Limiting Duplication of Specified Columns - r

Question

I'm trying to find a way to add some constraints into a linear programme to force the solution to have a certain level of uniqueness to it. I'll try explain what I mean here. Take the example below, the linear programme returns the max possible Score for a combination of 2 males and 1 female.
Looking at the Team/Grade/Rep columns however we can see that there is a lot of duplication from row to row. In fact Shana and Jason are identical.
Name<-c("Jane","Brad","Harry","Shana","Debra","Jason")
Sex<-c("F","M","M","F","F","M")
Score<-c(25,50,36,40,39,62)
Team<-c("A","A","A","B","B","B")
Grade<-c(1,2,1,2,1,2)
Rep<-c("C","D","C","D","D","D")
df<-data.frame(Name,Sex,Score,Team,Grade,Rep)
df
Name Sex Score Team Grade Rep
1 Jane F 25 A 1 C
2 Brad M 50 A 2 D
3 Harry M 36 A 1 C
4 Shana F 40 B 2 D
5 Debra F 39 B 1 D
6 Jason M 62 B 2 D
library(Rglpk)
num <- length(df$Name)
obj<-df$Score
var.types<-rep("B",num)
matrix <- rbind(as.numeric(df$Sex == "M"),as.numeric(df$Sex == "F"))
direction <- c("==","==")
rhs<-c(2,1)
sol <- Rglpk_solve_LP(obj = obj, mat = matrix, dir = direction, rhs = rhs,types = var.types, max = TRUE)
df[sol$solution==1,]
Name Sex Score Team Grade Rep
2 Brad M 50 A 2 D
4 Shana F 40 B 2 D
6 Jason M 62 B 2 D
What I am trying to work out is how to limit say the level of randomness across those last three columns. For example I would like there to no more than ie 2 columns the same across any two rows. So this would mean that either the Shana row or Jason row would be replaced in the model with an alternative.
I'm not sure if this is something that can be easily added into the Rglpk model? Appreciate any help that can be offered.

Answer 1

It sounds like you're asking how to prevent having a pair of individuals who are "too similar" from being returned by your optimization model. Once you have determined a rule for what makes a pair of people "too similar", you can simply add a constraint for each pair, limiting your solution to have no more than one of those two people.
For instance, if we use your rule of having no more than 2 columns the same, we could easily identify all pairs that we want to block:
pairs <- t(combn(nrow(df), 2))
(blocked <- pairs[rowSums(sapply(df[,c("Team", "Grade", "Rep")], function(x) {
x[pairs[,1]] == x[pairs[,2]]
})) >= 3,])
# [,1] [,2]
# [1,] 1 3
# [2,] 4 6
We want to block the pairs Jane/Harry and Shana/Jason. This is easy to do with linear constraints:
library(Rglpk)
num <- length(df$Name)
obj<-df$Score
var.types<-rep("B",num)
matrix <- rbind(as.numeric(df$Sex == "M"), as.numeric(df$Sex == "F"),
outer(blocked[,1], seq_len(num), "==") + outer(blocked[,2], seq_len(num), "=="))
direction <- rep(c("==", "<="), c(2, nrow(blocked)))
rhs<-c(2, 1, rep(1, nrow(blocked)))
sol <- Rglpk_solve_LP(obj = obj, mat = matrix, dir = direction, rhs = rhs,types = var.types, max = TRUE)
df[sol$solution==1,]
# Name Sex Score Team Grade Rep
# 2 Brad M 50 A 2 D
# 5 Debra F 39 B 1 D
# 6 Jason M 62 B 2 D
The approach of computing every pair to block is attractive because we could have a much more complicated rule for which pairs to block, since we don't need to encode the rule into the linear program. All we need to be able to do is to compute every pair that needs to be blocked.

Answer 2

For each group of rows having the same last 3 columns we construct a constraint such that at most one of those rows may appear. If a is an indictor vector of the rows of such a group then the constraint would look like this:
a'x <= 1
To do that split the row numbers by the last 3 columns into a list of vectors s each of whose components is a vector of row numbers for rows having the same last 3 columns. Only keep those conponents having more than 1 row number giving s1. In this case the first component of s1 is c(1, 3) referring to the Jane and Harry rows and the second component is c(4, 6) referring to the Shana and Jason rows. In this particular data there were 2 rows in each of the groups but in other data there could be more than 2 rows in a group. excl has one row (constraint) for each element of s1.
The data in the question only has groups of size 2 but in general if there were k rows in some group one would need k choose 2 constraint rows to ensure that only one of the k were chosen if this were done pairwise whereas the approach here only requires one constraint row for the entire group. For example, if k = 10 then choose(10, 2) = 45 so this uses 1 constraint in place of 45.
Finally rbind excl to matrix giving matrix2 and adjust the other Rglpk_solve_LP arguments accordingly giving:
nr <- nrow(df)
s <- split(1:nr, df[4:6])
s1 <- s[lengths(s) > 1]
excl <-t(sapply(s1, "%in%", x = 1:nr)) + 0
matrix2 <- rbind(matrix, excl)
direction2 <- c(direction, rep("<=", nrow(excl)))
rhs2 <- c(rhs, rep(1, nrow(excl)))
sol2 <- Rglpk_solve_LP(obj = obj, mat = matrix2,
dir = direction2, rhs = rhs2, types = "B", max = TRUE)
df[ sol2$solution == 1, ]
giving:
Name Sex Score Team Grade Rep
2 Brad M 50 A 2 D
5 Debra F 39 B 1 D
6 Jason M 62 B 2 D