Question

A company that produces pleasure boats has decided to expand one of its lines. Current facilities are insufficient to handle the increased workload, so the company is considering three a (new location), B (subcontract), and C (expand existing facilities). lternatives, A Alternative A would involve substantial fixed costs but relatively low v costs would be $250,000 per year, and variable costs would be $500 per boat. Subcontracting would involve a cost per boat of $2,500, and expansion would require an annual fixed cost of S50,000 and a variable cost of $1,000 per boat.

Find the range of output for each alternative that would yield the lowest total cost.

Answer 1

To find the range of output that yields the lowest total cost for producing pleasure boats, calculate the total cost for different production levels and compare the alternatives. For fewer than 25 boats, subcontracting is optimal; for 25 to 166 boats, expanding current facilities is best; and for 167 to 400 boats, a new location is the most cost-effective. Beyond 400 boats, expansion becomes optimal again.

Step-by-step explanation:

To determine the range of output for each alternative for the production of pleasure boats which yields the lowest total cost, we will need to set up and compare the total costs for different levels of output. The total cost function for each alternative is the sum of the fixed costs and the variable costs multiplied by the number of boats produced. We will perform a break-even analysis to find at what output levels one option becomes more beneficial than the others.

For Alternative A (new location): Fixed costs = $250,000; Variable cost per boat = $500.
For Alternative B (subcontract): Cost per boat = $2,500; no fixed cost.
For Alternative C (expand existing facilities): Fixed costs = $50,000; Variable cost per boat = $1,000.

To find the range of output for each alternative, we need to calculate the total cost for different quantities of boats and determine where one alternative's cost becomes less than the other's costs. We do this by setting up inequalities and solving for the quantity of boats (Q).

Comparing A and B: 250,000 + 500Q < 2,500Q; solving yields Q < 166.67 (meaning, if the company produces less than 167 boats, A is better).
Comparing A and C: 250,000 + 500Q < 50,000 + 1,000Q; solving yields Q < 400 (meaning, if the company produces between 167 and 400 boats, C is better).
Comparing B and C: By setting up 2,500Q < 50,000 + 1,000Q and solving, we find that for Q > 25, option C is better than B.

Thus, for fewer than 25 boats, subcontracting (B) is the best option. For 25 to 166 boats, expanding existing facilities (C) is the best option. For 167 to 400 boats, a new location (A) is optimal. However, once surpassing 400 boats, expansion (C) once again becomes the best option because of its lower variable cost impact.