1. A cooperative society of farmers has 50 hectare of land to grow two crops X and Y. The profit from crops X and Y per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at rates of 20 litres and 10 litres per hectare. Further, no more than 800 litres of herbicide should be used in order to protect fish and wild life using a pond which collects drainage from this land. How much land should be allocated to each crop so as to maximise the total profit of the society? Formulate this as a LPP. Use graphical solution approach to conclude whether this LPP has a unbounded solution/ feasible solution/infeasible solution? If problem is feasible obtain the optimal solution to the corresponding LPP.
2. Prove that the open half space is convex.
3. Arbitrary intersection of convex sets is convex! Is it true? Justify your answer.
4. Suppose C is a convex set. If x is an extreme point of C, then prove that x is on the boundary of C.
5. Find all the basic solutions corresponding to the system of equations
2x1 + 3x2 − 2x3 − 7x4 = 1 x1 + x2 + x3 + 3x4 = 6 x1 − x2 + x3 + 5x4 = 4.
Are all the solutions basic feasible solutions?
6. Solve the following LPP using simplex method in tabular form
maximize7x1 + 6x2
s.t.
3x11 + x22 ≤ 120 x + 2x ≤ 160 x1 ≤ 35
x1,x2 ≥ 0
7. Conclude the nature of following LPP using simplex method in tabular form
maximize7x1 + 6x2
s.t.
3x111 + x22 ≤ 120 x + 2x ≤ 160 x ≤ 35
471x1 +2 x2 ≤ 100 x ,x ≥ 0
