





'''simple docstring'''




from typing import Any

import numpy as np




class       UpperCamelCase__			:




					"""simple docstring"""


					def __init__(		self							:	Dict     ,		lowerCamelCase_							:	np.ndarray     ,		lowerCamelCase_							:	int		):

							'''simple docstring'''



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

							SCREAMING_SNAKE_CASE				:							Union[str, Any]						=	tableau
							SCREAMING_SNAKE_CASE,    SCREAMING_SNAKE_CASE				:							List[Any]						=	tableau.shape

							# Number of decision variables x1, x2, x3...
							SCREAMING_SNAKE_CASE				:							Optional[Any]						=	n_vars

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

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

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

							# Objectives for each stage
							SCREAMING_SNAKE_CASE				:							Optional[Any]						=	["""max"""]

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

							SCREAMING_SNAKE_CASE				:							Optional[Any]						=	[""""""]

							# Index of current pivot row and column
							SCREAMING_SNAKE_CASE				:							Dict						=	None
							SCREAMING_SNAKE_CASE				:							List[Any]						=	None

							# Does objective row only contain (non)-negative values?
							SCREAMING_SNAKE_CASE				:							Optional[int]						=	False


					@staticmethod
					def lowerCamelCase_	(		*lowerCamelCase_							:	int		):

							'''simple docstring'''



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

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

							# decision | slack | artificial
							SCREAMING_SNAKE_CASE				:							List[Any]						=	["""x""", """s""", """a"""]
							SCREAMING_SNAKE_CASE				:							Dict						=	[]
							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 lowerCamelCase_	(		self							:	Any     ,		lowerCamelCase_							:	np.ndarray		):

							'''simple docstring'''



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

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

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

							# Elements of pivot column within slice
							SCREAMING_SNAKE_CASE				:							Union[str, Any]						=	tableau[s, col_idx]

							# Array filled with nans
							SCREAMING_SNAKE_CASE				:							Optional[Any]						=	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
							SCREAMING_SNAKE_CASE				:							int						=	np.divide(lowerCamelCase_     ,		lowerCamelCase_     ,		out=lowerCamelCase_     ,		where=divisor > 0		)

							# Arg of minimum quotient excluding the nan values. n_stages is added
							# to compensate for earlier exclusion of objective columns
							SCREAMING_SNAKE_CASE				:							int						=	np.nanargmin(lowerCamelCase_		) + self.n_stages
							return row_idx, col_idx


					def lowerCamelCase_	(		self							:	Any     ,		lowerCamelCase_							:	np.ndarray     ,		lowerCamelCase_							:	int     ,		lowerCamelCase_							:	int		):

							'''simple docstring'''



							SCREAMING_SNAKE_CASE				:							Tuple						=	tableau[row_idx].copy()

							SCREAMING_SNAKE_CASE				:							str						=	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
							SCREAMING_SNAKE_CASE				:							List[str]						=	piv_row
							return tableau


					def lowerCamelCase_	(		self							:	Optional[Any]     ,		lowerCamelCase_							:	np.ndarray		):

							'''simple docstring'''



							self.objectives.pop()

							if not self.objectives:
									return tableau

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

							# Delete the artificial variable columns
							SCREAMING_SNAKE_CASE				:							Union[str, Any]						=	np.delete(lowerCamelCase_     ,		lowerCamelCase_     ,		axis=1		)

							# Delete the objective row of the first stage
							SCREAMING_SNAKE_CASE				:							Tuple						=	np.delete(lowerCamelCase_     ,		0     ,		axis=0		)

							SCREAMING_SNAKE_CASE				:							Tuple						=	1
							self.n_rows -= 1
							SCREAMING_SNAKE_CASE				:							Optional[Any]						=	0
							SCREAMING_SNAKE_CASE				:							Any						=	False
							return tableau


					def lowerCamelCase_	(		self							:	Tuple		):

							'''simple docstring'''



							for _ in range(1_00		):
									# Completion of each stage removes an objective. If both stages
									# are complete, then no objectives are left
									if not self.objectives:
											SCREAMING_SNAKE_CASE				:							List[Any]						=	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		)

									SCREAMING_SNAKE_CASE,    SCREAMING_SNAKE_CASE				:							Dict						=	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
											SCREAMING_SNAKE_CASE				:							int						=	self.change_stage(self.tableau		)
									else:
											SCREAMING_SNAKE_CASE				:							Optional[Any]						=	self.pivot(self.tableau     ,		lowerCamelCase_     ,		lowerCamelCase_		)
							return {}



					def lowerCamelCase_	(		self							:	Tuple     ,		lowerCamelCase_							:	np.ndarray     ,		lowerCamelCase_							:	list[str]		):

							'''simple docstring'''



							SCREAMING_SNAKE_CASE				:							Optional[Any]						=	{"""P""": abs(tableau[0, -1]		)}

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

									# First entry in the nonzero ids
									SCREAMING_SNAKE_CASE				:							Union[str, Any]						=	nonzero[0][0]
									SCREAMING_SNAKE_CASE				:							List[str]						=	tableau[nonzero_rowidx, i]

									# If there is only one nonzero value in column, which is one
									if n_nonzero == nonzero_val == 1:
											SCREAMING_SNAKE_CASE				:							str						=	tableau[nonzero_rowidx, -1]
											SCREAMING_SNAKE_CASE				:							str						=	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(lowerCamelCase_     ,		0		)
							return output_dict


if __name__ == "__main__":
							import doctest

							doctest.testmod()










'''simple docstring'''




from __future__ import annotations



def 						__A						(       lowerCamelCase_	,	lowerCamelCase_	,	lowerCamelCase_	,	lowerCamelCase_					):


		"""simple docstring"""



		SCREAMING_SNAKE_CASE,    SCREAMING_SNAKE_CASE				:							List[Any]						=	m - 1, 0
		while i >= 0 and j < n:
				if key == mat[i][j]:
						print(f'''Key {key} found at row- {i + 1} column- {j + 1}'''					)
						return
				if key < mat[i][j]:
						i -= 1
				else:
						j += 1
		print(f'''Key {key} not found'''					)



def 						__A						(       ):


		"""simple docstring"""



		SCREAMING_SNAKE_CASE				:							Optional[Any]						=	[[2, 5, 7], [4, 8, 13], [9, 11, 15], [12, 17, 20]]
		SCREAMING_SNAKE_CASE				:							Dict						=	int(input("""Enter the element to be searched:"""					)					)
		print(lowerCamelCase_					)
		search_in_a_sorted_matrix(lowerCamelCase_	,	len(lowerCamelCase_					)	,	len(mat[0]					)	,	lowerCamelCase_					)


if __name__ == "__main__":
							import doctest

							doctest.testmod()
							main()



