A Transportation Problem (TRNSPORT)

trnsport.py

"""
## GAMSSOURCE: https://www.gams.com/latest/gamslib_ml/libhtml/gamslib_trnsport.html
## LICENSETYPE: Demo
## MODELTYPE: LP
## KEYWORDS: linear programming, transportation problem, scheduling


A Transportation Problem (TRNSPORT)

This problem finds a least cost shipping schedule that meets
requirements at markets and supplies at factories.


Dantzig, G B, Chapter 3.3. In Linear Programming and Extensions.
Princeton University Press, Princeton, New Jersey, 1963.

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.

The line numbers will not match those in the book because of these
comments.
"""

from __future__ import annotations

import os
import sys

from gamspy import (
    Container,
    Equation,
    Model,
    Parameter,
    Sense,
    Set,
    Sum,
    Variable,
)


def main():
    m = Container(
        system_directory=os.getenv("SYSTEM_DIRECTORY", None),
    )

    # Prepare data
    distances = [
        ["seattle", "new-york", 2.5],
        ["seattle", "chicago", 1.7],
        ["seattle", "topeka", 1.8],
        ["san-diego", "new-york", 2.5],
        ["san-diego", "chicago", 1.8],
        ["san-diego", "topeka", 1.4],
    ]

    capacities = [["seattle", 350], ["san-diego", 600]]
    demands = [["new-york", 325], ["chicago", 300], ["topeka", 275]]

    # Set
    i = Set(
        m,
        name="i",
        records=["seattle", "san-diego"],
        description="canning plants",
    )
    j = Set(
        m,
        name="j",
        records=["new-york", "chicago", "topeka"],
        description="markets",
    )

    # Data
    a = Parameter(
        m,
        name="a",
        domain=i,
        records=capacities,
        description="capacity of plant i in cases",
    )
    b = Parameter(
        m,
        name="b",
        domain=j,
        records=demands,
        description="demand at market j in cases",
    )
    d = Parameter(
        m,
        name="d",
        domain=[i, j],
        records=distances,
        description="distance in thousands of miles",
    )
    c = Parameter(
        m,
        name="c",
        domain=[i, j],
        description="transport cost in thousands of dollars per case",
    )
    c[i, j] = 90 * d[i, j] / 1000

    # Variable
    x = Variable(
        m,
        name="x",
        domain=[i, j],
        type="Positive",
        description="shipment quantities in cases",
    )

    # Equation
    supply = Equation(
        m,
        name="supply",
        domain=i,
        description="observe supply limit at plant i",
    )
    demand = Equation(
        m, name="demand", domain=j, description="satisfy demand at market j"
    )

    supply[i] = Sum(j, x[i, j]) <= a[i]
    demand[j] = Sum(i, x[i, j]) >= b[j]

    transport = Model(
        m,
        name="transport",
        equations=m.getEquations(),
        problem="LP",
        sense=Sense.MIN,
        objective=Sum((i, j), c[i, j] * x[i, j]),
    )
    transport.solve(output=sys.stdout)

    import math

    assert math.isclose(transport.objective_value, 153.675000, rel_tol=0.001)

    print(x.records)
    print(transport.objective_value)
    print(transport.status)


if __name__ == "__main__":
    main()