Optimization in JuMP using GLPK

Question

I am trying to find an optimal number of employees to satisfy a scheduling problem, this is my code:

using JuMP
using GLPK

# Constants and parameters
Wjk = [[19, 19, 14]  [16, 16, 11]  [22, 22, 16]  [22, 22, 16]  [22, 22, 16]  [22, 22, 16]  [22, 22, 16]]   # Workload requirement
Ajk = 8  # Some constant value for Ajk
M = 395  # Total number of shifts

# Create a JuMP model
model = Model(GLPK.Optimizer)

# Decision variables for shift allocation
@variable(model, x[1:N, 1:3, 1:7] >= 0, Int)  # Decision variable for shift allocation


# Objective function
@objective(model, Min, sum(x[i, j, k] for i in 1:N, j in 1:3, k in 1:7))


# Objective function
#@objective(model, Min, sum(x[i, j, k] for i in 1:N for j in 1:3 for k in 1:7))


# Constraints
for i in 1:N
    for k in 1:7
        @constraint(model, sum(x[i, j, k] for j in 1:3) <= 2)  # Constraints (8) - (10)
        @constraint(model, sum(x[i, j, k]) + sum(x[i, 2, mod(k + 1, 7) + 1] for j in 2:3) <= 2)  # Constraint (9)
        @constraint(model, sum(x[i, 3, k]) + sum(x[i, 1, mod(k + 1, 7) + 1]) +
                    sum(x[i, 2, (mod(k + 1, 7)) + 1]) <= 2)  # Constraint (10)
    end
end


for j in 1:3
    for k in 1:7
        @constraint(model, sum(x[i, j, k] for i in 1:N) >= Wjk[k, j])  # Constraint (11)
        @constraint(model, sum(x[i, j, k] for i in 1:N) <= Wjk[k, j] + Ajk)  # Constraint (12)
    end
end

for i in 1:N
    @constraint(model, sum(x[i, j, ((i - 1) % 7) + 1] + x[i, j, i % 7 + 1] for j in 1:3) == 0)  # Constraint (13)
end

for i in 1:N
    for k in 1:5
        @constraint(model, sum(x[i, j, (i + k) % 7 + 1] for j in 1:3) >= 1)  # Constraint (14)
    end
end

@constraint(model, sum(x[i, j, k] for i in 1:N for j in 1:3 for k in 1:7) == M)

# Solve the optimization problem
optimize!(model)

# Display the results
println("Optimal Number of Employees: ", value(N))
println("Optimal Shift Allocation:")
for i in 1:value(N)
    for j in 1:3
        for k in 1:7
            if value(x[i, j, k]) > 0.5
                println("Employee $i, Shift $j on Day $k")
            end
        end
    end
end

However I cannot seem to get past this error:

> ERROR: MethodError: Cannot `convert` an object of type VariableRef to an object of type Float64
> Closest candidates are:
> convert(::Type{T}, ::Base.TwicePrecision) where T<:Number at twiceprecision.jl:250
> convert(::Type{T}, ::AbstractChar) where T<:Number at char.jl:180
> convert(::Type{T}, ::CartesianIndex{1}) where T<:Number at multidimensional.jl:136
> >   ...
> Stacktrace:
> [1] MathOptInterface.GreaterThan{Float64}(lower::VariableRef)
> MathOptInterface C:\Users\doriiido\.julia\packages\MathOptInterface\IiXiU\src\sets.jl:171
> [2] _moi_constrain_variable(moi_backend::MathOptInterface.Utilities.CachingOptimizer{MathOptInterface.Bridges.LazyBridgeOptimizer{GLPK.Optimize                                                                                                                                          er}, MathOptInterface.Utilities.UniversalFallback{MathOptInterface.Utilities.Model{Float64}}}, index::MathOptInterface.VariableIndex, info::VariableInfo{VariableRef, Float64, Float64, Float64}, #unused#::Type{Float64})
> JuMP C:\Users\doriiido\.julia\packages\JuMP\ToPd2\src\variables.jl:1754
> [3] _moi_add_variable(moi_backend::MathOptInterface.Utilities.CachingOptimizer{MathOptInterface.Bridges.LazyBridgeOptimizer{GLPK.Optimizer}, MathOptInterface.Utilities.UniversalFallback{MathOptInterface.Utilities.Model{Float64}}}, model::Model, v::ScalarVariable{VariableRef, Float64, Float64, Float64}, name::String)
> JuMP C:\Users\doriiido\.julia\packages\JuMP\ToPd2\src\variables.jl:1737
> [4] add_variable(model::Model, v::ScalarVariable{VariableRef, Float64, Float64, Float64}, name::String)
> JuMP C:\Users\doriiido\.julia\packages\JuMP\ToPd2\src\variables.jl:1726
> [5] macro expansion
> C:\Users\doriiido\.julia\packages\JuMP\ToPd2\src\macros.jl:1213 [inlined]
> [6] top-level scope
> Untitled-2:13

All of the parameters, constraints and variables stated within the code are taken directly from the mathematical model for this optimiziation (i.e., finding the optimal number of employees required to meet the scheduling requirements based on the defined workloads, Wjk, where j refers to the number of shifts from 1 to 3 and k the number of days from 1 to 7)

I am expecting to return the total number of employees to be allocated per shift per day

Answer 1

You have a number of typos. For example, you do not define N, and in the line starting with @constraint(model, sum(x[i, j, k]), you do not define what j is. You must have previously defined some of these values.

This question has also been asked on the JuMP community forum, so let's keep discussing over there: https://discourse.julialang.org/t/optimization-using-glpk/107210