Chapter 11.8, Problem 26AYU

Ice Cream The Mom and Pop Ice Cream Company makes two kinds of chocolate ice cream: regular and premium. The properties of 1 gallon (gal) of each type are shown in the table:

In addition, current commitments require the company to make at least 1 gal of premium for every 4 gal of regular. Each day, the company has available 725 pounds (lb) of flavoring and 425 lb of milk-fat products. If the company can ship no more than 3000 lb of product per day, how many gallons of each type should be produced daily to maximize profit?

Source: www.scitoys.com/ingredients/ice_cream.html

To determine

To solve: The given linear programming problem.

Answer to Problem 26AYU

Solution:

The company has to produce 400 regular and 100 premium ice creams.

Explanation of Solution

Given:

• A company produces regular and premium ice creams.
• It uses 24 oz and 20 oz of flavoring agents for regular and premium ice creams respectively.
• It uses 12 oz and 20 oz of milk fat products for regular and premium ice creams respectively.
• The shipping weights are 5 lbs and 6 lbs for regular and premium ice creams respectively.
• The company earns a profit of $\$0.75\text{and}\$0.90$ for regular and premium ice creams respectively.
• The company has to make at least 1 gal of premium ice cream for every 4 gals of regular ice cream.
• 725 lbs (11600 oz) of flavoring agent and 425 lbs (6800 oz) of milk fat products are available to the company every day.

Calculation:

Begin by assigning symbols for the two variables.

$x$ be the amount of regular ice-cream produced.

$y$ be the amount of premium ice-cream produced.

(a) If $P$ is the total profit then,

$P=0.75x+0.90y$

The goal is to maximize $P$ subject to certain constraints on $x\text{and}y$. Because $x\text{and}y$ represents amount of ice-cream produced, 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,

$y\ge x/4$

$24x+20y\le 11600$   $\Rightarrow 6x+5y\le 2900$

$12x+20y\le 6800$   $\Rightarrow 3x+5y\le 1700$

$5x+6y\le 3000$

Therefore, the linear programming problem may be stated as,

Maximize, $P=0.75x+0.90y$.

Subject to,

$x\ge 0$

$y\ge 0$

$y\ge x/4$

$6x+5y\le 2900$

$3x+5y\le 1700$

$5x+6y\le 3000$

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

The corner point that satisfies the constraints is (400,100) and the value of objective function is 390.

The company has to produce 400 regular and 100 premium ice creams.