Linear Programm...

Consider the following Linear Programming Problem (LPP).

Maximise \(Z=x_{1}+2 x_{2}\)

Subject to:

\(x_{1} \leq 2\)

\(x_{2} \leq 2\)

\(x_{1}+x_{2} \leq 2\)

\(x_{1}, x_{2} \geq 0\) (i.e., +ve decision variables)

Minimize \(\mathrm{Z}=7 \mathrm{x}+\mathrm{y}\) subject to \(5 x+y \geq 5, x+y \geq 3, x \geq 0, y \geq 0\).

Find graphically, the maximum value of \(\mathrm{z}=2 \mathrm{x}+5 \mathrm{y}\), subject to constraints given below:

\(2 \mathrm{x}+4 \mathrm{y} \leq 8 \)

\(3 \mathrm{x}+\mathrm{y} \leq 6 \)

\(\mathrm{x}+\mathrm{y} \leq 4 \)

\(\mathrm{x} \geq 0, \mathrm{y} \geq 0\)

Linear programming \(( LP )\) problem can be formulated in terms of \(X\) and \(Y\) and Profit \((P)\). Maximize \(P=\) \(7 X+5 Y\) (Objective function) subject to \(4 X+3 Y \leq 240\) (hours of carpentry constraint) \(2 X+Y \leq 100\) (hours of painting constraint) \(X \geq 0, Y \geq 0\) (Non-negativity constraint).

The maximum value of\(z = ~3 x+2 y\), subject to the constraints:\(x+2y \leq 10 \);\(~3 x+y \leq 15 \) and\(x ,\) \(y \geq 0\) is:

In order that linear programming techniques provide valid results:

If \(\mathrm{x}=(7+4 \sqrt{3})^{2 \mathrm{n}}=[\mathrm{x}]+\mathrm{f}\), then \(\mathrm{x}(1-\mathrm{f})\) is equal to:

Solve the linear programming problem by graphical method.

Minimize \(Z=20 x_{1}+40 x_{2}\) subject to the constraints \(36 x_{1}+6 x_{2} \geq 108 ; 3 x_{1}+12 x_{2} \geq 36\) \(20 x_{1}+10 x_{2} \geq 100\) and \(x_{1}, x_{2} \geq 0\).

Solve the following using graphical method:

Minimize: \(\mathrm{Z}=3 \mathrm{x}+5\) y subject to \(2 \mathrm{x}+3 \mathrm{y} \geq 12,-\mathrm{x}+\mathrm{y} \leq 3, \mathrm{x} \leq 4, \mathrm{y} \geq 3, \mathrm{x} \geq 0, \mathrm{y} \geq 0\).

Solve the linear programming problem by graphical method:

Minimize \(Z=3 x+5 y\) Subject to the constraints \(x+3 y \geq 3, x+y \geq 2\) and \(x \geq 0, y \geq 0\).