Corner points of the feasible region determined by the system of linear constraints are (0, 3), (1,1) and (3, 0). Let Z = px + qy, where p, q > 0. Condition on p and q so that minimum value of Z occurs at (3, 0) and (1,1) is
The feasible region in linear programming is the set of all possible solutions that satisfy the given constraints. It is typically represented as a polygon or polyhedron in the coordinate system. The optimal solution is found at one of the vertices or edges of the feasible region.
Linear Programming is a mathematical technique used to optimize a linear objective function subject to linear constraints. It helps in making optimal decisions by determining the best solution within the feasible region formed by the constraints. The solution is often found at the vertices of the feasible region for maximum or minimum values.
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To find the condition on p and q, we substitute the coordinates of the corner points 1, 1 and 3, 0 into the objective function Z = px + qy.
1. For point (1, 1):
Z = p(1) + q(1) = p + q
2. For point (3, 0)
Z = p(3) + q(0) = 3p
For the minimum value of Z to occur at (3, 0) and (1, 1), the objective function value at (1, 1) must be greater than at (3, 0). So we require,
p + q ≥ 3p
q ≥ 2p
Therefore, the constraint on p and q is q = 2p.