"""simple docstring"""







from typing import Any

import numpy as np






class 							__snake_case:


    '''simple docstring'''





    def __init__(  self					,       __lowerCamelCase					,       __lowerCamelCase	):




         '''simple docstring'''



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

         __A					:					List[str]								=		tableau
         __A						,		__A					:					Dict								=		tableau.shape

         # Number of decision variables x1, x2, x3...
         __A					:					int								=		n_vars

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

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

         # Number of slack variables added to make inequalities into equalities
         __A					:					Optional[Any]								=		self.n_rows - self.n_stages

         # Objectives for each stage
         __A					:					Optional[Any]								=		['max']

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

         __A					:					str								=		['']

         # Index of current pivot row and column
         __A					:					Any								=		None
         __A					:					Union[str, Any]								=		None

         # Does objective row only contain (non)-negative values?
         __A					:					Tuple								=		False
    @staticmethod
    def 						_a							(  *__lowerCamelCase	):




         '''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
         __A					:					List[str]								=		['x', 's', 'a']
         __A					:					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 						_a							(  self					,       __lowerCamelCase	):




         '''simple docstring'''



         __A					:					Union[str, Any]								=		self.objectives[-1]

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

         # RHS
         __A					:					Tuple								=		tableau[s, -1]

         # Elements of pivot column within slice
         __A					:					int								=		tableau[s, col_idx]

         # Array filled with nans
         __A					:					List[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
         __A					:					Tuple								=		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
         __A					:					List[Any]								=		np.nanargmin(__lowerCamelCase	) + self.n_stages
         return row_idx, col_idx
    def 						_a							(  self					,       __lowerCamelCase					,       __lowerCamelCase					,       __lowerCamelCase	):




         '''simple docstring'''



         __A					:					int								=		tableau[row_idx].copy()

         __A					:					Any								=		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
         __A					:					List[str]								=		piv_row
         return tableau
    def 						_a							(  self					,       __lowerCamelCase	):




         '''simple docstring'''



         self.objectives.pop()

         if not self.objectives:
              return tableau

         # Slice containing ids for artificial columns
         __A					:					Dict								=		slice(-self.n_art_vars - 1					,       -1	)

         # Delete the artificial variable columns
         __A					:					Optional[Any]								=		np.delete(__lowerCamelCase					,       __lowerCamelCase					,       axis=1	)

         # Delete the objective row of the first stage
         __A					:					Dict								=		np.delete(__lowerCamelCase					,       0					,       axis=0	)

         __A					:					Dict								=		1
         self.n_rows -= 1
         __A					:					Any								=		0
         __A					:					List[Any]								=		False
         return tableau
    def 						_a							(  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:
                   __A					:					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	)

              __A						,		__A					:					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
                   __A					:					Union[str, Any]								=		self.change_stage(self.tableau	)
              else:
                   __A					:					Dict								=		self.pivot(self.tableau					,       __lowerCamelCase					,       __lowerCamelCase	)
         return {}

    def 						_a							(  self					,       __lowerCamelCase					,       __lowerCamelCase	):




         '''simple docstring'''



         __A					:					Optional[int]								=		{'P': abs(tableau[0, -1]	)}

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

              # First entry in the nonzero ids
              __A					:					int								=		nonzero[0][0]
              __A					:					Optional[int]								=		tableau[nonzero_rowidx, i]

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


if __name__ == "__main__":
       import doctest

       doctest.testmod()




"""simple docstring"""







def 						_lowercase				(  _SCREAMING_SNAKE_CASE				:				list[int]						)    ->     tuple[int, int]:







     '''simple docstring'''


     if len(_SCREAMING_SNAKE_CASE						) == 1:
          return a[0], a[0]
     else:
          # split A into half.
          __A					:					List[str]								=		a[: len(_SCREAMING_SNAKE_CASE						) // 2]
          __A					:					Optional[Any]								=		a[len(_SCREAMING_SNAKE_CASE						) // 2 :]

          # 2 sub problems, 1/2 of original size.
          __A						,		__A					:					int								=		max_difference(_SCREAMING_SNAKE_CASE						)
          __A						,		__A					:					str								=		max_difference(_SCREAMING_SNAKE_CASE						)

          # get min of first and max of second
          # linear time
          __A					:					str								=		min(_SCREAMING_SNAKE_CASE						)
          __A					:					Optional[int]								=		max(_SCREAMING_SNAKE_CASE						)

          # 3 cases, either (small1, big1),
          # (min_first, max_second), (small2, big2)
          # constant comparisons
          if biga - smalla > max_second - min_first and biga - smalla > biga - smalla:
               return smalla, biga
          elif biga - smalla > max_second - min_first:
               return smalla, biga
          else:
               return min_first, max_second


if __name__ == "__main__":
       import doctest

       doctest.testmod()



