Fix simplex.py (TheAlgorithms#8843)

* changes to accommodate special case

* changed n_slack calculation method

* fix precommit typehints

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* n_art_vars inputs

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* fix: docstrings and typehints

* fix: doctest issues when running code

* additional check and doctests

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* fix ruff

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* fix whitespace

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>

Author: 2 people

Diff for: linear_programming/simplex.py

@@ -20,40 +20,60 @@
 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)
+    >>> 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
     """
 
-    def __init__(self, tableau: np.ndarray, n_vars: int) -> None:
+    # 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 np.any(tableau[:, -1], where=tableau[:, -1] < 0):
+        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, _ = tableau.shape
+        self.n_rows, n_cols = 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])
+        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_art_vars > 0) + 1
+        self.n_stages = (self.n_artificial_vars > 0) + 1
 
         # Number of slack variables added to make inequalities into equalities
-        self.n_slack = self.n_rows - self.n_stages
+        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_art_vars:
+        if self.n_artificial_vars:
             self.objectives.append("min")
 
-        self.col_titles = [""]
+        self.col_titles = self.generate_col_titles()
 
         # Index of current pivot row and column
         self.row_idx = None
@@ -62,48 +82,39 @@ def __init__(self, tableau: np.ndarray, n_vars: int) -> None:
         # Does objective row only contain (non)-negative values?
         self.stop_iter = False
 
-    @staticmethod
-    def generate_col_titles(*args: int) -> list[str]:
+    def generate_col_titles(self) -> 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")
+        >>> 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']
 
-        if not all(x >= 0 and isinstance(x, int) for x in args):
-            raise ValueError("All arguments must be non-negative integers")
+        >>> 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 | artificial
-        string_starts = ["x", "s", "a"]
+        # decision | slack
+        string_starts = ["x", "s"]
         titles = []
-        for i in range(3):
+        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, tableau: np.ndarray) -> tuple[Any, Any]:
+    def find_pivot(self) -> 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)
+        >>> 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 * tableau[0, : self.n_vars])
+        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:
@@ -117,15 +128,15 @@ def find_pivot(self, tableau: np.ndarray) -> tuple[Any, Any]:
         s = slice(self.n_stages, self.n_rows)
 
         # RHS
-        dividend = tableau[s, -1]
+        dividend = self.tableau[s, -1]
 
         # Elements of pivot column within slice
-        divisor = tableau[s, col_idx]
+        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 zeron_stages, return
+        # If element in pivot column is greater than zero, return
         # quotient or nan otherwise
         quotients = np.divide(dividend, divisor, out=nans, where=divisor > 0)
 
@@ -134,67 +145,66 @@ def find_pivot(self, tableau: np.ndarray) -> tuple[Any, Any]:
         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:
+    def pivot(self, 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()
+        >>> 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 = tableau[row_idx].copy()
+        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(tableau[:, col_idx]):
-            tableau[idx] += -coeff * piv_row
-        tableau[row_idx] = piv_row
-        return tableau
+        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, tableau: np.ndarray) -> np.ndarray:
+    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.
 
-        >>> t = Tableau(np.array([
+        >>> 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()
+        ... ]), 2, 2).change_stage().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]]
+        [[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 tableau
+            return self.tableau
 
         # Slice containing ids for artificial columns
-        s = slice(-self.n_art_vars - 1, -1)
+        s = slice(-self.n_artificial_vars - 1, -1)
 
         # Delete the artificial variable columns
-        tableau = np.delete(tableau, s, axis=1)
+        self.tableau = np.delete(self.tableau, s, axis=1)
 
         # Delete the objective row of the first stage
-        tableau = np.delete(tableau, 0, axis=0)
+        self.tableau = np.delete(self.tableau, 0, axis=0)
 
         self.n_stages = 1
         self.n_rows -= 1
-        self.n_art_vars = 0
+        self.n_artificial_vars = 0
         self.stop_iter = False
-        return tableau
+        return self.tableau
 
     def run_simplex(self) -> dict[Any, Any]:
         """Operate on tableau until objective function cannot be
@@ -205,15 +215,29 @@ def run_simplex(self) -> dict[Any, Any]:
         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()
+        ... 2, 0).run_simplex()
         {'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
+        >>> 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() # doctest: +ELLIPSIS
+        {'P': 5.4, 'x1': 0.199..., 'x3': 1.6}
+
+
         # 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()
+        ... ]), 2, 0).run_simplex()
         {'P': 2.0, 'x1': 1.0, 'x2': 1.0}
 
         # Non-standard: >= constraints
@@ -227,81 +251,84 @@ def run_simplex(self) -> dict[Any, Any]:
         ... [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()
+        ... ]), 3, 2).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()
+        ... [8, 6, 0, 0, 52],
+        ... [1, 1, 0, 0, 0],
+        ... [2, 1, 1, 0, 12],
+        ... [6, 5, 0, 1, 40.],
+        ... ]), 2, 2).run_simplex()
         {'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
+        >>> 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() # doctest: +ELLIPSIS
+        {'P': 132.0, 'x1': 12.000... 'x2': 5.999...}
         """
         # Stop simplex algorithm from cycling.
-        for _ in range(100):
+        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:
-                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)
+                return self.interpret_tableau()
 
-            row_idx, col_idx = self.find_pivot(self.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(self.tableau)
+                self.tableau = self.change_stage()
             else:
-                self.tableau = self.pivot(self.tableau, row_idx, col_idx)
+                self.tableau = self.pivot(row_idx, col_idx)
         return {}
 
-    def interpret_tableau(
-        self, tableau: np.ndarray, col_titles: list[str]
-    ) -> dict[str, float]:
+    def interpret_tableau(self) -> dict[str, float]:
         """Given the final tableau, add the corresponding values of the basic
         decision variables to the `output_dict`
-        >>> tableau = np.array([
+        >>> 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"])
+        ... ]),2, 0).interpret_tableau()
         {'P': 5.0, 'x1': 1.0, 'x2': 1.0}
         """
         # P = RHS of final tableau
-        output_dict = {"P": abs(tableau[0, -1])}
+        output_dict = {"P": abs(self.tableau[0, -1])}
 
         for i in range(self.n_vars):
-            # Gives ids of nonzero entries in the ith column
-            nonzero = np.nonzero(tableau[:, i])
+            # 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 ids
+            # First entry in the nonzero indices
             nonzero_rowidx = nonzero[0][0]
-            nonzero_val = tableau[nonzero_rowidx, i]
+            nonzero_val = self.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)
+            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