Question

A company manufactures two articles $A$ and $B$. There are two department through which these article are processed : (i) assembly and (ii) finishing department. The maximum capacity of the first department is $60$ hours a week and that of other department. $48$ hours, per week. The product of each unit of article $A$ requires $4$ hours in assembly and $2$ hours in finishing and that of each unit of $B$ requires $2$ hours in assembly and $4$ hours in finishing. If the profit is Rs. $6$ for each unit $A$ and Rs. $8$ for each unit of $A$ and Rs. $8$ for each unit of $B$, find the number of units of $A$ and $B$ to be produced per week in order to have maximum profit.

Answer 1

Let the number of type $A$ and $B$ articles produced be $x$ and $y$ respectively.

Each unit of article $A$ requires $4$ hours in assembly and $2$ hours in the finishing department.

Total time taken by $x$ units of article $A$ in assembly department $=4x$

Total time taken by $x$ units of article $A$ in finishing department $=2x$

Each unit of article $B$ requires $2$ hours in assembly and $4$ hours in the finishing department.

Total time taken by $y$ units of article $B$ in assembly department $=2y$

Total time taken by $y$ units of article $B$ in finishing department $=4y$

Maximum capacity of assembly department = $60$hours/week

Maximum capacity of finishing department = $48$hours/week

Profit on article $A$ = $Rs.6/$unit

Profit on article $B$ = $Rs.8/$unit

Total profit in a week = $6x+8y$

Thus the required constraints are-

$4x+2y\le 60$ $.............\left(i\right)$

$2x+4y\le 48$ $.............\left(ii\right)$

$x\ge 0,y\ge 0$ $.............\left(iii\right)$

Max $Z=6x+8y$

Plotting these inequations on the graph we get the following corner points-

$A(0,12)$ ; $B(12,6)$ ; $C(15,0)$

At $A$, $Z=6\times 0+8\times 12=96$

At $B$, $Z=6\times 12+8\times 6=72+48=120$

At $C$, $Z=6\times 15+8\times 0=90$

Thus, $Z$ is maximum at $B(12,6)$

Thus, 12 units of article $A$ and 6 units of article $B$ should be produced to get the maximum profit per week.