Consider the following parametrized linear programming probl

Consider the following parametrized linear programming problem: Maximize z = (3-6t)x1 + (2-2t)x2 + (5+5t)x3 subject to x1 + 2x2 + x3 40, 3x1 + 2x3 60, x1 + 4x2 30, x1 , x2, x3 0 and assume that t0 . (b) Determine the first critical value t= t1 for which remains optimal

Solution

x1 + 2x2 + x3 = 40 is equetion of plane with 30,20,30 as x1,x2 & x3 axes

3x1 + 2x3= 60 is eqn of line with 20 &30 as x1 &x3 intercepts

x1 + 4x2 = 30   is eqn of line with 30 &30/4 as x1 &x2 intercepts

as x1 , x2, x3 0 we are in fist octant

3x1 + 2x3= 60 and x1 + 4x2 = 30 intersects in point x2 =7.5 ,x3= 30 and x1= -5

so solution is at this point puting these values we get given function which is to be maximise 165 t +150 we choose t=20 as we want it in feasible region and it should be criticl and optimal so it is required solution