if $A$ and $C$ become $100A $ and $100C$ , what is the value of $z$?

Question

i have a maximization problem

$max z=Cx$

s.t $ Ax\leq b$

$x\geq0 $

if A and C become $100*A$ and$ 100*C$

what is the value of new $x=x'$ and new $z=z'$ ?

i think new z not changed and new x become $(1/100)*x$

because new$B'^{-1}$ is $(1/00)*B^{-1}$

is this true?

Answer 1

The effect of increasing C to 100C has a direct impact on Z in that sense that it will be 100Z. The effect of increasing A to 100A has an indirect impact of reducing X and thus reducing Z. But this cannot be quantified as you would in the case of C. The simultaneous increasing of C and A will have an increasing effect on z directionally but not directly quantifiable ( as the amount by which x reduces is not one to one).