The integer linear program (ILP) max 40x15x2 + 60x3 +8x4 s.t. 18x13x2 +20x3+5x4 25 X1, X2, X3, X4 {0,1} has linear programming (LP) relaxation optimum x = 5 18, X2 = (1) (2) (3) 0, x3 = 1, and x4 = 0. Determine whether each of the following is a valid inequality for the ILP, and if so, whether it would strengthen the LP relaxation to add the inequality as a constraint. (a) x1 + x3 1 (b) x1 + x2 x3 + x4 4 (c) x2 + x4 1 (d) 18x120x3 20 (e) Based on your answers to parts (a), (b), (c), and (d), find the tightest bounds you can on the optimal ILP objective function value.