Chapter 3.4, Problem 14PPSE

Solution

To State:

What are the unknowns in the given situation.

What are the constraints due to each given condition.

If there are any implicit constraints.

The number of samples of Type II Bacteria that should be used to minimize the cost.

The number of samples used of Type I (say $x$ ) and Type II (say $y$ ) are the unknowns here.

  $4x+3y\ge 240$ is the constraint for the condition that there must be at least $240$ new viable bacteria. $30\le x\text{ and }x\le 60$ are the constraints for the condition that the number of type I samples must be at least $30$ but not more than $60$ . $y\le 70$ is the constraint for the condition that the number of type II constraints is not more than $70$ .

There are implicit constraints $x\ge 0$ and $y\ge 0$ which impose the condition that the number of samples cannot be negative.

The minimum cost is $\$300$ when no samples of type II are used.

Given:

Each sample of Type I bacteria produces four new viable bacteria.

Each sample of Type II bacteria produces three new viable bacteria.

At least $240$ new viable bacteria must be produced.

The number of type I samples must be at least $30$ but not more than $60$ .

A sample of type I costs $\$5$ and that of type II costs $\$7$ .

Concepts Used:

Modelling a situation as a Linear Programming Problem (LPP).

Solving a LPP graphically.

Calculations:

Summarize the given information in a table:

	Type I Sample	Type II Sample	Total
Number	$x$	$y$
New Viable Bacteria	$4x$	$3y$	$4x+3y$
Cost	$5x$	$7y$	$5x+7y$

Write the constraints:

  $4x+3y\ge 240$ is the constraint for the condition that there must be at least $240$ new viable bacteria. $30\le x\text{ and }x\le 60$ are the constraints for the condition that the number of type I samples must be at least $30$ but not more than $60$ . $y\le 70$ is the constraint for the condition that the number of type II constraints is not more than $70$ . $x\ge 0$ and $y\ge 0$ are implicit constraints which impose the condition that the number of samples cannot be negative.

  $\left\{\begin{array}{l}4x+3y\ge 240\hfill \\ 30\le x\hfill \\ x\le 60\hfill \\ y\le 70\hfill \\ x\ge 0\hfill \\ y\ge 0\hfill \end{array}\right.$

Write the objective function: the cost $C=5x+7y$ to minimize.

Determine the feasible region by graphing the constraint inequalities. Also determine the corner points of the feasible region.

  

Evaluate the objective function $C=5x+7y$ at the corner points.

  $\begin{array}{cc}\text{Corner Point }\left(x,y\right)& \text{Value of Objective Function }\left(C=5x+7y\right)\\ \left(60,0\right)& 5\times 60+7\times 0=300\\ \left(30,40\right)& 5\times 30+7\times 40=430\\ \left(30,70\right)& 5\times 30+7\times 70=640\\ \left(60,70\right)& 5\times 60+7\times 70=790\end{array}$

The minimum cost is $\$300$ when no samples of type II are used.

Conclusion:

The minimum cost is $\$300$ when no samples of type II are used.