Social Golfer Problem (SGOLFER)#

`sgolfer.py`

```"""
## GAMSSOURCE: https://www.gams.com/latest/gamslib_ml/libhtml/gamslib_sgolfer.html
## MODELTYPE: MINLP, MIP
## KEYWORDS: mixed integer linear programming, mixed integer nonlinear programming, social golfer problem, combinatorial optimization

Social Golfer Problem (SGOLFER)

In a golf club, there are 32 social golfers, each of whom play golf once a
week,
and always in groups of 4. The problem is to build a schedule of play for 10
weeks with ''maximum socialisation''; that is, as few repeated meetings as
possible. More generally the problem is to schedule m groups of n golfers over
p weeks, with maximum socialisation.

Warwick, H, The Fully Social Golfer Problem. In Smith, B, and Warwick, H, Eds,
Proceedings of the Third International Workshop on Symmetry in Constraint
Satisfaction Problems (SymCon 2003). 2003, pp. 75-85.
"""

from __future__ import annotations

import os

from gamspy import (
Alias,
Container,
Equation,
Model,
Ord,
Sense,
Set,
Sum,
Variable,
)
from gamspy.math import Max as gams_max

def main(gr_c=8, gg_c=4, nw_c=10, mip=False):
cont = Container(
system_directory=os.getenv("SYSTEM_DIRECTORY", None),
)

gf_c = gr_c * gg_c

# Set
gf = Set(
cont,
name="gf",
records=[str(i) for i in range(1, gf_c + 1)],
description="golfers",
)
gr = Set(
cont,
name="gr",
records=[str(i) for i in range(1, gr_c + 1)],
description="groups",
)
w = Set(
cont,
name="w",
records=[str(i) for i in range(1, nw_c + 1)],
description="weeks",
)

# Alias
gf1 = Alias(cont, name="gf1", alias_with=gf)
gf2 = Alias(cont, name="gf2", alias_with=gf)

mgf = Set(
cont, name="mgf", domain=[gf1, gf2], description="possible meeting"
)
mgf[gf1, gf2] = Ord(gf2) > Ord(gf1)

# Variable
x = Variable(
cont,
name="x",
type="binary",
domain=[w, gr, gf],
description="golfer gf is in group gr on week w",
)
m = Variable(
cont,
name="m",
type="free",
domain=[w, gr, gf, gf],
description="golfers meet in week w in some group",
)
numm = Variable(
cont,
name="numm",
type="free",
domain=[gf, gf],
description="number of meetings",
)
redm = Variable(
cont,
name="redm",
type="free",
domain=[gf, gf],
description="number of redundant meetings",
)

# Equation
defx = Equation(
cont,
name="defx",
domain=[w, gf],
description="each golfer is assigned to exactly one group",
)
defgr = Equation(
cont,
name="defgr",
domain=[w, gr],
description="each group contains exactly |gg| golfers",
)
defm = Equation(
cont,
name="defm",
domain=[w, gr, gf, gf],
description="meet in group gr on week w",
)
defnumm = Equation(
cont, name="defnumm", domain=[gf, gf], description="number of meetings"
)
defredm = Equation(
cont,
name="defredm",
domain=[gf, gf],
description="number of redundant meetings",
)

if not isinstance(mip, bool):
raise Exception(
f"Argument <mip> should be a boolean. Not {type(mip)}."
)

if mip:
m.type = "binary"
redm.type = "positive"

defm2 = Equation(
cont,
name="defm2",
domain=[w, gr, gf, gf],
description="meet in group gr on week w",
)
defm3 = Equation(
cont,
name="defm3",
domain=[w, gr, gf, gf],
description="meet in group gr on week w",
)

defm[w, gr, mgf[gf1, gf2]] = m[w, gr, mgf] <= x[w, gr, gf1]

defm2[w, gr, mgf[gf1, gf2]] = m[w, gr, mgf] <= x[w, gr, gf2]

defm3[w, gr, mgf[gf1, gf2]] = (
m[w, gr, mgf] >= x[w, gr, gf1] + x[w, gr, gf2] - 1
)

defredm[mgf] = redm[mgf] >= numm[mgf] - 1

else:
defm[w, gr, mgf[gf1, gf2]] = (
m[w, gr, mgf] == x[w, gr, gf1] & x[w, gr, gf2]
)

defredm[mgf] = redm[mgf] == gams_max(0, numm[mgf] - 1)

defx[w, gf] = Sum(gr, x[w, gr, gf]) == 1

defgr[w, gr] = Sum(gf, x[w, gr, gf]) == gg_c

defnumm[mgf] = numm[mgf] == Sum((w, gr), m[w, gr, mgf])

x.l[w, gr, gf].where[
((Ord(gr) - 1) * gg_c + 1 <= Ord(gf)) & (Ord(gf) <= (Ord(gr)) * gg_c)
] = 1

defobj = Sum(mgf, redm[mgf])  # minimize redundant meetings

social_golfer_mip = Model(
cont,
name="social_golfer_mip",
equations=cont.getEquations(),
problem="mip",
sense=Sense.MIN,
objective=defobj,
)

social_golfer_minlp = Model(
cont,
name="social_golfer_minlp",
equations=cont.getEquations(),
problem="minlp",
sense=Sense.MIN,
objective=defobj,
)

if mip:
social_golfer_mip.solve()
obj_val = social_golfer_mip.objective_value
else:
social_golfer_minlp.solve()
obj_val = social_golfer_minlp.objective_value

print("Objective Function Variable: ", obj_val)

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