Questions: Customers buy 12 units of regular beer and 14 units of light beer monthly. The brewery decides to produce extra beer, beyond that needed to satisfy the customers. The cost per unit of regular beer is 28,000 and the cost per unit of light beer is 47,000. Every unit of regular beer brings in 100,000 in revenue, while every unit of light beer brings in 200,000 in revenue. The brewery wants at least 10,000,000 in revenue. At least 62 additional units of beer can be sold. Complete parts (a) and (b). (a) How much of each type of beer should be made so as to minimize total production costs? Let y1 represent units of regular beer and y2 represent units of light beer. Find the objective function, w, used to minimize production costs. w=28000 y1+47000 y2 The brewery should make units of regular beer and units of light beer so as to minimize total production costs.

Customers buy 12 units of regular beer and 14 units of light beer monthly. The brewery decides to produce extra beer, beyond that needed to satisfy the customers. The cost per unit of regular beer is 28,000 and the cost per unit of light beer is 47,000. Every unit of regular beer brings in 100,000 in revenue, while every unit of light beer brings in 200,000 in revenue. The brewery wants at least 10,000,000 in revenue. At least 62 additional units of beer can be sold. Complete parts (a) and (b).
(a) How much of each type of beer should be made so as to minimize total production costs?

Let y1 represent units of regular beer and y2 represent units of light beer. Find the objective function, w, used to minimize production costs.
w=28000 y1+47000 y2
The brewery should make units of regular beer and units of light beer so as to minimize total production costs.

Transcript text: Customers buy 12 units of regular beer and 14 units of light beer monthly. The brewery decides to produce extra beer, beyond that needed to satisfy the customers. The cost per unit of regular beer is $28,000 and the cost per unit of light beer is $47,000. Every unit of regular beer brings in $100,000 in revenue, while every unit of light beer brings in $200,000 in revenue. The brewery wants at least $10,000,000 in revenue. At least 62 additional units of beer can be sold. Complete parts (a) and (b). (a) How much of each type of beer should be made so as to minimize total production costs? Let $y_{1}$ represent units of regular beer and $y_{2}$ represent units of light beer. Find the objective function, $w$, used to minimize production costs. \[ w=28000 y_{1}+47000 y_{2} \] The brewery should make $\square$ units of regular beer and $\square$ units of light beer so as to minimize total production costs.

Solution

Solution Steps

To solve this problem, we need to minimize the total production costs while satisfying the constraints on revenue and additional units sold. The objective function is given as the cost function, and we need to set up the constraints based on the revenue requirement and the minimum additional units. We can use linear programming to find the optimal number of units for each type of beer.

Define the objective function to minimize: $ w = 28000y_1 + 47000y_2 $.
Set up the constraints:
- Revenue constraint: $ 100000y_1 + 200000y_2 \geq 10000000 $.
- Additional units constraint: $ y_1 + y_2 \geq 62 $.
Use a linear programming solver to find the values of $ y_1 $ and $ y_2 $ that minimize the cost while satisfying the constraints.

Step 1: Define the Objective Function

The objective function to minimize the total production costs is given by:

\[ w = 28000y_1 + 47000y_2 \]

where $ y_1 $ represents the units of regular beer and $ y_2 $ represents the units of light beer.

Step 2: Set Up the Constraints

The constraints based on the problem requirements are: