A firm manufactures two types of products *A* and *B* and sells them at a profit of Rs 2 on type *A* and Rs 3 on type *B*. Each product is processed on two machines *M*_{1} and *M*_{2}. Type *A* requires one minute of processing time on *M*_{1} and two minutes of *M*_{2}; type *B* requires one minute on *M*_{1} and one minute on *M*_{2}. The machine *M*_{1} is available for not more than 6 hours 40 minutes while machine *M*_{2} is available for 10 hours during any working day. Formulate the problem as a LPP.

#### Solution

Let the firm produces *x *units of product A and *y* units of product B.

Since, each unit of product A requires one minute on machine \[M_1\] and two minutes on machine \[M_2\] Therefore, *x* units of product A will require product x minutes on machine \[M_1\] and 2x minutes on machine \[M_2\]

Also,

Since each unit of product B requires one minute on machine \[M_1\] and one minute on machine \[M_2\] Therefore, y units of product A will require product y minutes on machine \[M_1\] and y minutes on machine \[M_2\]

It is given that the machine \[M_1\] is available for \[6 \text{ hours and 40 minutes} \] i.e. 400 minutes and machine \[M_2\] is available for 10 hours i.e. 600 minutes

\[2x + y \leq 600\]

Hence, the required LPP is as follows:

Maximize Z = 2

*x*

*+ 3*

*y*

subject to \[x + y \leq 400\]

\[2x + y \leq 600\]