1. Solve the following problem (P) by adding artificial variables with cost M, where M is large. Minimize 3.x - 2y subject to - x +y > 2 x+y > 4 y 0,4 > 0. (Hint: Convert the above inequality constraints into equalities by subtracting si and S2 from the first two constraints and adding s3 to the last. Then add artificial variables in the first and the second constraint only to get the initial basic feasible solution.) (a) Show that if M 2 3x + 3y 0, y > 0. By using dual simplex method find the direction along which the objective function of the dual is unbounded. 3. Consider the following linear programming problem and its optimal final tableau shown below. Maximize 2.21 + x2 13 subject to x1 + 2x2 + I3 0, x2 > 0, 33 > 0. Zj - Cj 21 B-la B-la2 B-laz B-'s B-1s2 B-lb 0 3 3 2 0 1 2 1 1 0 8 0 3 -1 1 12 S2 (a) If the coefficient of x2 in the objective function is changed from 1 to 6, then find the new optimal solution. (b) If the coefficient of x2 in the first constraint is changed from 2 to , find the new optimal solution. (c) if the coefficient of x1 in the first constraint is changed from 1 to 0, find the new optimal solution. (d) If the constraint x2 + x3 = 3 is added to the problem then find the new optimal solution. (e) If you were to choose between increasing the right hand side of the first and second constraints, which one would you choose? Why? What will be the effect of this change in the value of the objective function. (f) Suppose a new variable 34 is added with c4 4 and a4 = (1,2), find the new optimal solution. =