mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-11-23 21:11:08 +00:00
340 lines
12 KiB
Python
340 lines
12 KiB
Python
"""
|
|
Python implementation of the simplex algorithm for solving linear programs in
|
|
tabular form with
|
|
- `>=`, `<=`, and `=` constraints and
|
|
- each variable `x1, x2, ...>= 0`.
|
|
|
|
See https://gist.github.com/imengus/f9619a568f7da5bc74eaf20169a24d98 for how to
|
|
convert linear programs to simplex tableaus, and the steps taken in the simplex
|
|
algorithm.
|
|
|
|
Resources:
|
|
https://en.wikipedia.org/wiki/Simplex_algorithm
|
|
https://tinyurl.com/simplex4beginners
|
|
"""
|
|
|
|
from typing import Any
|
|
|
|
import numpy as np
|
|
|
|
|
|
class Tableau:
|
|
"""Operate on simplex tableaus
|
|
|
|
>>> Tableau(np.array([[-1,-1,0,0,1],[1,3,1,0,4],[3,1,0,1,4]]), 2, 2)
|
|
Traceback (most recent call last):
|
|
...
|
|
TypeError: Tableau must have type float64
|
|
|
|
>>> Tableau(np.array([[-1,-1,0,0,-1],[1,3,1,0,4],[3,1,0,1,4.]]), 2, 2)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: RHS must be > 0
|
|
|
|
>>> Tableau(np.array([[-1,-1,0,0,1],[1,3,1,0,4],[3,1,0,1,4.]]), -2, 2)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: number of (artificial) variables must be a natural number
|
|
"""
|
|
|
|
# Max iteration number to prevent cycling
|
|
maxiter = 100
|
|
|
|
def __init__(
|
|
self, tableau: np.ndarray, n_vars: int, n_artificial_vars: int
|
|
) -> None:
|
|
if tableau.dtype != "float64":
|
|
raise TypeError("Tableau must have type float64")
|
|
|
|
# Check if RHS is negative
|
|
if not (tableau[:, -1] >= 0).all():
|
|
raise ValueError("RHS must be > 0")
|
|
|
|
if n_vars < 2 or n_artificial_vars < 0:
|
|
raise ValueError(
|
|
"number of (artificial) variables must be a natural number"
|
|
)
|
|
|
|
self.tableau = tableau
|
|
self.n_rows, n_cols = tableau.shape
|
|
|
|
# Number of decision variables x1, x2, x3...
|
|
self.n_vars, self.n_artificial_vars = n_vars, n_artificial_vars
|
|
|
|
# 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
|
|
self.n_stages = (self.n_artificial_vars > 0) + 1
|
|
|
|
# Number of slack variables added to make inequalities into equalities
|
|
self.n_slack = n_cols - self.n_vars - self.n_artificial_vars - 1
|
|
|
|
# Objectives for each stage
|
|
self.objectives = ["max"]
|
|
|
|
# In two stage simplex, first minimise then maximise
|
|
if self.n_artificial_vars:
|
|
self.objectives.append("min")
|
|
|
|
self.col_titles = self.generate_col_titles()
|
|
|
|
# Index of current pivot row and column
|
|
self.row_idx = None
|
|
self.col_idx = None
|
|
|
|
# Does objective row only contain (non)-negative values?
|
|
self.stop_iter = False
|
|
|
|
def generate_col_titles(self) -> list[str]:
|
|
"""Generate column titles for tableau of specific dimensions
|
|
|
|
>>> Tableau(np.array([[-1,-1,0,0,1],[1,3,1,0,4],[3,1,0,1,4.]]),
|
|
... 2, 0).generate_col_titles()
|
|
['x1', 'x2', 's1', 's2', 'RHS']
|
|
|
|
>>> Tableau(np.array([[-1,-1,0,0,1],[1,3,1,0,4],[3,1,0,1,4.]]),
|
|
... 2, 2).generate_col_titles()
|
|
['x1', 'x2', 'RHS']
|
|
"""
|
|
args = (self.n_vars, self.n_slack)
|
|
|
|
# decision | slack
|
|
string_starts = ["x", "s"]
|
|
titles = []
|
|
for i in range(2):
|
|
for j in range(args[i]):
|
|
titles.append(string_starts[i] + str(j + 1))
|
|
titles.append("RHS")
|
|
return titles
|
|
|
|
def find_pivot(self) -> tuple[Any, Any]:
|
|
"""Finds the pivot row and column.
|
|
>>> tuple(int(x) for x in Tableau(np.array([[-2,1,0,0,0], [3,1,1,0,6],
|
|
... [1,2,0,1,7.]]), 2, 0).find_pivot())
|
|
(1, 0)
|
|
"""
|
|
objective = self.objectives[-1]
|
|
|
|
# Find entries of highest magnitude in objective rows
|
|
sign = (objective == "min") - (objective == "max")
|
|
col_idx = np.argmax(sign * self.tableau[0, :-1])
|
|
|
|
# Choice is only valid if below 0 for maximise, and above for minimise
|
|
if sign * self.tableau[0, col_idx] <= 0:
|
|
self.stop_iter = True
|
|
return 0, 0
|
|
|
|
# Pivot row is chosen as having the lowest quotient when elements of
|
|
# the pivot column divide the right-hand side
|
|
|
|
# Slice excluding the objective rows
|
|
s = slice(self.n_stages, self.n_rows)
|
|
|
|
# RHS
|
|
dividend = self.tableau[s, -1]
|
|
|
|
# Elements of pivot column within slice
|
|
divisor = self.tableau[s, col_idx]
|
|
|
|
# Array filled with nans
|
|
nans = np.full(self.n_rows - self.n_stages, np.nan)
|
|
|
|
# If element in pivot column is greater than zero, return
|
|
# quotient or nan otherwise
|
|
quotients = np.divide(dividend, divisor, out=nans, where=divisor > 0)
|
|
|
|
# Arg of minimum quotient excluding the nan values. n_stages is added
|
|
# to compensate for earlier exclusion of objective columns
|
|
row_idx = np.nanargmin(quotients) + self.n_stages
|
|
return row_idx, col_idx
|
|
|
|
def pivot(self, row_idx: int, col_idx: int) -> np.ndarray:
|
|
"""Pivots on value on the intersection of pivot row and column.
|
|
|
|
>>> Tableau(np.array([[-2,-3,0,0,0],[1,3,1,0,4],[3,1,0,1,4.]]),
|
|
... 2, 2).pivot(1, 0).tolist()
|
|
... # doctest: +NORMALIZE_WHITESPACE
|
|
[[0.0, 3.0, 2.0, 0.0, 8.0],
|
|
[1.0, 3.0, 1.0, 0.0, 4.0],
|
|
[0.0, -8.0, -3.0, 1.0, -8.0]]
|
|
"""
|
|
# Avoid changes to original tableau
|
|
piv_row = self.tableau[row_idx].copy()
|
|
|
|
piv_val = piv_row[col_idx]
|
|
|
|
# Entry becomes 1
|
|
piv_row *= 1 / piv_val
|
|
|
|
# Variable in pivot column becomes basic, ie the only non-zero entry
|
|
for idx, coeff in enumerate(self.tableau[:, col_idx]):
|
|
self.tableau[idx] += -coeff * piv_row
|
|
self.tableau[row_idx] = piv_row
|
|
return self.tableau
|
|
|
|
def change_stage(self) -> np.ndarray:
|
|
"""Exits first phase of the two-stage method by deleting artificial
|
|
rows and columns, or completes the algorithm if exiting the standard
|
|
case.
|
|
|
|
>>> Tableau(np.array([
|
|
... [3, 3, -1, -1, 0, 0, 4],
|
|
... [2, 1, 0, 0, 0, 0, 0.],
|
|
... [1, 2, -1, 0, 1, 0, 2],
|
|
... [2, 1, 0, -1, 0, 1, 2]
|
|
... ]), 2, 2).change_stage().tolist()
|
|
... # doctest: +NORMALIZE_WHITESPACE
|
|
[[2.0, 1.0, 0.0, 0.0, 0.0],
|
|
[1.0, 2.0, -1.0, 0.0, 2.0],
|
|
[2.0, 1.0, 0.0, -1.0, 2.0]]
|
|
"""
|
|
# Objective of original objective row remains
|
|
self.objectives.pop()
|
|
|
|
if not self.objectives:
|
|
return self.tableau
|
|
|
|
# Slice containing ids for artificial columns
|
|
s = slice(-self.n_artificial_vars - 1, -1)
|
|
|
|
# Delete the artificial variable columns
|
|
self.tableau = np.delete(self.tableau, s, axis=1)
|
|
|
|
# Delete the objective row of the first stage
|
|
self.tableau = np.delete(self.tableau, 0, axis=0)
|
|
|
|
self.n_stages = 1
|
|
self.n_rows -= 1
|
|
self.n_artificial_vars = 0
|
|
self.stop_iter = False
|
|
return self.tableau
|
|
|
|
def run_simplex(self) -> dict[Any, Any]:
|
|
"""Operate on tableau until objective function cannot be
|
|
improved further.
|
|
|
|
# Standard linear program:
|
|
Max: x1 + x2
|
|
ST: x1 + 3x2 <= 4
|
|
3x1 + x2 <= 4
|
|
>>> {key: float(value) for key, value in Tableau(np.array([[-1,-1,0,0,0],
|
|
... [1,3,1,0,4],[3,1,0,1,4.]]), 2, 0).run_simplex().items()}
|
|
{'P': 2.0, 'x1': 1.0, 'x2': 1.0}
|
|
|
|
# Standard linear program with 3 variables:
|
|
Max: 3x1 + x2 + 3x3
|
|
ST: 2x1 + x2 + x3 ≤ 2
|
|
x1 + 2x2 + 3x3 ≤ 5
|
|
2x1 + 2x2 + x3 ≤ 6
|
|
>>> {key: float(value) for key, value in Tableau(np.array([
|
|
... [-3,-1,-3,0,0,0,0],
|
|
... [2,1,1,1,0,0,2],
|
|
... [1,2,3,0,1,0,5],
|
|
... [2,2,1,0,0,1,6.]
|
|
... ]),3,0).run_simplex().items()} # doctest: +ELLIPSIS
|
|
{'P': 5.4, 'x1': 0.199..., 'x3': 1.6}
|
|
|
|
|
|
# Optimal tableau input:
|
|
>>> {key: float(value) for key, value in Tableau(np.array([
|
|
... [0, 0, 0.25, 0.25, 2],
|
|
... [0, 1, 0.375, -0.125, 1],
|
|
... [1, 0, -0.125, 0.375, 1]
|
|
... ]), 2, 0).run_simplex().items()}
|
|
{'P': 2.0, 'x1': 1.0, 'x2': 1.0}
|
|
|
|
# Non-standard: >= constraints
|
|
Max: 2x1 + 3x2 + x3
|
|
ST: x1 + x2 + x3 <= 40
|
|
2x1 + x2 - x3 >= 10
|
|
- x2 + x3 >= 10
|
|
>>> {key: float(value) for key, value in Tableau(np.array([
|
|
... [2, 0, 0, 0, -1, -1, 0, 0, 20],
|
|
... [-2, -3, -1, 0, 0, 0, 0, 0, 0],
|
|
... [1, 1, 1, 1, 0, 0, 0, 0, 40],
|
|
... [2, 1, -1, 0, -1, 0, 1, 0, 10],
|
|
... [0, -1, 1, 0, 0, -1, 0, 1, 10.]
|
|
... ]), 3, 2).run_simplex().items()}
|
|
{'P': 70.0, 'x1': 10.0, 'x2': 10.0, 'x3': 20.0}
|
|
|
|
# Non standard: minimisation and equalities
|
|
Min: x1 + x2
|
|
ST: 2x1 + x2 = 12
|
|
6x1 + 5x2 = 40
|
|
>>> {key: float(value) for key, value in Tableau(np.array([
|
|
... [8, 6, 0, 0, 52],
|
|
... [1, 1, 0, 0, 0],
|
|
... [2, 1, 1, 0, 12],
|
|
... [6, 5, 0, 1, 40.],
|
|
... ]), 2, 2).run_simplex().items()}
|
|
{'P': 7.0, 'x1': 5.0, 'x2': 2.0}
|
|
|
|
|
|
# Pivot on slack variables
|
|
Max: 8x1 + 6x2
|
|
ST: x1 + 3x2 <= 33
|
|
4x1 + 2x2 <= 48
|
|
2x1 + 4x2 <= 48
|
|
x1 + x2 >= 10
|
|
x1 >= 2
|
|
>>> {key: float(value) for key, value in Tableau(np.array([
|
|
... [2, 1, 0, 0, 0, -1, -1, 0, 0, 12.0],
|
|
... [-8, -6, 0, 0, 0, 0, 0, 0, 0, 0.0],
|
|
... [1, 3, 1, 0, 0, 0, 0, 0, 0, 33.0],
|
|
... [4, 2, 0, 1, 0, 0, 0, 0, 0, 60.0],
|
|
... [2, 4, 0, 0, 1, 0, 0, 0, 0, 48.0],
|
|
... [1, 1, 0, 0, 0, -1, 0, 1, 0, 10.0],
|
|
... [1, 0, 0, 0, 0, 0, -1, 0, 1, 2.0]
|
|
... ]), 2, 2).run_simplex().items()} # doctest: +ELLIPSIS
|
|
{'P': 132.0, 'x1': 12.000... 'x2': 5.999...}
|
|
"""
|
|
# Stop simplex algorithm from cycling.
|
|
for _ in range(Tableau.maxiter):
|
|
# Completion of each stage removes an objective. If both stages
|
|
# are complete, then no objectives are left
|
|
if not self.objectives:
|
|
# Find the values of each variable at optimal solution
|
|
return self.interpret_tableau()
|
|
|
|
row_idx, col_idx = self.find_pivot()
|
|
|
|
# If there are no more negative values in objective row
|
|
if self.stop_iter:
|
|
# Delete artificial variable columns and rows. Update attributes
|
|
self.tableau = self.change_stage()
|
|
else:
|
|
self.tableau = self.pivot(row_idx, col_idx)
|
|
return {}
|
|
|
|
def interpret_tableau(self) -> dict[str, float]:
|
|
"""Given the final tableau, add the corresponding values of the basic
|
|
decision variables to the `output_dict`
|
|
>>> {key: float(value) for key, value in Tableau(np.array([
|
|
... [0,0,0.875,0.375,5],
|
|
... [0,1,0.375,-0.125,1],
|
|
... [1,0,-0.125,0.375,1]
|
|
... ]),2, 0).interpret_tableau().items()}
|
|
{'P': 5.0, 'x1': 1.0, 'x2': 1.0}
|
|
"""
|
|
# P = RHS of final tableau
|
|
output_dict = {"P": abs(self.tableau[0, -1])}
|
|
|
|
for i in range(self.n_vars):
|
|
# Gives indices of nonzero entries in the ith column
|
|
nonzero = np.nonzero(self.tableau[:, i])
|
|
n_nonzero = len(nonzero[0])
|
|
|
|
# First entry in the nonzero indices
|
|
nonzero_rowidx = nonzero[0][0]
|
|
nonzero_val = self.tableau[nonzero_rowidx, i]
|
|
|
|
# If there is only one nonzero value in column, which is one
|
|
if n_nonzero == 1 and nonzero_val == 1:
|
|
rhs_val = self.tableau[nonzero_rowidx, -1]
|
|
output_dict[self.col_titles[i]] = rhs_val
|
|
return output_dict
|
|
|
|
|
|
if __name__ == "__main__":
|
|
import doctest
|
|
|
|
doctest.testmod()
|