is of size (v,k,b). A recommended reference on the subject is Chapter 5 of John & Williams (1995).

rcd is a Gendex program for constructing optimal or near-optimal row-column designs. The approach adopted by rcd is to permute the treatments within the blocks of an optimal or near-optimal block design used as the column component of the RCD. The optimality criterion and the algorithm used in rcd are discussed in Nguyen (1997).

## Using rcd

Let's assume all Gendex class files are in the directory c:\gendex and suppose you want to construct an RCD of size (v,k,b)=(9,3,6). The following is the file bib.txt in the working directory which contains an IBD of size (v,k,r)=(9,3,2) with blocks as columns:

```5     6     3     4     8     5
0     7     7     8     1     1
4     0     2     3     6     2```

At the working directory, issue the following command at the Command Prompt (case is important and no space is allowed before and after the equal sign):

`java -cp c:\gendex rcd in=bib.txt`

rcd will start running and after try 10, the plan of the constructed design for this try will be displayed at the terminal window (as the ratio E/U reaches 1) and rcd stops:

```rcd 10.0: Program for Constructing an rcd of size (v,k,b)
(c) 2018 Design Computing (http://designcomputing.net/)

Note: Best rcd for v=9, r=2, k=3 and b=6

Try	iter	f	E(col)	E(row)	E	E/U	Concurrences
1	2	223	0.667	0.923	0.576	0.912	1(13) 2(19) 3(4)
2	1	219	0.667	0.923	0.593	0.939	1(12) 2(21) 3(3)
16	2	207	0.667	0.923	0.632	1.001	1(9) 2(27)

Plan (as a k x b matrix) for try 16 using seed 481900595:

4	0	7	8	1	2
0	6	2	3	8	5
5	7	3	4	6	1

Note: rcd used 0.166 seconds.
Note: rcd.htm has been created.
```

Note that the default random seed is the one obtained from the system clock and the default number of tries is 1000. You can change these default values by specifying the seed number and the number of tries, e.g.

`java -cp c:\gendex rcd=bib.txt seed=1234 tries=1000`

The START button has changed to RESET button. If you click this RESET button, the output will disappear and you can now start a new design problem. Note that the default random seed is the one obtained from the system clock and the default number of tries is 1000. You can change these default values if you wish to.

## Output

The result of the best try is displayed at the terminal window window and is also saved in the file rcd.htm in the working directory. This file can be read by a browser such as IE or Google Chrome. Information for this try includes:

1. Try number;
2. The number of iterations;
3. The objective function f.
4. The row, column and row-column efficiency E of the constructed design and the ratio E/U where U is the upper bound of an RCD of size (v,k,b). The program automatically stops if this ratio reaches 1.
5. The distribution of the concurrences of this design;
6. The design plan and the associated random seed;
7. The time in seconds rcd used to construct the above design.

## Examples

1. An RCD of size (v,k,b)=(4,6,6) (http://designcomputing.net/gendex/rcd/r1.html).
2. An RCD of size (v,k,b)=(10,5,6) (http://designcomputing.net/gendex/rcd/r2.html).
3. An RCD of size (v,k,b)=(8,4,4) (http://designcomputing.net/gendex/rcd/r3.html).
4. An RCD of size (v,k,b)=(9,3,6) (http://designcomputing.net/gendex/rcd/r4.html).
5. An RCD of size (v,k,b)=(12,4,6) (http://designcomputing.net/gendex/rcd/r5.html).
6. An RCD of size (v,k,b)=(12,4,9) (http://designcomputing.net/gendex/rcd/r6.html).
7. An RCD of size (v,k,b)=(16,4,12) (http://designcomputing.net/gendex/rcd/r7.html).
8. An RCD of size (v,k,b)=(6,4,9) (http://designcomputing.net/gendex/rcd/r8.html).
9. An RCD of size (v,k,b)=(9,6,6) (http://designcomputing.net/gendex/rcd/r9.html).
10. An RCD of size (v,k,b)=(15,6,10) (http://designcomputing.net/gendex/rcd/r10.html).

Notes:

The IBDs used as column components of the constructed RCDs in Examples 1-10 are in the files r1.txt, r2.txt, r3.txt, r4.txt, r5.txt, r6.txt, r7.txt, r8.txt, r9.txt and r10.txt.

## References

John, J.A. & Williams E.R. (1987) Cyclic designs and computer-generated designs. New York: Chapman & Hall.
Nguyen, N-K (1997) Construction of optimal row-column designs by computer. Computing Science & Statistics 28, 471-475.