How do I add a constraint to ompr linear programming constraint in R so that each person can only be put in 1 role?

Question

I am trying to add a constraint to a mixed integer programming model in R so that each person is only assigned to a single role.

I have a data frame that looks like this:

ID    Name     Role     PreferenceScore  
 ----- ------- --------- ------------------ 
    1   Abby    Chef                    10  
    1   Abby    Waiter                   8  
    1   Abby    Greeter                  9  
    2   Bob     Chef                     7  
    2   Bob     Waiter                   8  
    2   Bob     Greeter                  3  
    3   Carly   Chef                     5  
    3   Carly   Waiter                   8  
    3   Carly   Greeter                  4  
   ...  ...     ...                      ...
   20   David   Chef                     2  
   20   David   Waiter                   3  
   20   David   Greeter                  8  

I am trying to assign each person to a role based on their preferences (larger number is better) using MIPmodel. Each role can have up to 8 people, and there are 20 total people in total.

Here is what I have so far:

library(dplyr)
library(ompr)
library(ompr.roi)
library(ROI)
library(ROI.plugin.glpk)

teamData <- read.csv("filename")
teamData$wait <- if_else(teamData$jobType == "Waiter", 1, 0)
teamData$chef <- if_else(teamData$jobType == "Chef", 1, 0)
teamData$greet <- if_else(teamData$jobType == "Greeter", 1, 0)

p <- nrow(teamData)
v <- as.numeric(teamData$PreferenceScore)
maxTeamSize <- 8
role <- teamData$Role

chef_job <- teamData$chef 
waiter_job <- teamData$wait
greeter_job <- teamData$greet
name <- teamData$Name

# Build the model
model <- MIPModel() %>%
  add_variable(x[i], i=1:p, type = "binary") %>%
  set_objective(sum_expr(x[i] * v[i], i=1:p)) %>%
  add_constraint(sum_expr(chef_job[i], i=1:p) <= 8) %>%
  add_constraint(sum_expr(waiter_job[i], i=1:p) <= 8) %>%
  add_constraint(sum_expr(greeter_job[i], i=1:p) <= 8) # %>%
#   add_constraint(sum_expr(count(name[i])) == 1)

solved <- solve_model(model, with_ROI("glpk"))

result <- solved %>%
  get_solution(x[i]) %>%
  select(i) %>% 
  rowwise() %>% 
  mutate(Pref = v[i], Role = role[i], teamData$Name[i]) %>%
  ungroup

result

My main problem right now is that I can't figure out how to add a constraint so that each person only has one role in the solution (i.e. each person can only be either a chef, waiter, or greeter)

Any pointers would be greatly appreciated.

Answer 1

This is a typical assignment problem where you are assigning person p to job j. So, you need to update your formulation to have a double indexed decision variable, x[p, j]. Then things will start to make more sense... :).

You can then sum across all jobs to ensure each person has no more than one assignment (in pseudocode... my R syntax is terrible):

sum (x[p, j] for j in Jobs) <=1  for p in Persons

and you can make make sure no position is overfilled if you have a maximum of spots[j] fills in each role:

sum (x[p, j] for p in People) <= spots[j] for j in Jobs

and your objective function will fall into place similarly by multiplying by preference[p, j]