The knapsack problem

Formulate and solve a simple knapsack problem:

max sum(p_j x_j)
 st sum(w_j x_j) <= C
    x binary

using JuMP
import GLPK
import Test

function example_knapsack(; verbose = true)
    profit = [5, 3, 2, 7, 4]
    weight = [2, 8, 4, 2, 5]
    capacity = 10
    model = Model(GLPK.Optimizer)
    @variable(model, x[1:5], Bin)
    # Objective: maximize profit
    @objective(model, Max, profit' * x)
    # Constraint: can carry all
    @constraint(model, weight' * x <= capacity)
    # Solve problem using MIP solver
    optimize!(model)
    if verbose
        println("Objective is: ", objective_value(model))
        println("Solution is:")
        for i in 1:5
            print("x[$i] = ", value(x[i]))
            println(", p[$i]/w[$i] = ", profit[i] / weight[i])
        end
    end
    Test.@test termination_status(model) == OPTIMAL
    Test.@test primal_status(model) == FEASIBLE_POINT
    Test.@test objective_value(model) == 16.0
    return
end

example_knapsack()

Objective is: 16.0
Solution is:
x[1] = 1.0, p[1]/w[1] = 2.5
x[2] = 0.0, p[2]/w[2] = 0.375
x[3] = 0.0, p[3]/w[3] = 0.5
x[4] = 1.0, p[4]/w[4] = 3.5
x[5] = 1.0, p[5]/w[5] = 0.8