Login/Sign Up
Home » Minimize     subject to the constraints:     .
Minimize

    \[\mathbf{Z}\text{ }=\text{ }\mathbf{13}x~\text{ }\mathbf{15}y~\]

subject to the constraints:

    \[x~+~y~\text{£}\text{ }\mathbf{7},\text{ }\mathbf{2}x~\text{ }\mathbf{3}y~+\text{ }\mathbf{6}\text{ }{}^\text{3}\text{ }\mathbf{0},~x~{}^\text{3}\text{ }\mathbf{0},~y~{}^\text{3}\text{ }\mathbf{0}\]

.
CBSE | Class 12 | Exercise 1 | Linear Programming | Maths | NCERT Exemplar
Minimize

    \[\mathbf{Z}\text{ }=\text{ }\mathbf{13}x~\text{ }\mathbf{15}y~\]

subject to the constraints:

    \[x~+~y~\text{£}\text{ }\mathbf{7},\text{ }\mathbf{2}x~\text{ }\mathbf{3}y~+\text{ }\mathbf{6}\text{ }{}^\text{3}\text{ }\mathbf{0},~x~{}^\text{3}\text{ }\mathbf{0},~y~{}^\text{3}\text{ }\mathbf{0}\]

.

According to the question:

    \[\mathbf{Z}\text{ }=\text{ }\mathbf{13}x~\text{ }\mathbf{15}y~\]

and the constraints 

    \[x~+~y~\text{£}\text{ }\mathbf{7},\text{ }\mathbf{2}x~\text{ }\mathbf{3}y~+\text{ }\mathbf{6}\text{ }{}^\text{3}\text{ }\mathbf{0},~x~{}^\text{3}\text{ }\mathbf{0},~y~{}^\text{3}\text{ }\mathbf{0}\]

.

Take

    \[x\text{ }+\text{ }y\text{ }=\text{ }7\]

, we have

take 

    \[2x\text{ }\text{ }3y\text{ }+\text{ }6\text{ }=\text{ }0\]

we have

Now, plotting all the constrain equations we see that the shaded area OABC is the feasible region determined by the constraints.

The feasible region is bounded with four corners

    \[O\left( 0,\text{ }0 \right),\text{ }A\left( 7,\text{ }0 \right),\text{ }B\left( 3,\text{ }4 \right)\text{ }and\text{ }C\left( 0,\text{ }2 \right)\]

.

So, the maximum value can occur at any corner.

On evaluating the value of Z, we get

From the above table it’s seen that the minimum value of Z is

    \[-30\]

.

Therefore, the minimum value of the function Z is

    \[-30\]

at

    \[(0,2)\]

.

i

More Material