A company produces two products that are produced on 2 assembly lines. Assembly line 1 has 100 available hours and assembly line 2 has 42 available hours. Each product requires 10 hours of processing time on line one, while on line 2, product 1 requires 7 hours and product 2 requires 3 hours. The profit of product 1 is \$6 per unit, and the profit for product 2 is \$4 per unit.

Formulate a linear programming model for this problem.

#### Solution

Let the company produces $x_1$ and $x_2$ units of First type product and second type of product respectively.

Given information can be summarized in the table form as :

Assembly \ Product | First Type Product | Second Type Product | Availability |
---|---|---|---|

Assembly Line 1 | 10 hours | 10 hours | 100 hours |

Assembly Line 2 | 7 hours | 3 hours | 42 hours |

Profit per unit | \$6 | $4 |

Profit per unit for first type product is \$6, so for $x_1$ units of first type product the profit will be \$ $6x_1$.

Profit per unit for second type product is \$4, so for $x_2$ units of second type product will be \$4 $x_2$.

So the total profit will be \$ $(6x_1 + 4x_2)$. The objective of the company is to maximize the profit. Hence the objective function is

`$$Max Z = 6x_1 + 4x_2$$`

First type product requires 10 hours on assembly line 1 and second type of product require 10 hours on assembly line 1. So for $x_1$ units of first type product and $x_2$ units of second type product requires $10x_1 + 10x_2$ hours on assembly line 1. But assembly line 1 is available for 100 hours only. Hence the constraint for first assembly line is

`$$10x_1 + 10x_2 \leq 100$$`

First type product requires 7 hours on assembly line 2 and second type of product require 3 hours on assembly line 2. So for $x_1$ units of first type product and $x_2$ units of second type product requires `$7x_1 + 4x_2$`

hours on assembly line 2. But assembly line 2 is available for 42 hours only. Hence the constraint for assembly line 2 is

`$$7x_1 + 4x_2 \leq 42$$`

It is not possible to produce any product in negative quantity, hence we have `$x_1 \geq 0$`

and `$x_2 \geq 0$`

.

Hence the complete formulated Linear Programming Model is

` $$ \begin{eqnarray*} Max\; z= 6x_1 +4x_2 & & \\ \mbox{s.t. } 10x_1+10x_2&\leq& 100 \\ 7x_1 +4x_2 &\leq & 42 \\ \mbox{and }x_1 , x_2 & \geq & 0 \end{eqnarray*} $$ `