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 S = X : CT X > Z 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.
3X1 + X2 ≤ 120
X1 + 2X2 ≤ 160
X1≤35
X1,X2 ≥ 0
7. Conclude the nature of following LPP using simplex method in tabular form
maximize7X1 + 6X2
s.t.
3X1 + X2 ≤ 120
X +2X ≤160
1 2
X1≤35
7X1 + X2 ≤ 100
4
X1,X2 ≥ 0
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