


from typing import Any

import numpy as np



class      snake_case__	:
				def __init__(	self  ,   _UpperCAmelCase  ,   _UpperCAmelCase   ):




										"""simple docstring"""




										if np.any(tableau[:, -1]  ,   where=tableau[:, -1] < 0   ):
																raise ValueError('RHS must be > 0'   )

										__magic_name__					:				int				=   tableau
										__magic_name__		,			__magic_name__					:				Optional[Any]				=   tableau.shape

										# Number of decision variables x1, x2, x3...
										__magic_name__					:				Union[str, Any]				=   n_vars

										# Number of artificial variables to be minimised
										__magic_name__					:				List[Any]				=   len(np.where(tableau[self.n_vars : -1] == -1   )[0]   )

										# 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
										__magic_name__					:				int				=   (self.n_art_vars > 0) + 1

										# Number of slack variables added to make inequalities into equalities
										__magic_name__					:				Tuple				=   self.n_rows - self.n_stages

										# Objectives for each stage
										__magic_name__					:				List[str]				=   ['max']

										# In two stage simplex, first minimise then maximise
										if self.n_art_vars:
																self.objectives.append('min'   )

										__magic_name__					:				Optional[Any]				=   ['']

										# Index of current pivot row and column
										__magic_name__					:				Tuple				=   None
										__magic_name__					:				int				=   None

										# Does objective row only contain (non)-negative values?
										__magic_name__					:				Dict				=   False
				@staticmethod
				def 			UpperCamelCase_    (	*_UpperCAmelCase   ):




										"""simple docstring"""




										if len(_UpperCAmelCase   ) != 3:
																raise ValueError('Must provide n_vars, n_slack, and n_art_vars'   )

										if not all(x >= 0 and isinstance(_UpperCAmelCase  ,   _UpperCAmelCase   ) for x in args   ):
																raise ValueError('All arguments must be non-negative integers'   )

										# decision | slack | artificial
										__magic_name__					:				Any				=   ['x', 's', 'a']
										__magic_name__					:				str				=   []
										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 			UpperCamelCase_    (	self  ,   _UpperCAmelCase   ):




										"""simple docstring"""




										__magic_name__					:				Optional[int]				=   self.objectives[-1]

										# Find entries of highest magnitude in objective rows
										__magic_name__					:				List[str]				=   (objective == 'min') - (objective == 'max')
										__magic_name__					:				Optional[Any]				=   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:
																__magic_name__					:				Union[str, Any]				=   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
										__magic_name__					:				Optional[int]				=   slice(self.n_stages  ,   self.n_rows   )

										# RHS
										__magic_name__					:				List[str]				=   tableau[s, -1]

										# Elements of pivot column within slice
										__magic_name__					:				List[str]				=   tableau[s, col_idx]

										# Array filled with nans
										__magic_name__					:				str				=   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
										__magic_name__					:				Optional[int]				=   np.divide(_UpperCAmelCase  ,   _UpperCAmelCase  ,   out=_UpperCAmelCase  ,   where=divisor > 0   )

										# Arg of minimum quotient excluding the nan values. n_stages is added
										# to compensate for earlier exclusion of objective columns
										__magic_name__					:				List[str]				=   np.nanargmin(_UpperCAmelCase   ) + self.n_stages
										return row_idx, col_idx
				def 			UpperCamelCase_    (	self  ,   _UpperCAmelCase  ,   _UpperCAmelCase  ,   _UpperCAmelCase   ):




										"""simple docstring"""




										__magic_name__					:				Optional[int]				=   tableau[row_idx].copy()

										__magic_name__					:				Optional[int]				=   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
										__magic_name__					:				List[Any]				=   piv_row
										return tableau
				def 			UpperCamelCase_    (	self  ,   _UpperCAmelCase   ):




										"""simple docstring"""




										self.objectives.pop()

										if not self.objectives:
																return tableau

										# Slice containing ids for artificial columns
										__magic_name__					:				Union[str, Any]				=   slice(-self.n_art_vars - 1  ,   -1   )

										# Delete the artificial variable columns
										__magic_name__					:				List[str]				=   np.delete(_UpperCAmelCase  ,   _UpperCAmelCase  ,   axis=1   )

										# Delete the objective row of the first stage
										__magic_name__					:				str				=   np.delete(_UpperCAmelCase  ,   0  ,   axis=0   )

										__magic_name__					:				List[Any]				=   1
										self.n_rows -= 1
										__magic_name__					:				str				=   0
										__magic_name__					:				Dict				=   False
										return tableau
				def 			UpperCamelCase_    (	self   ):




										"""simple docstring"""




										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:
																						__magic_name__					:				Dict				=   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   )

																__magic_name__		,			__magic_name__					:				List[Any]				=   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
																						__magic_name__					:				Optional[Any]				=   self.change_stage(self.tableau   )
																else:
																						__magic_name__					:				int				=   self.pivot(self.tableau  ,   _UpperCAmelCase  ,   _UpperCAmelCase   )
										return {}





				def 			UpperCamelCase_    (	self  ,   _UpperCAmelCase  ,   _UpperCAmelCase   ):




										"""simple docstring"""




										__magic_name__					:				List[str]				=   {'P': abs(tableau[0, -1]   )}

										for i in range(self.n_vars   ):
																# Gives ids of nonzero entries in the ith column
																__magic_name__					:				List[str]				=   np.nonzero(tableau[:, i]   )
																__magic_name__					:				str				=   len(nonzero[0]   )

																# First entry in the nonzero ids
																__magic_name__					:				Optional[Any]				=   nonzero[0][0]
																__magic_name__					:				List[Any]				=   tableau[nonzero_rowidx, i]

																# If there is only one nonzero value in column, which is one
																if n_nonzero == nonzero_val == 1:
																						__magic_name__					:				int				=   tableau[nonzero_rowidx, -1]
																						__magic_name__					:				Optional[Any]				=   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(_UpperCAmelCase  ,   0   )
										return output_dict


if __name__ == "__main__":
			import doctest

			doctest.testmod()






import math
def 			__SCREAMING_SNAKE_CASE							(					lowercase__	:int					)							->	list:
						if n <= 0:
												raise ValueError('Only positive integers have prime factors'					)
						__magic_name__					:				Union[str, Any]				=   []
						while n % 2 == 0:
												pf.append(2					)
												__magic_name__					:				Union[str, Any]				=   int(n / 2					)
						for i in range(3   ,    int(math.sqrt(lowercase__					)					) + 1   ,    2					):
												while n % i == 0:
																		pf.append(lowercase__					)
																		__magic_name__					:				List[str]				=   int(n / i					)
						if n > 2:
												pf.append(lowercase__					)
						return pf
def 			__SCREAMING_SNAKE_CASE							(					lowercase__	:int					)							->	int:
						if n <= 0:
												raise ValueError('Only positive numbers are accepted'					)
						__magic_name__					:				Union[str, Any]				=   1
						__magic_name__					:				Tuple				=   1
						while n % 2 == 0:
												temp += 1
												__magic_name__					:				Any				=   int(n / 2					)
						div *= temp
						for i in range(3   ,    int(math.sqrt(lowercase__					)					) + 1   ,    2					):
												__magic_name__					:				List[Any]				=   1
												while n % i == 0:
																		temp += 1
																		__magic_name__					:				Any				=   int(n / i					)
												div *= temp
						if n > 1:
												div *= 2
						return div
def 			__SCREAMING_SNAKE_CASE							(					lowercase__	:int					)							->	int:
						if n <= 0:
												raise ValueError('Only positive numbers are accepted'					)
						__magic_name__					:				int				=   1
						__magic_name__					:				List[str]				=   1
						while n % 2 == 0:
												temp += 1
												__magic_name__					:				List[Any]				=   int(n / 2					)
						if temp > 1:
												s *= (2**temp - 1) / (2 - 1)
						for i in range(3   ,    int(math.sqrt(lowercase__					)					) + 1   ,    2					):
												__magic_name__					:				int				=   1
												while n % i == 0:
																		temp += 1
																		__magic_name__					:				Optional[Any]				=   int(n / i					)
												if temp > 1:
																		s *= (i**temp - 1) / (i - 1)
						return int(lowercase__					)
def 			__SCREAMING_SNAKE_CASE							(					lowercase__	:int					)							->	int:
						__magic_name__					:				Dict				=   n
						for x in set(prime_factors(lowercase__					)					):
												s *= (x - 1) / x
						return int(lowercase__					)


if __name__ == "__main__":
			import doctest

			doctest.testmod()


