AE job alloc.jl

using JuMP,  GLPK, GLPKMathProgInterface

########## Inputs
N = 4 # number of departments
n = 300 # number of candidates

# defines preferences over each department (columns) for each candidate (rows)
# each cell indicates utility of getting job (from 0 to 1). 
# TODO: use nonlinear (exp) preferences 
# importance of matching each candidates preferences can be altered by multiplying a row by a constant (0..1), 
# e.g. may not want to give full weight to those who got their first choice in a previous round
prefs = vcat([collect(linspace(0,1,N))[randperm(N)]' for i = 1:n]...)
jobheld = rand(1:N, n)

########## Model
begin
    mod = Model(solver=GLPKSolverMIP())

    # choice array `x` where xij is binary indicating whether to give the job `j` to candidate `i`
    @variable(mod, x[1:n,1:N], Bin)

    for i = 1:n
        @constraint(mod, sum(x[i,1:N]) == 1) # only 1 job per person
        @constraint(mod, x[i,jobheld[i]] == 0) # can't move to a department you're already in
    end

    # limit of jobs at department 1 and 2 to 5 and 15 respec.
    @constraint(mod, sum(x[1:n,1]) <= 5)
    @constraint(mod, sum(x[1:n,2]) <= 15)

    @objective(mod, Max, sum(x.*prefs))
    solve(mod)
    val =getvalue(x)
end

firstchoiceperdepartment = sum(prefs.==1.0,1)
firstchoice = [indmax(prefs[i,:]) for i = 1:n]


jobafter = [indmax(val[i,:]) for i = 1:n]

sum(firstchoice .== jobafter)/n

T = [sum((jobheld.==i) .& (jobafter.==j)) for i = 1:N, j = 1:N]
println("No. of first choices by departments:")
println(firstchoiceperdepartment)
println()
println("No. assigned to each department:")
println(sum(val.>0, 1))
println()
println("Transition matrix: ")
display(T)

# Example output
# No. of first choices by departments:
# [76 74 65 85]

# No. assigned to each department:
# [5 15 123 157]

# Transition matrix:
# 4×4 Array{Int64,2}:
#  0  0  32  53
#  0  0  25  41
#  4  7   0  63
#  1  8  66   0