Linear Programm...

Let R be the feasible region (convex polygon) for a linear programming problem and let Z = ax + by be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities,

Maximise Z = 3x + 4y subject to the constraints: x + y ≤4, x ≥0, y ≥0.

A factory manufactures two types of screws, A and B. Each type of screw requires the use of two machines, an automatic and a hand operated. It takes 4 minutes on the automatic and 6 minutes on hand operated machines tomanufacture a package of screws A, while it takes 6 minutes on automatic and 3 minutes on the hand operated machines to manufacture a package of screws B. Each machine is available for at the most 4 hours on any day. The manufacturer can sell a package of screws A at a profit of Rs 7 and screws B at a profit of Rs 10. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce in a day in order to maximize his profit? Determine the maximum profit.

Determine the maximum value of Z = 11x + 7y subject to the constraints :2x + y ≤6, x ≤2, x ≥0, y ≥0.

The feasible region for a LPP is shown in Figure. Find the minimum value of Z = 11x + 7y.

Let R be the feasible region for a linear programming problem, and let Z = ax + by be the objective function. If R is bounded, then

Minimise Z = –3x + 4 y subject to x + 2y ≤8, 3x + 2y ≤12, x ≥0, y ≥0.

A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The profit from the sale of a lamp is Rs 5 and that from a shade is Rs 3. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximize his profit?

Maximize Z = 3x + 4y, subject to the constraints: x + y ≤1, x ≥0, y ≥0.

The feasible region for a LPP is shown in Figure. Find the maximum value of Z = 11x + 7y.