Written Assignment – 1

Description

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 : C^T X > Z is convex.

3. Arbitrary intersection of convex sets is convex! Is it true? Justify your answer.

3. Suppose C is a convex set. If X is an extreme point of C, then prove that X is on the boundary of C.

3. 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

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