Question

A brewery produces regular beer and a lower-carbohydrate "light" beer. Steady customers of the brewery buy 10 unils of tegular beer and 16 unils of bght beer monthly. Whie soting up the brewery to produce the beers, the management decides to produce extra beer, beyond that needed to satisty customers. The cost per unit of regular beer is $29,000 and the cost per unit of light beer is $35,000. Every unit of regular beer brings in $100,000 in fovenue, while overy unit of ight beer beings in $300,000 in teverue. The brewery needs at least $9,000,000 in revenue, and has determined that the total demand wil be at least 50 units of boer. Complete parts (a) and (b). (a) How much of each type of boer should be made so as to minimize lotal production conts? units of rogudar beer and units of loght beor thould be mado, for a minimum total cost of : (

Answer 1

To minimize production costs, the brewery should meet the customers' demand for regular beer of 10 units and add sufficient units of light beer to exceed the $9,000,000 revenue requirement. Producing 30 units of light beer, including those bought by steady customers, would meet the revenue requirements and minimize the total production cost.

Step-by-step explanation:

To minimize total production costs, we need to find the optimal mix of regular and light beer units that satisfy customer demand and revenue requirements, using cost and revenue data provided. We have to satisfy the condition that the brewery needs at least $9,000,000 in revenue and the total demand will be at least 50 units of beer.

The cost of producing one unit of regular beer is $29,000 with a revenue of $100,000, and for the light beer, it's $35,000 cost per unit with a revenue of $300,000. To minimize costs while meeting the required revenue, we should prioritize producing the type of beer that offers the higher profit margin per unit.

In this case, each unit of light beer generates a profit of $300,000 - $35,000 = $265,000, and each unit of regular beer generates a profit of $100,000 - $29,000 = $71,000. Since the light beer yields a higher profit, the brewery should produce more units of light beer. However, we have to factor in the minimum requirement of total units and revenue.

If the brewery purely produced light beer to meet the $9,000,000 revenue requirement, it would need to produce at least 30 units (since 30 units × $300,000 per unit = $9,000,000). Producing 30 units of light beer would also exceed the minimum total units requirement of 50 because there are steady customers buying 16 units of light beer already.

Therefore, the brewery should produce 16 units of regular beer for the steady customers and an additional 14 units of light beer (for a cost of $490,000), which totals to 30 units of light beer. This strategy surpasses the revenue requirement with a total revenue of $9,000,000 and minimizes the total production costs. The total production cost of light beer would be 30 units × $35,000 = $1,050,000 plus 10 units of regular beer at $29,000 each (for existing customers) equals $290,000, so the minimum total cost would be $1,340,000.