Question

QUESTION 2 a) Consider the following linear programming model. Maximize profit, Z = 5x + 6X2 + 4X3 Subject to: 3X, +4X2 + 2X3 = 120 X1 + 2X2 + X3 s 50 X1 + 2X2 + 3X3 = 30 X1, X2, X3 20 Obtain the dual of the above linear programming model. (4 marks) b) The following is the simplex tableau for a profit maximization linear programming problem where X1, X2, and X3 represent different ingredients and S, is the slack for resources i where i=1, 2, 3. Basis Quantity S3 X3 X1 z X1 5 0 0 1 5 X2 6 4 2 0 8 X3 4 0 1 0 4 S1 0 -2 -1 1 1 S2 0 7 3 -2 2 S3 0 1 0 0 0 80 30 20 i) Find the maximum profit for the above model. (1 mark) ii) is the above simplex tableau optimal? Explain. (2 marks) iii) State the optimal solution for the above problem. (2 marks) iv) Does this problem have alternative optimal solution? Explain. (2 marks) v) Which resource(s) is/are not fully utilized? State the value. (1 mark) vi) State the shadow price for each resource. What does a zero-shadow price mean? How could this happen