design setup
The design fixed parameters
To get the design I used a few assumptions. The number of rows is twice the number of columns. This is because I remember reading there are no solutions with the same number of rows as columns. The first row reads 1:n. The first column is all elements of 1:n twice.Filling in the design
The approach is actually very simple.- Look at the first not yet know point.
- Examine which value it could have.
- not yet used in current row
- at most once used in current column
- at maximum once used after the product in current row, one column to the left (carry over)
- if this is the second column and the first column of this row is the same as the first column of the previous row, then the current value must be higher than the value of one row up.
- Try the possible values in a loop
- Within that loop go back to step one
Tricks
Compared to the version with an even number of products I implemented some things a bit smarter. Removed some if within a for loop by selecting the objects over which the loop sequences. I also noticed some things which surprised me in making things faster. For instance, creating temporary variables (e.g. row <- desconst$todo[desobject$count,2L]) is faster than using desconst$todo[desobject$count,2L] to index matrices later on. Merging two simple lines of code into one complex line of code is slower. Apparently it is cheaper to store a small result than have the interpreter figure out a complex line of code. Hence I used quite a lot of local variables, certainly more than I find 'nice'.Designs
The result if 90 Williams designs for five products. Without doubt there is quite some duplication from permutations of the objects. The designs can be separated in three classes. Designs symmetric in columns (72 of those), designs with four rows symmetric in columns (five of those) and designs where are not symmetric in columns at all (13 of those).
design symmetric in columns
The first design has five unique rows each copied reversed to make the ten rows. For ease I colored the matching rows.
[,1] [,2] [,3] [,4] [,5]
[1,] 1 2 3 4 5
[2,] 1 3 2 5 4
[3,] 2 1 4 3 5
[4,] 2 4 1 5 3
[5,] 3 1 5 2 4
[6,] 3 5 1 4 2
[7,] 4 2 5 1 3
[8,] 4 5 2 3 1
[9,] 5 3 4 1 2
[10,] 5 4 3 2 1
designs four rows symmetric in columns
The colored rows have a reversed row in the same color.
[,1] [,2] [,3] [,4] [,5]
[1,] 1 2 3 4 5
[2,] 1 3 4 2 5
[3,] 2 3 5 1 4
[4,] 2 4 1 5 3
[5,] 3 1 5 4 2
[6,] 3 5 2 1 4
[7,] 4 1 2 5 3
[8,] 4 5 1 3 2
[9,] 5 2 4 3 1
[10,] 5 4 3 2 1
design not symmetric in columns
[,1] [,2] [,3] [,4] [,5]
[1,] 1 2 3 4 5
[2,] 1 3 5 2 4
[3,] 2 4 1 5 3
[4,] 2 5 3 4 1
[5,] 3 1 4 2 5
[6,] 3 2 1 5 4
[7,] 4 3 5 1 2
[8,] 4 5 2 1 3
[9,] 5 1 4 3 2
[10,] 5 4 2 3 1
R code
generation
gendesign <- function(n=3) {
nc <- as.integer(n)
nr <- nc+nc
desmat <- matrix(NA,nrow=nr,ncol=nc)
desmat[1,] <- 1L:nc
desmat[,1] <- rep(1L:nc,each=2)
carover <- matrix(0L,nrow=nr,ncol=nc)
for (i in 1L:(nc-1L)) carover[i,i+1] <- 1L
todo <- which(is.na(t(desmat)),arr.ind=TRUE)
desobject <- list(desmat=desmat,carover = carover,count=0)
desconst <- list(todo=todo,ntodo=nrow(todo),nc=nc,totest = 1L:nc)
desresult <- list()
addpoint(desobject,desresult,desconst)
}
addpoint <- function(desobject,desresult,desconst) {
desobject$count <- desobject$count+1L
if (desobject$count > desconst$ntodo) {
l <- length(desresult)
desresult[[l+1]] <- desobject$desmat
return(desresult)
}
row <- desconst$todo[desobject$count,2L]
col <- desconst$todo[desobject$count,1L]
dob <- desobject
currow <- desobject$desmat[row,]
currow <- currow[!is.na(currow)]
totest <- desconst$totest[-currow]
prev <- desobject$desmat[row,col-1L]
totest <- totest[desobject$carover[prev,totest]<2L]
counts <- rowSums(outer(totest,desobject$desmat[,col],function(x,y) as.integer(x==y)),na.rm=TRUE)
totest <- totest[counts<2L]
if (col==2 & desobject$desmat[row,1]==desobject$desmat[row-1L,1L]) {
totest <- totest[totest > (desobject$desmat[row-1L,2])]
}
for (i in totest) {
desobject$carover[prev,i] <- desobject$carover[prev,i] + 1L
desobject$desmat[row,col] <- i
desresult <- addpoint(desobject,desresult,desconst)
desobject <- dob
}
desresult
}
g5 <- gendesign(5)
checking
# check that each column has each object twice
sapply(g5,function(sub) all(sapply(1:5,function(x)
all(table(sub[,x])==rep(2,5)))))
# check that each product is followed by each other product twice
sub <- g5[[1]]
target <- Reduce('+',lapply(1:4,function(x) table(sub[,x],sub[,x+1])))
target # shows a symmetric matrix with 0 on diagonal, 2 elsewhere.
sapply(g5,function(sub) identical(target,
Reduce('+',lapply(1:4,function(x) table(sub[,x],sub[,x+1])))))
#check the number of reversed rows
sa <- sapply(g5,function(sub1) {
10-sum(1:10 %in% sapply(1:10,function(i) {
r1 <- rev(sub1[i,])
mat <- matrix(rep(r1,each=10),nrow=10,ncol=5)
which(rowSums(sub1==mat)==5)
}))
})
table(sa)
# number of reversed rows
# 0 4 10
#13 5 72
prints
g5[[1]]
g5[[4]]
g5[[8]]
