# The multi-commodity flow problem

JuMP implementation of the multicommodity transportation model AMPL: A Modeling Language for Mathematical Programming, 2nd ed by Robert Fourer, David Gay, and Brian W. Kernighan 4-1.

Originally contributed by Louis Luangkesorn, February 26, 2015.

```
using JuMP
import HiGHS
import Test
function example_multi(; verbose = true)
orig = ["GARY", "CLEV", "PITT"]
dest = ["FRA", "DET", "LAN", "WIN", "STL", "FRE", "LAF"]
prod = ["bands", "coils", "plate"]
numorig = length(orig)
numdest = length(dest)
numprod = length(prod)
# supply(prod, orig) amounts available at origins
supply = [
400 700 800
800 1600 1800
200 300 300
]
# demand(prod, dest) amounts required at destinations
demand = [
300 300 100 75 650 225 250
500 750 400 250 950 850 500
100 100 0 50 200 100 250
]
# limit(orig, dest) of total units from any origin to destination
limit = [625.0 for j in 1:numorig, i in 1:numdest]
# cost(dest, orig, prod) Shipment cost per unit
cost = reshape(
[
[
[30, 10, 8, 10, 11, 71, 6]
[22, 7, 10, 7, 21, 82, 13]
[19, 11, 12, 10, 25, 83, 15]
]
[
[39, 14, 11, 14, 16, 82, 8]
[27, 9, 12, 9, 26, 95, 17]
[24, 14, 17, 13, 28, 99, 20]
]
[
[41, 15, 12, 16, 17, 86, 8]
[29, 9, 13, 9, 28, 99, 18]
[26, 14, 17, 13, 31, 104, 20]
]
],
7,
3,
3,
)
# DECLARE MODEL
multi = Model(HiGHS.Optimizer)
# VARIABLES
@variable(multi, trans[1:numorig, 1:numdest, 1:numprod] >= 0)
# OBJECTIVE
@objective(
multi,
Max,
sum(
cost[j, i, p] * trans[i, j, p] for i in 1:numorig, j in 1:numdest,
p in 1:numprod
)
)
# CONSTRAINTS
# Supply constraint
@constraint(
multi,
supply_con[i in 1:numorig, p in 1:numprod],
sum(trans[i, j, p] for j in 1:numdest) == supply[p, i]
)
# Demand constraint
@constraint(
multi,
demand_con[j in 1:numdest, p in 1:numprod],
sum(trans[i, j, p] for i in 1:numorig) == demand[p, j]
)
# Total shipment constraint
@constraint(
multi,
total_con[i in 1:numorig, j in 1:numdest],
sum(trans[i, j, p] for p in 1:numprod) - limit[i, j] <= 0
)
optimize!(multi)
Test.@test termination_status(multi) == OPTIMAL
Test.@test primal_status(multi) == FEASIBLE_POINT
Test.@test objective_value(multi) == 225_700.0
if verbose
println("RESULTS:")
for i in 1:length(orig)
for j in 1:length(dest)
for p in 1:length(prod)
print(
" $(prod[p]) $(orig[i]) $(dest[j]) = $(value(trans[i, j, p]))\t",
)
end
println()
end
end
end
return
end
example_multi()
```

```
Presolving model
44 rows, 60 cols, 165 nonzeros
41 rows, 60 cols, 149 nonzeros
Presolve : Reductions: rows 41(-10); columns 60(-3); elements 149(-40)
Solving the presolved LP
Using EKK dual simplex solver - serial
Iteration Objective Infeasibilities num(sum)
0 -1.6599957764e+03 Ph1: 41(149); Du: 60(1660) 0s
60 -2.2570000000e+05 Pr: 0(0) 0s
Solving the original LP from the solution after postsolve
Model status : Optimal
Simplex iterations: 60
Objective value : 2.2570000000e+05
HiGHS run time : 0.00
RESULTS:
bands GARY FRA = 25.0 coils GARY FRA = 500.0 plate GARY FRA = 100.0
bands GARY DET = 125.0 coils GARY DET = 0.0 plate GARY DET = 50.0
bands GARY LAN = 0.0 coils GARY LAN = 0.0 plate GARY LAN = 0.0
bands GARY WIN = 0.0 coils GARY WIN = 0.0 plate GARY WIN = 50.0
bands GARY STL = 250.0 coils GARY STL = 300.0 plate GARY STL = 0.0
bands GARY FRE = 0.0 coils GARY FRE = 0.0 plate GARY FRE = 0.0
bands GARY LAF = 0.0 coils GARY LAF = 0.0 plate GARY LAF = 0.0
bands CLEV FRA = 275.0 coils CLEV FRA = 0.0 plate CLEV FRA = 0.0
bands CLEV DET = 0.0 coils CLEV DET = 300.0 plate CLEV DET = 50.0
bands CLEV LAN = 100.0 coils CLEV LAN = 0.0 plate CLEV LAN = -0.0
bands CLEV WIN = 0.0 coils CLEV WIN = 0.0 plate CLEV WIN = 0.0
bands CLEV STL = 0.0 coils CLEV STL = 625.0 plate CLEV STL = 0.0
bands CLEV FRE = 225.0 coils CLEV FRE = 400.0 plate CLEV FRE = 0.0
bands CLEV LAF = 100.0 coils CLEV LAF = 275.0 plate CLEV LAF = 250.0
bands PITT FRA = 0.0 coils PITT FRA = 0.0 plate PITT FRA = 0.0
bands PITT DET = 175.0 coils PITT DET = 450.0 plate PITT DET = -0.0
bands PITT LAN = 0.0 coils PITT LAN = 400.0 plate PITT LAN = 0.0
bands PITT WIN = 75.0 coils PITT WIN = 250.0 plate PITT WIN = 0.0
bands PITT STL = 400.0 coils PITT STL = 25.0 plate PITT STL = 200.0
bands PITT FRE = 0.0 coils PITT FRE = 450.0 plate PITT FRE = 100.0
bands PITT LAF = 150.0 coils PITT LAF = 225.0 plate PITT LAF = 0.0
```

This tutorial was generated using Literate.jl. View the source `.jl`

file on GitHub.