Simplex algorithm

linear_programming/simplex.py

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+"""
+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
+
+    >>> t = Tableau(np.array([[-1,-1,0,0,-1],[1,3,1,0,4],[3,1,0,1,4.]]), 2)
+    Traceback (most recent call last):
+    ...
+    ValueError: RHS must be > 0
+    """
+
+    def __init__(self, tableau: np.ndarray, n_vars: int) -> None:
+        # Check if RHS is negative
+        if np.any(tableau[:, -1], where=tableau[:, -1] < 0):
+            raise ValueError("RHS must be > 0")
+
+        self.tableau = tableau
+        self.n_rows, _ = tableau.shape
+
+        # Number of decision variables x1, x2, x3...
+        self.n_vars = n_vars
+
+        # Number of artificial variables to be minimised
+        self.n_art_vars = len(np.where(tableau[self.n_vars : -1] == -1)[0])
+
+        # 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
+        self.n_stages = (self.n_art_vars > 0) + 1
+
+        # Number of slack variables added to make inequalities into equalities
+        self.n_slack = self.n_rows - self.n_stages
+
+        # Objectives for each stage
+        self.objectives = ["max"]
+
+        # In two stage simplex, first minimise then maximise
+        if self.n_art_vars:
+            self.objectives.append("min")
+
+        self.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
+
+    @staticmethod
+    def generate_col_titles(*args: int) -> list[str]:
+        """Generate column titles for tableau of specific dimensions
+
+        >>> Tableau.generate_col_titles(2, 3, 1)
+        ['x1', 'x2', 's1', 's2', 's3', 'a1', 'RHS']
+
+        >>> Tableau.generate_col_titles()
+        Traceback (most recent call last):
+        ...
+        ValueError: Must provide n_vars, n_slack, and n_art_vars
+        >>> Tableau.generate_col_titles(-2, 3, 1)
+        Traceback (most recent call last):
+        ...
+        ValueError: All arguments must be non-negative integers
+        """
+        if len(args) != 3:
+            raise ValueError("Must provide n_vars, n_slack, and n_art_vars")
+
+        if not all(x >= 0 and isinstance(x, int) for x in args):
+            raise ValueError("All arguments must be non-negative integers")
+
+        # decision | slack | artificial
+        string_starts = ["x", "s", "a"]
+        titles = []
+        for i in range(3):
+            for j in range(args[i]):
+                titles.append(string_starts[i] + str(j + 1))
+        titles.append("RHS")
+        return titles
+
+    def find_pivot(self, tableau: np.ndarray) -> tuple[Any, Any]:
+        """Finds the pivot row and column.
+        >>> t = Tableau(np.array([[-2,1,0,0,0], [3,1,1,0,6], [1,2,0,1,7.]]), 2)
+        >>> t.find_pivot(t.tableau)
+        (1, 0)
+        """
+        objective = self.objectives[-1]
+
+        # Find entries of highest magnitude in objective rows
+        sign = (objective == "min") - (objective == "max")
+        col_idx = np.argmax(sign * tableau[0, : self.n_vars])
+
+        # 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 = tableau[s, -1]
+
+        # Elements of pivot column within slice
+        divisor = 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 zeron_stages, 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, tableau: np.ndarray, row_idx: int, col_idx: int) -> np.ndarray:
+        """Pivots on value on the intersection of pivot row and column.
+
+        >>> t = Tableau(np.array([[-2,-3,0,0,0],[1,3,1,0,4],[3,1,0,1,4.]]), 2)
+        >>> t.pivot(t.tableau, 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 = 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(tableau[:, col_idx]):
+            tableau[idx] += -coeff * piv_row
+        tableau[row_idx] = piv_row
+        return tableau
+
+    def change_stage(self, tableau: np.ndarray) -> 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.
+
+        >>> t = 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)
+        >>> t.change_stage(t.tableau).tolist()
+        ... # doctest: +NORMALIZE_WHITESPACE
+        [[2.0, 1.0, 0.0, 0.0, 0.0, 0.0],
+        [1.0, 2.0, -1.0, 0.0, 1.0, 2.0],
+        [2.0, 1.0, 0.0, -1.0, 0.0, 2.0]]
+        """
+        # Objective of original objective row remains
+        self.objectives.pop()
+
+        if not self.objectives:
+            return tableau
+
+        # Slice containing ids for artificial columns
+        s = slice(-self.n_art_vars - 1, -1)
+
+        # Delete the artificial variable columns
+        tableau = np.delete(tableau, s, axis=1)
+
+        # Delete the objective row of the first stage
+        tableau = np.delete(tableau, 0, axis=0)
+
+        self.n_stages = 1
+        self.n_rows -= 1
+        self.n_art_vars = 0
+        self.stop_iter = False
+        return 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
+        >>> Tableau(np.array([[-1,-1,0,0,0],[1,3,1,0,4],[3,1,0,1,4.]]),
+        ... 2).run_simplex()
+        {'P': 2.0, 'x1': 1.0, 'x2': 1.0}
+
+        # Optimal tableau input:
+        >>> 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).run_simplex()
+        {'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
+        >>> 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).run_simplex()
+        {'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
+        >>> Tableau(np.array([
+        ... [8, 6, 0, -1, 0, -1, 0, 0, 52],
+        ... [1, 1, 0, 0, 0, 0, 0, 0, 0],
+        ... [2, 1, 1, 0, 0, 0, 0, 0, 12],
+        ... [2, 1, 0, -1, 0, 0, 1, 0, 12],
+        ... [6, 5, 0, 0, 1, 0, 0, 0, 40],
+        ... [6, 5, 0, 0, 0, -1, 0, 1, 40.]
+        ... ]), 2).run_simplex()
+        {'P': 7.0, 'x1': 5.0, 'x2': 2.0}
+        """
+        # Stop simplex algorithm from cycling.
+        for _ in range(100):
+            # Completion of each stage removes an objective. If both stages
+            # are complete, then no objectives are left
+            if not self.objectives:
+                self.col_titles = self.generate_col_titles(
+                    self.n_vars, self.n_slack, self.n_art_vars
+                )
+
+                # Find the values of each variable at optimal solution
+                return self.interpret_tableau(self.tableau, self.col_titles)
+
+            row_idx, col_idx = self.find_pivot(self.tableau)
+
+            # 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(self.tableau)
+            else:
+                self.tableau = self.pivot(self.tableau, row_idx, col_idx)
+        return {}
+
+    def interpret_tableau(
+        self, tableau: np.ndarray, col_titles: list[str]
+    ) -> dict[str, float]:
+        """Given the final tableau, add the corresponding values of the basic
+        decision variables to the `output_dict`
+        >>> tableau = np.array([
+        ... [0,0,0.875,0.375,5],
+        ... [0,1,0.375,-0.125,1],
+        ... [1,0,-0.125,0.375,1]
+        ... ])
+        >>> t = Tableau(tableau, 2)
+        >>> t.interpret_tableau(tableau, ["x1", "x2", "s1", "s2", "RHS"])
+        {'P': 5.0, 'x1': 1.0, 'x2': 1.0}
+        """
+        # P = RHS of final tableau
+        output_dict = {"P": abs(tableau[0, -1])}
+
+        for i in range(self.n_vars):
+            # Gives ids of nonzero entries in the ith column
+            nonzero = np.nonzero(tableau[:, i])
+            n_nonzero = len(nonzero[0])
+
+            # First entry in the nonzero ids
+            nonzero_rowidx = nonzero[0][0]
+            nonzero_val = tableau[nonzero_rowidx, i]
+
+            # If there is only one nonzero value in column, which is one
+            if n_nonzero == nonzero_val == 1:
+                rhs_val = tableau[nonzero_rowidx, -1]
+                output_dict[col_titles[i]] = rhs_val
+
+        # Check for basic variables
+        for title in col_titles:
+            # Don't add RHS or slack variables to output dict
+            if title[0] not in "R-s-a":
+                output_dict.setdefault(title, 0)
+        return output_dict
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()