Question

A firm produces an alloy having the following specifications:
$\left(i\right)$ specific gravity$\le 0.98$
$\left(ii\right)$chromium$\ge 8$percent
$\left(iii\right)$melting point$\ge {450}^0$C
Raw materials $A,B$ and $C$ having the properties shown in the table can be used to make the alloy.

Property	Property of raw material A	Property of raw material B	Property of raw material C
Specific gravity	$0.92$	$0.97$	$1.04$
Chromium	$7$ percent	$13$ percent	$16$ percent
Melting point	${440}^0$ Celsius	${490}^0$ Celsius	${480}^0$ Celsius

Costs of the various raw materials per ton are:Rs.$90$ for $A$,Rs.$280$ for $B$ and Rs.$40$ for $C$. Formulate the L.P model to find the proportions in which $A,B$ and $C$ be used to obtain an alloy of desired properties while the cost of raw materials is minimum.

Answer 1

Formulation of Linear Programming Model:
Let the proportions of raw materials $A,B$ and $C$ to be used for making the alloy be $x_1,x_2$ and $x_3$ respectively.
Objective is to minimize the cost
i.e., minimize $Z=90x_1+280x_2+40x_3$
Constraints are imposed by the specifications required for the alloy.
They are
$0.92x_1+0.97x_2+1.04x_3\le 0.98,$
$7x_1+13x_2+16x_3\ge 8,$
$440x_1+490x_2+480x_3\ge 450$,
and $x_1+x_2+x_3=1,$
as $x_1,x_2$ and $x_3$ are the proportions of materials $A,B$ and $C$ in making the alloy.
Also, $x_1,x_2,x_3,$ each $\ge 0$