Question

A company has two grades of inspectors,$I$ and $II$ to undertake quality control inspection.At least $1,500$ pieces must be inspected in an $8-$hour day.Grade$I$ inspector can check $20$ pieces in an hour with an accuracy of $96$percent.
Grade$II$ inspector checks $14$ pieces an hour with an accuracy of $92$ percent.
Wages of grade $I$ inspector are Rs.$5$ per hour while those of grade $II$ inspector are Rs.$4$ per hour. Any error made by an inspector costs Rs.$3$ to the company.If there are,in all, $10$ grade $I$ inspectors and $15$ grade $II$ inspectors in the company,find the optimal assignment of inspectors that minimizes the daily inspection cost.

Answer 1

Formulation of L.P model:
Let $x_1$ and $x_2$ denote the number of grade$I$ and grade$II$ inspectors that may be assigned the job of quality control inspection.
The objective is to minimize the daily cost of inspection.Now the company has to incur two types of costs:wages paid to the inspectors and the cost of their inspection errors.The cost of a grade $I$ inspector/hour is
Rs.$(5+3\times 0.04\times 20)=$Rs.$7.40$
Similarly,cost of a grade$II$ inspector/hour is
Rs.$(4+3\times 0.08\times 14)=$Rs.$\mathrm{7.36.}$
$\therefore$ The objective function is
minimize $Z=8(7.40x_1+7.36x_2)=59.20x_1+58.88x_2.$
Constraints are
on the number of grade$I$ inspectors$:=x_1\le 10,$
on the number of grade$II$ inspectors$:=x_2\le 15,$
on the number of pieces to be inspected daily$:20\times 8x_1+14\times 8x_2\ge 1,500$
or $160x_1+112x_2\ge 1,500,$
where $x_1,x_2\ge 0$