Optimization function applied to table of values in R - r

Question

`values <- matrix(c(0.174,0.349,1.075,3.1424,0.173,0.346,1.038,3.114,0.171,0.343,1.03,3.09,0.17,0.34,1.02,3.06),ncol=4) `
I am attempting to maximize the total value for the dataset taking only one value from each row, and with associated costs for each column
subject to:
One value column used per row.
cost of each use of column 1 is 4
cost of each use of column 2 is 3
cost of each use of column 3 is 2
cost of each use of column 4 is 1
total cost <= 11
These are stand in values for a larger dataset. I need to be able to apply it directly to all the rows of a dataset.
I have been trying to use the lpSolve package, with no success.
`f.obj <- values
f.con <- c(4,3,2,1)
f.dir <- "<="
f.rhs <- 11
lp("max", f.obj, f.con, f.dir, f.rhs)`
I am getting a solution of "0"
I do not know how to model this in a way that chooses one value per row and then uses a different value in calculating the constraints.

Answer 1

Looks like the problem is as follows:
We have a matrix a[i,j] with values, and a vector c[j] with costs.
We want to select one value for each row such that:
a. total cost <= 11
b. total value is maximized
To develop a mathematical model, we introduce binary variables x[i,j] ∈ {0,1}. With this, we can write:
max sum((i,j), a[i,j]*x[i,j])
subject to
sum((i,j), c[j]*x[i,j]) <= 11
sum(j, x[i,j]) = 1 ∀i
x[i,j] ∈ {0,1}
Implement in R. I use here CVXR.
#
# data
# A : values
# C : cost
#
A <- matrix(c(0.174,0.349,1.075,3.1424,0.173,0.346,1.038,3.114,0.171,0.343,1.03,3.09,0.17,0.34,1.02,3.06),ncol=4)
C <- c(4,3,2,1)
maxcost <- 11
#
# form a matrix cmat[i,j] indicating the cost of element i,j
#
cmat <- matrix(C,nrow=dim(A)[1],ncol=dim(A)[2],byrow=T)
#
# problem:
# pick one value from each row
# such that total value of selected cells is maximized
# and cost of selected cells is limited to maxcost
#
# model:
# min sum((i,j), a[i,j]*x[i,j])
# subject to
# sum((i,j), c[j]*x[i,j]) <= maxcost
# sum(j,x[i,j]) = 1 ∀i
# x[i,j] ∈ {0,1}
#
#
library(CVXR)
x = Variable(dim(A), name="x", boolean=T)
p <- Problem(Maximize(sum_entries(A*x)),
constraints=list(
sum_entries(cmat*x) <= maxcost,
sum_entries(x,axis=1) == 1
))
res <- solve(p,verbose=T)
res$status
res$value
res$getValue(x)*A
The output looks like:
> res$status
[1] "optimal"
> res$value
[1] 4.7304
> res$getValue(x)*A
[,1] [,2] [,3] [,4]
[1,] 0.0000 0 0.000 0.17
[2,] 0.0000 0 0.343 0.00
[3,] 1.0750 0 0.000 0.00
[4,] 3.1424 0 0.000 0.00
The description in the original post is not very precise. For instance, I assumed that we need to select precisely one cell from each row. If we just want "select at most one cell from each row", then replace
sum(j, x[i,j]) = 1 ∀i
by
sum(j, x[i,j]) <= 1 ∀i

Answer 2

As mentioned by Steve, the lpSolve package expects a single objective function not a matrix. You could reformulate as maximize(sum(RowSums(values*xij)) given constraint
Eg, change the matrix to a vector, and change the problem to a integer optimization problem
obj <- as.vector(values)
f.con <- rep(f.con, each = 4)
r <- lp('max', obj, matrix(f.con, nrow = 1), f.dir, f.rhs, int.vec = seq_along(obj))
#' Success: the objective function is 9.899925