# ibd: Program for Constructing Optimal and Near-Optimal Incomplete Block Designs

## Introduction

An incomplete block design (IBD) of size (v,k,r) is an arrangement of v treatments set out in blocks of size k (<v) such that each treatment is replicated r times. We confine to binary IBDs, i.e. IBDs in which no treatment occurs more than once in a block. An IBD is said to be a balanced IBD, or BIBD, if every pair of treatments occurs in exactly blocks. IBDs in Examples 2, 4 and 9 are examples of BIBDs. An IBD is said to be t-resolvable if its blocks can be divided into subsets and each subset is an IBD of size (v,k,t). IBDs in Examples 1, 6, 7, 8 and 9 are examples of 1-resolvable IBDs or resolvable IBDs. IBDs in Examples 2 and 10 are examples of 2-resolvable IBDs. Some recommended references on IBDs are John (1980), John & Williams (1995) and Raghavarao & Padgett (2005). See Nguyen & Blagoeva (2010) for a quick reference on this subject.

ibd is a Gendex program for constructing optimal or near-optimal IBDs (resolvable and non-resolvable). The optimality criterion and the algorithm used in IBD is discussed in Nguyen (1993, 1994). Designs produced by ibd are comparable to α-designs of Patterson & Williams (1976) and generalized cyclic designs of Hall & Jarrett (1981) in terms of the efficiency factor E of the design. These designs, in turn, can be used as column components of row-column designs (Nguyen & Williams 1993; Nguyen 1997).

## Using ibd

Let's assume all Gendex class files are in the directory c:\gendex and suppose you want to construct a resolvable IBD of size (v,k,r)=(6,2,4) (Example 1). At the working directory, issue the following command at the Command Prompt (case is important and no space is allowed before/after the equal sign):

`java -cp c:\gendex ibd v=6  k=2  r=4`

Since the combination (v,k,r)=(6,2,4) allows resolvable IBD, the following question will pop up: Click Yes, ibd will start running and after try 1, the plan of the constructed design will be displayed in the terminal window (as the ratio E/U reaches 1) and ibd stops:

```ibd 10.0: Program for Constructing Incomplete Block Designs
(c) 2018 Design Computing (http://designcomputing.net/)

Note: Best 1-resolvable IBD for v=6, k=2, r=4 and b=12

Try	iter	E	E/U	Concurrences
1	4	0.577	1	0(3) 1(12)

Plan for try 1 using seed 1533442519479 (blocks are columns):

2	1	4
5	3	0

2	4	3
0	1	5

0	2	4
1	3	5

2	4	0
1	3	5

Note: ibd used 4.952 seconds.
Note: ibd.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 ibd v=6 k=2 r=4 seed=1234 tries=1000`

## Output

The result of the best try is displayed at the terminal window and is also saved in the file ibd.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. E, the efficiency factor of this design.
4. The ratio E/U where U is the upper bound of an IBD. U=UJ for non-resolvable IBDs and U=min(UJ,UWP) for resolvable IBDs. UWP is the bound of Williams & Patterson (1977) good for any resolvable IBDs with v≥b. UJ is the bound of Jarrett (1989) good for any regular graph design (RGD). RGDs are IBDs with concurrences differing by at most 1. IBDs in Examples 1-6 and 8-10 are examples of RGDs. The program automatically stops if this ratio reaches 1.
5. The distribution of the concurrences of this design;
6. The design plan with blocks as columns and the associated random seed. If you want to print the blocks as rows, please specify the option -r on the command line.
7. The time in seconds ibd used to construct this design.

## Examples

1. A resolvable IBD of size (v,k,r)=(6,2,4) (http://designcomputing.net/gendex/ibd/b1.html).
2. A 2-resolvable BIBD of size (v,k,r)=(6,4,10) (http://designcomputing.net/gendex/ibd/b2.html).
3. An IBD of size (v,k,r)=(14,5,10) (http://designcomputing.net/gendex/ibd/b3.html).
4. A BIBD of size (v,k,r)=(15,5,14) (http://designcomputing.net/gendex/ibd/b4.html).
5. An IBD of size (v,k,r)=(60,9,3) (http://designcomputing.net/gendex/ibd/b5.html).
6. A resolvable IBD of size (v,k,r)=(30,5,4) (http://designcomputing.net/gendex/ibd/b6.html).
7. A resolvable IBD of size (v,k,r)=(36,6,4) (http://designcomputing.net/gendex/ibd/b7.html).
8. A resolvable IBD of size (v,k,r)=(98,7,2) (http://designcomputing.net/gendex/ibd/b8.html).
9. A resolvable BIBD of size (v,k,r)=(15,3,7) (http://designcomputing.net/gendex/ibd/b9.html).
10. A 2-resolvable IBD of size (v,k,r)=(21,6,10) (http://designcomputing.net/gendex/ibd/b10.html).

Notes:

• Example 2: See Box, Hunter & Hunter (1987) p. 271 for an alternative solution to the BIBD.
• Example 4: This BIBD has been listed as unsolved in Raghavarao (1971) p. 95.
• Example 9: See Street & Street (1987) p. 164 for an alternative solution to this BIBD, also known as Kirkman's school girl problem.
• Example 10: See Clatworthy (1973) plan T69 for an alternative solution. Clearly, T69 is a very poor design as it does not makes sense for certain pairs of treatments to appear together in five blocks while certain pairs never appear together in any of the blocks.

## References

Box, G.E.P, Hunter, W.G. & Hunter, J.S. (1978) Statistics for experimenters. New York: John Wiley.
Clatworthy, W.H. (1973) Tables of two-associates-class partially balanced designs. Appl. Math. Ser. 63. National Bureau of Standards, Washington.
Hall, W.B. & Jarrett, R.G. (1981) Non-resolvable incomplete block designs with few replications. Biometrika 68, 617-627.
Jarrett, R.G. (1989) A review of bounds for the efficiency factor of block designs. Austral. J. Statist. 31, 118-129.
John, J.A. & Williams E.R. (1987) Cyclic designs and computer-generated designs. New York: Chapman & Hall.
John, P.W.M. (1980) Incomplete block designs. New York: Marcel Decker, Inc.
Nguyen, N-K. (1993) An algorithm for constructing optimal resolvable block designs. Commun. Statist. B 22, 911-923.
Nguyen, N-K. (1994) Construction of optimal incomplete block designs by computer. Technometrics 36, 300-307.
Nguyen, N-K. and Williams, E.R. (1993) An algorithm for constructing optimal resolvable row-column designs. Austral. J. Statist. 35, 363-370.
Nguyen, N-K. (1997) Construction of optimal row-column designs by computer. Computing Science & Statistics 28, 471-475.
Nguyen, N-K. & Blagoeva K.L. (2010) Incomplete block designs, in Encyclopedia of Statistical Science, Edited by L. Miodrag. Springer, 653-655.
Patterson, H.D. & Williams, E.R. (1976) A class of resolvable incomplete block designs. Biometrika 63, 83-92.
Raghavarao, D. & Padgett L.V. (2005). Block Designs: Analysis, Combinatorics and Applications. World Scientific.
Williams & Patterson (1977) Upper bound for efficiency factors in block designs. Austral. J. Statist. 19, 194-201.