How to work with binary contraints in linear optimization? - r

Question

I have two input matrices, dt(10,3) & wt(3,3), that i need to use to find the optimal decision matrix (same dimension), Par(10,3) so as to maximize an objective function. Below R code would give some direction into the problem (used Sample inputs here) -
#Input Matrices
dt <- matrix(runif(300),100,3)
wt <- matrix(c(1,0,0,0,2,0,0,0,1),3,3) #weights
#objective function
Obj <- function(Par) {
P = matrix(Par, nrow = 10, byrow=F) # Reshape
X = t((dt%*%wt)[,1])%*%P[,1]
Y = t((dt%*%wt)[,2])%*%P[,2]
Z = t((dt%*%wt)[,3])%*%P[,3]
as.numeric(X+Y+Z) #maximize
}
Now I am struggling to apply the following constraints to the problem :
1) Matrix, Par can only have binary values (0 or 1)
2) rowSums(Par) = 1 (Basically a row can only have 1 in one of the three columns)
3) colSums(Par[,1]) <= 5, colSums(Par[,2]) <= 6, & colSums(Par[,3]) <= 4
4) X/(X+Y+Z) < 0.35, & Y/(X+Y+Z) < 0.4 (X,Y,Z are defined in the objective function)
I tried coding the constraints in constrOptim, but not sure how to input binary & integer constraints. I am reading up on lpSolve, but not able to figure out. Any help much appreciated. Thanks!

Answer 1

I believe this is indeed a MIP so no issues with convexity. If I am correct the model can look like:
This model can be easily transcribed into R. Note that LP/MIP solvers do not use functions for the objective and constraints (opposed to NLP solvers). In R typically one builds up matrices with the LP coefficients.
Note: I had to make the limits on the column sums much larger (I used 50,60,40).

Answer 2

Based on Erwin's response, I am able to formulate the model using lpSolve in R. However still struggling to add the final constraint to the model (4th constraint in my question above). Here's what I am able to code so far :
#input dimension
r <- 10
c <- 3
#input matrices
dt <- matrix(runif(r*c),r,c)
wt <- matrix(c(1,0,0,0,2,0,0,0,1),3,3) #weights
#column controller
c.limit <- c(60,50,70)
#create structure for lpSolve
ncol <- r*c
lp.create <- make.lp(ncol=ncol)
set.type(lp.create, columns=1:ncol, type = c("binary"))
#create objective values
obj.vals <- as.vector(t(dt%*%wt))
set.objfn(lp.create, obj.vals)
lp.control(lp.create,sense='max')
#Add constraints to ensure sum of parameters for every row (rowSum) <= 1
for (i in 1:r){
add.constraint(lp.create, xt=c(1,1,1),
indices=c(3*i-2,3*i-1,3*i), rhs=1, type="<=")
}
#Add constraints to ensure sum of parameters for every column (colSum) <= column limit (defined above)
for (i in 1:c){
add.constraint(lp.create, xt=rep(1,r),
indices=seq(i,ncol,by=c), rhs=c.limit[i], type="<=")
}
#Add constraints to ensure sum of column objective (t((dt%*%wt)[,i])%*%P[,i) <= limits defined in the problem)
#NOT SURE HOW TO APPLY A CONSTRAINT THAT IS DEPENDENT ON THE OBJECTIVE FUNCTION
solve(lp.create)
get.objective(lp.create) #20
final.par <- matrix(get.variables(lp.create), ncol = c, byrow=T) # Reshape
Any help that can get me to the finish line is much appreciated :)
Thanks