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Transposition in JuMP of the basic transport model used in the GAMS tutorial
 
This problem finds a least cost shipping schedule that meets
requirements at markets and supplies at factories.
 
- Original formulation: Dantzig, G B, Chapter 3.3. In Linear Programming and Extensions.
Princeton University Press, Princeton, New Jersey, 1963.
- Gams implementation: This formulation is described in detail in:
Rosenthal, R E, Chapter 2: A GAMS Tutorial. In GAMS: A User's Guide.
The Scientific Press, Redwood City, California, 1988.
- JuMP implementation: Antonello Lobianco
 
 
using JuMP, DataFrames
 
# Sets
plants  = ["seattle","san_diego"]          # canning plants
markets = ["new_york","chicago","topeka"]  # markets
 
# Parameters
a = Dict(              # capacity of plant i in cases
  "seattle"   => 350,
  "san_diego" => 600,
)
b = Dict(              # demand at market j in cases
  "new_york"  => 325,
  "chicago"   => 300,
  "topeka"    => 275,
)
 
#  distance in thousands of miles
d_table = wsv"""
plants     new_york  chicago  topeka
seattle    2.5       1.7      1.8
san_diego  2.5       1.8      1.4
"""
d = Dict( (r[:plants],m) => r[Symbol(m)] for r in eachrow(d_table), m in markets)
 
f = 90 # freight in dollars per case per thousand miles
 
c = Dict() # transport cost in thousands of dollars per case ;
[ c[p,m] = f * d[p,m] / 1000 for p in plants, m in markets]
 
# Model declaration
trmodel = Model() # transport model
 
# Variables
@variables trmodel begin
    x[p in plants, m in markets] >= 0 # shipment quantities in cases
end
 
# Constraints
@constraints trmodel begin
    supply[p in plants],   # observe supply limit at plant p
        sum(x[p,m] for m in markets)  <=  a[p]
    demand[m in markets],  # satisfy demand at market m
        sum(x[p,m] for p in plants)  >=  b[m]
end
 
# Objective
@objective trmodel Min begin
    sum(c[p,m]*x[p,m] for p in plants, m in markets)
end
 
print(trmodel)
 
status = solve(trmodel)
 
if status == :Optimal
    println("Objective value: ", getobjectivevalue(trmodel))
    println("Shipped quantities: ")
    println(getvalue(x))
    println("Shadow prices of supply:")
    [println("$p = $(getdual(supply[p]))") for p in plants]
    println("Shadow prices of demand:")
    [println("$m = $(getdual(demand[m]))") for m in markets]
 
else
    println("Model didn't solved")
    println(status)
end
 
# Expected result:
# obj= 153.675
#['seattle','new-york']   = 50
#['seattle','chicago']    = 300
#['seattle','topeka']     = 0
#['san-diego','new-york'] = 275
#['san-diego','chicago']  = 0
#['san-diego','topeka']   = 275