Complex budgets

library(om)
#> Loading required package: ROI.plugin.glpk
#> Loading required package: ompr
#> Loading required package: ompr.roi

In the introductory vignette, we saw how Omland could optimise the national budget given the following modelled cost and impact estimates

omland_output <- data.frame(
  unit = rep(1:10, each = 5),
  option = rep(1:5, 10),
  impact = rep(0:4, 10),
  cost = rep(c(0, 5, 10, 15, 20), 10)  * rep(c(rep(1, 5), rep(5, 5)), each = 5)
)
impact_matrix <- matrix(omland_output$impact, nrow = 10, ncol =  5, byrow = TRUE)
cost_matrix <- matrix(omland_output$cost, nrow = 10, ncol =  5, byrow = TRUE)

Breaking news!!! A regional initiative has been set up to increase funding in two of Omland’s sub-national units, units number 9 and 10. An additional $200 has been made available to Omland, but spending must be restricted units 9 and 10.

Fortunately for Omland, om can deal with multiple budget levels. Let’s look at how we can set this up. We now specify two budgets, the national and regional

budget <- c(100, 200)

We also need to specify which sub-national units have access to which parts of the budget. We can do this with another matrix. In the recipients matrix, each row is a sub-national unit, and each column corresponds to a level in our budget vector. A 1 indicates that a sub-national unit may receive a share of that budget, a 0 indicates it can not receive any of it.

recipients <- cbind(
  rep(1, 10),
  c(rep(0, 8), rep(1, 2))
)
recipients
#>       [,1] [,2]
#>  [1,]    1    0
#>  [2,]    1    0
#>  [3,]    1    0
#>  [4,]    1    0
#>  [5,]    1    0
#>  [6,]    1    0
#>  [7,]    1    0
#>  [8,]    1    0
#>  [9,]    1    1
#> [10,]    1    1

Here, we see that only rows 9 and 10 have access to both budget levels. Now we can optimise again

optimised <- om(z = impact_matrix, cost = cost_matrix, budget = budget,
                recipients = recipients)
optimised
#>     i j cost z budget_level_1 budget_level_2
#> 1   1 5   20 4             20              0
#> 2   2 5   20 4             20              0
#> 3   3 5   20 4             20              0
#> 4   4 5   20 4             20              0
#> 5   5 5   20 4             20              0
#> 6   6 1    0 0              0              0
#> 7   7 1    0 0              0              0
#> 8   8 1    0 0              0              0
#> 9   9 5  100 4              0            100
#> 10 10 5  100 4              0            100

Now we see that the first budget level, available to all of Omland, is shared across the first 5 sub-national units, just like in the introductory example. However, thanks to the regional initiative, Omland can now spend an additional $200 in sub-national units 9 and 10, improving the overall impact. The columns budget_level_1 and budget_level_2 show us how each section of Omland’s budget is distributed.

You can add more budget levels by modifying the budget and recipients inputs.