$1.$ A small firm intends to increase the capacity of a bottleneck operation by adding a new machine. Two alternatives, A and B$,$ have been identified, and the associated costs and revenues have been estimated. Annual fixed costs would be $$50,000$ for A and $$40,000$ for B; variable costs per unit would be $$13$ for A and $$15$ for B; and revenue per unit would be $$20.$

a$.$ Determine each alternative$$s break$-$even point in units.

b$.$ At what volume of output would the two alternatives yield the same profit?

c$.$ If expected annual demand is $12,000$ units, which alternative would yield the higher profit? $2.$ A manager is trying to decide whether to purchase a certain part or to have it produced internally. Internal production could use either of two processes. One would entail a variable cost of $$16$ per unit and an annual fixed cost of $$300,000$; the other would entail a variable cost of $$19$ per unit and an annual fixed cost of $$290,000.$ Three vendors are willing to provide the part. Vendor A has a price of $$21$ per unit for any volume up to $30,000$ units. Vendor B has a price of $$24$ per unit for demand of $1,000$ units or less, and $$19$ per unit for larger quantities. Vendor C offers a price of $$22$ per unit for the first $1,000$ units, and $$18$ per unit for additional units. a$.$ If the manager anticipates an annual volume of $20,000$ units, which alternative would be best from a cost standpoint? For $30,000$ units, which alternative would be best?

b$.$ Determine the range for which each alternative is best. Are there any alternatives that are never best? Which?