Spatial Equilibrium (SPATEQU)#

```"""
## GAMSSOURCE: https://www.gams.com/latest/gamslib_ml/libhtml/gamslib_spatequ.html
## MODELTYPE: LP, MCP, NLP
## DATAFILES: spatequ.gdx
## KEYWORDS: linear programming, nonlinear programming, mixed complementarity problem, spatial equilibrium model

Spatial Equilibrium (SPATEQU)

This program is written for the spatial equilibrium model with linear supply
and demand having two products and three regions.

The model contains multiple approaches (LP, NLP, and MCP) for solving this
problem.

Phan, S H, Policy option to promote the wood-processing industry in
northern Vietnam, forth coming. PhD thesis,
The University of Queensland, Australia, 2011.

Phan, S H, and Harrison, S, A Review of the Formulation and
Application of the Spatial Equilibrium Models to Analyze
Policy. Journal of Forestry Research 22, 4 (2011).

The numerical example has been taken from:
Takayama, T, and Judge, G G, Spatial Equilibrium and Quadratic
Programming. Journal of Farm Economics 46, 1 (1964), 67-93

Contributed by: Phan Sy Hieu, November 2010
"""

from __future__ import annotations

import os
from pathlib import Path

import gamspy.math as gams_math
from gamspy import Container, Model, Problem, Sense, Sum

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

# Sets
c, r, rr, cc = m.getSymbols(["c", "r", "rr", "cc"])

# Parameters
(
AlphaS,
BetaS,
BetasSq,
TCost,
) = m.getSymbols(
[
"AlphaS",
"BetaS",
"BetasSq",
"TCost",
]
)

# Variables
DINT, SINT, TC, Qd, Qs, X, P, OBJ = m.getSymbols(
["DINT", "SINT", "TC", "Qd", "Qs", "X", "P", "OBJ"]
)

# Equations
(
DEM,
DEMLOG,
DEMINT,
SUP,
SUPLOG,
SUPINT,
SDBAL,
PDIF,
TRANSCOST,
SX,
DX,
OBJECT,
IN_OUT,
) = m.getSymbols(
[
"DEM",
"DEMLOG",
"DEMINT",
"SUP",
"SUPLOG",
"SUPINT",
"SDBAL",
"PDIF",
"TRANSCOST",
"SX",
"DX",
"OBJECT",
"IN_OUT",
]
)

DEM[r, c] = AlphaD[r, c] + Sum(cc, (BetaD[r, c, cc] * P[r, c])) == Qd[r, c]

DEMLOG[r, c] = (
== Qd[r, c]
)

DEMINT[r, c] = (
DINT[r, c]
== AlphaD[r, c] * P[r, c]
+ Sum(cc, BetadSq[r, c, cc] * P[r, cc]) * P[r, c]
)

SUP[r, c] = AlphaS[r, c] + Sum(cc, (BetaS[r, c, cc] * P[r, c])) == Qs[r, c]
SUPLOG[r, c] = (
AlphaS[r, c] + Sum(cc, (BetaS[r, c, cc] * gams_math.log(P[r, c])))
== Qs[r, c]
)

SUPINT[r, c] = (
SINT[r, c]
== AlphaS[r, c] * P[r, c]
+ Sum(cc, BetasSq[r, c, cc] * P[r, cc]) * P[r, c]
)

SDBAL[c] = Sum(r, Qd[r, c]) == Sum(r, Qs[r, c])

TRANSCOST[...] = Sum((r, rr, c), X[r, rr, c] * TCost[r, rr, c]) == TC

OBJECT[...] = Sum([r, c], DINT[r, c] - SINT[r, c]) - TC == OBJ

PDIF[r, rr, c] = P[r, c] - P[rr, c] <= TCost[r, rr, c]

SX[r, c] = Sum(rr, X[r, rr, c]) == Qs[r, c]

DX[r, c] = Sum(rr, X[rr, r, c]) == Qd[r, c]

IN_OUT[r, c] = Qs[r, c] + Sum(rr, X[rr, r, c] - X[r, rr, c]) == Qd[r, c]

DOM_TRAD[r, rr, c] = P[r, c] + TCost[r, rr, c] >= P[rr, c]

P2R3_Linear = Model(
m,
name="P2R3_Linear",
equations=[DEM, SUP, SDBAL, PDIF, TRANSCOST, SX, DX],
problem="LP",
sense=Sense.MIN,
objective=TC,
)
P2R3_LinearLog = Model(
m,
name="P2R3_LinearLog",
equations=[DEMLOG, SUPLOG, SDBAL, PDIF, TRANSCOST, SX, DX],
problem=Problem.NLP,
sense=Sense.MIN,
objective=TC,
)
P2R3_NonLinear = Model(
m,
name="P2R3_NonLinear",
equations=P2R3_Linear.equations + [DEMINT, SUPINT, OBJECT],
problem=Problem.NLP,
sense=Sense.MAX,
objective=OBJ,
)
P2R3_MCP = Model(
m,
name="P2R3_MCP",
equations=[DEM, SUP],
problem="MCP",
)

P2R3_Linear.solve()
P2R3_LinearLog.solve()
P2R3_NonLinear.solve()

X.fx[r, r, c] = 0

P2R3_MCP.solve()

if __name__ == "__main__":
main()
```