Iâ€™ll repeat my answer from: Trying to solve a puzzle - #15 by odow

```
using JuMP, Cbc
function main(; houses::Vector{Tuple{Int,Int}}, ratio::Float64)
M = maximum(h[1] for h in houses)
N = maximum(h[2] for h in houses)
model = Model(Cbc.Optimizer)
@variable(model, cell_tower[1:M, 1:N], Bin)
@variable(model, is_covered[houses], Bin)
@objective(model, Min, sum(cell_tower))
stencil(i, j) = cell_tower[max(i-1,1):min(i+1,M), max(j-1,1):min(j+1,N)]
@constraint(model, [h in houses], is_covered[h] <= sum(stencil(h[1], h[2])))
@constraint(model, sum(is_covered) >= ratio * length(houses))
optimize!(model)
return value.(cell_tower)
end
houses = [(1, 9), (3, 2), (4, 4), (5, 6), (8, 9), (9, 1)]
main(houses, ratio = 0.7)
```

If the right-hand side of the `is_covered[h] <= `

constraint is 1, then `is_covered[h]`

can be 0 or 1. But the next constraint means that there is a benefit to it taking the value 1. Of course, is the `ratio`

constraint isnâ€™t binding in an optimal solution, then there may be some houses `h`

that are covered by the cell tower, but still have `is_covered[h] = 0`

.

The typical way to fix this is to add a small term to the objective to encourage the â€ścorrectâ€ť solution

```
@objective(model, Min, sum(cell_tower) - 0.001 * sum(is_covered))
```

OR modeling is a large field, and more of an art than a science. There are some good online courses to learn more: