I'm trying to select 5 values in the first row, 1 value in rows 2-9, and maximize the objective function. The only thing I can't figure out how to embed in the problem is that I can only choose a value once - which is to say, if value 49.27 is selected for row 1, I can't also choose value 49.27 for row 9.
obj <- c(
49.2723892,
31.4629786,
25.3265757,
23.3915431,
19.5677222,
19.2621303,
18.2912147,
16.7899278,
14.5498144,
12.7440869,
11.6331623,
11.0012557,
7.9473880,
7.8453557,
3.7182273,
0.2084929,
0.0100000
)
mat <- as.matrix(c(1, 0, 0, 0, 0, 0, 0, 0, 1)) %>%
cbind( c(0, 1, 1, 0, 0, 0, 0, 0, 1),
c(1, 0, 0, 0, 0, 0, 0, 0, 1),
c(0, 0, 1, 1, 0, 0, 0, 0, 1),
c(0, 0, 0, 0, 1, 1, 0, 0, 1),
c(0, 0, 0, 0, 0, 1, 1, 0, 1),
c(0, 0, 0, 0, 0, 0, 1, 1, 1),
c(0, 0, 0, 0, 1, 1, 0, 0, 1),
c(1, 0, 0, 0, 0, 0, 0, 0, 1),
c(1, 0, 0, 0, 0, 0, 0, 0, 1),
c(0, 1, 0, 1, 0, 0, 0, 0, 1),
c(0, 1, 1, 0, 0, 0, 0, 0, 1),
c(1, 0, 0, 0, 0, 0, 0, 0, 1),
c(1, 0, 0, 0, 0, 0, 0, 0, 1),
c(1, 0, 0, 0, 0, 0, 0, 0, 1),
c(0, 0, 0, 0, 0, 0, 0, 1, 1),
c(0, 0, 0, 0, 0, 0, 0, 1, 1)
)
const.dir <- c("==", "==", "==", "==","==","==","==", "==", "==")
const.rhs <- c(5, 1, 1, 1, 1, 1, 1, 1, 1)
sol <- lp(direction = "max", obj,
mat, const.dir, const.rhs)
Am I missing something?
Why is it returning the following:
"Error: no feasible solution found"
Let the decision variables be a 0-1 matrix x with the same dimensions as mat and reformulate as:
max sum(x %*% obj)
sumRows(mat * x) == c(5, 1, ..., 1)
sumRows(x) == c(5, 1, ..., 1)
sumCols(x) <= 1
It is a bit easier to specify this with the CVXR package. Note that sum_entries(..., axis = 1) is the row sums and sum_entries(..., axis = 2) is the column sums when using this package.
library(CVXR)
nr <- nrow(mat)
nc <- ncol(mat)
x <- Variable(nr, nc, boolean = TRUE)
objective <- Maximize(sum(x %*% obj))
constraints <- list(
sum_entries(mat * x, axis = 1) == const.rhs,
sum_entries(x, axis = 1) == const.rhs,
sum_entries(x, axis = 2) <= 1L)
prob <- Problem(objective, constraints)
res <- solve(prob)
res$status
## [1] "optimal"
res$value
## [1] 261.2402
r <- round(res$getValue(x))
res$getValue(x)
## ...snip...
colSums(row(r) * r) # for each col show which row is 1
## [1] 1 2 1 4 5 7 8 6 1 1 9 3 1 0 0 0 0
Inputs used
const.rhs <- c(5, 1, 1, 1, 1, 1, 1, 1, 1)
mat <- structure(c(1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0,
1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0,
0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0,
0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0,
1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1,
1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0,
0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1,
1, 0, 0, 0, 0, 0, 0, 0, 1, 1), dim = c(9L, 17L))