Chapter 11.8, Problem 29AYU

Product Design An entrepreneur is having a design group produce at least six samples of a new kind of fastener that he wants to market. It costs $\$9.00$ to produce each metal fastener and $\$4.00$ to produce each plastic fastener. He wants to have at least two of each version of the fastener and needs to have all the samples 24 hours (h) from now. It takes 4 h to produce each metal sample and 2 h to produce each plastic sample. To minimize the cost of the samples, how many of each kind should the entrepreneur order? What will be the cost of the samples?

To determine

To solve: The given linear programming problem.

Answer to Problem 29AYU

Solution:

The entrepreneur has to produce 2 metal and 4 plastic fasteners with cost of $\$34$.

Explanation of Solution

Given:

• A design group has to produce at least six samples of fasteners.
• Cost of producing metal and plastic fasteners are $\$9.00\text{ and }\$4.00$ respectively.
• Time of producing each metal and plastic fasteners are 4 hours and 2 hours respectively.

Calculation:

Begin by assigning symbols for the two variables.

$x$ be the number of metal fasteners produced.

$y$ be the number of plastic fasteners produced.

If $C$ be the total cost of producing fasteners,

$C=9x+4y$

The goal is to minimize $C$ subject to certain constraints on $x\text{and}y$. Because $x\text{and}y$ represents number of fasteners, the only meaningful values of $x\text{and}y$ are non-negative.

Therefore, $x\ge 0,y\ge 0$.

From the given data we get,

$x+y\ge 6$

$4x+2y\le 24\Rightarrow 2x+y\le 12$

$x\ge 2$

$y\ge 2$

Therefore, the linear programming problem may be stated as,

Maximize, $C=9x+4y$.

Subject to,

$x\ge 0$

$y\ge 0$

$x+y\ge 6$

$2x+y\le 12$

$x\ge 2$

$y\ge 2$

The graph of the constraints is illustrated in the figure below.

The corner points are as follows:

Corner points are	Value of objective function
$\left(2,4\right)$	$34$
$\left(2,8\right)$	$50$
$\left(4,2\right)$	$44$
$\left(5,2\right)$	$53$

Therefore, the entrepreneur has to produce 2 metal and 4 plastic fasteners with cost of $\text{\$34}$.