Chapter 11.8, Problem 25AYU

Managing a Meat Market A meat market combines ground beef and ground pork in a single package for meat loaf. The ground beef is $75\%$ lean ( $75\%$ beef, $25\%$ fat) and costs the market $\$0.75$ per pound (lb). The ground pork is $60\%$ lean and costs the market $\$0.45/\text{lb}$. The meat loaf must be at least $70\%$ lean. If the market wants to use at least 50 lb of its available pork, but no more than 200 lb of its available ground beef, how much ground beef should be mixed with ground pork so that the cost is minimized?

To determine

To solve: The given linear programming problem.

Answer to Problem 25AYU

Solution:

100 lb of ground beef should be mixed with 50 lb of ground pork so that cost is minimized.

Explanation of Solution

Given:

• Meat loaf is made by combining ground beef and ground pork.
• Ground beef is $75\%$ lean ($75\%$ beef and $25\%$ fat) and costs $\$0.75$ per lb.
• Ground pork is $60\%$ lean and costs $\$0.45\text{/lb}$.
• The meat loaf must be $70\%$ lean.

Calculation:

Begin by assigning symbols for the two variables.

$x$ be the amount of ground beef (in lbs).

$y$ be the amount of ground port (in lbs).

(a) If $C$ is the total cost of the meat loaf, then,

$C=0.75x+0.45y$

The goal is to minimize $C$ subject to certain constraints on $x\text{ and }y$. Because $x\text{ and }y$ represents cost, 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,

$0.75x+0.60y\ge 0.7\left(x+y\right)$

$\Rightarrow 0.05x\ge 0.1y\Rightarrow y\le 0.5x$

$x\le 200$

$y\ge 50$

Therefore, the linear programming problem may be stated as,

Minimize, $C=0.75x+0.45y$.

Subject to,

$y\le 0.5x$

$x\le 200$

$y\ge 50$

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

Corner points are	Value of objective function
(100,50)	$97.5$
(200,50)	$172.5$
(200,100)	195

Therefore, 100 lb of ground beef should be mixed with 50 lb of ground pork so that cost is minimized.