from pulp import $*$

# Here is a useful function to implement the LHS upper bound that we need for the encoding

def lhsUpperBound$($c$\_$list, bounds$)$:

n $=$ len$($c$\_$list$)$

assert len$($bounds$)==$ n

upper$\_$bnd $=$ sum$([($cj$*$lj$)$ if cj $<0$ else cj$*$uj for $($cj$,($lj$,$ uj$))$ in zip$($c$\_$list, bounds$)])$

return upper$\_$bnd

def solveForMaximumInequalitySatisfaction$($n$,$ m$,$ c$\_$matrix, d$\_$values, bounds$)$:

# always check pre$-$conditions: saves so much time later

assert len$($c$\_$matrix$)==$ m

assert all$($len$($c$\_$list$)==$ n for c$\_$list in c$\_$matrix$)$

assert len$($d$\_$values$)==$ m

assert len$($bounds$)==$ n

assert all $($lj $<=$ uj for $($lj$,$ uj$)$ in bounds$)$

## TODO: set up and solve the problem for satisfying the maximum number of inequalities