Question

A large city in mid-west needs to buy a street-cleaning machine. A used vehicle will cost $75000 and has a market value of $20000 after its five-year life. A new system cost $150000 and has a market value of $40000 after five-years.The new system has some features that reduce labor hours compared with used system. the used system requires labor hours of 8 hours per day and 20 days per month. the labor costs are $50 per hour. the MARR is 12%. if the new system is expected to be able to reduce labor hours by 20% compared with the used system, which system should the city purchase? and how many hours must the system be operated at the break even?

Answer 1

Solution:

Two alternatives: Used vehicle and new system

Doing the present worth analysis in order to compare the two alternatives:

Life years, N = 5 years, MARR = 12%, labor cost, w = $50 per hour. Let annual costs be denoted by A.

Present Worth, PW = -IC - A(P/A, MARR, N) + S((P/F, MARR, N)

(P/A, MARR, N) = ((1 + MARR)^N - 1)/(MARR)(1 + MARR)^N

(P/A, 12%, 5) = ((1 + 0.12)⁵ - 1)/(0.12)(1 + 0.12)⁵ = 0.762/0.211 = 3.611

(P/F, MARR, N) = (1 + MARR)^-N

(P/F, 12%, 5) = (1 + 0.12)^-5 = 0.567

Hence, PW = -IC - 3.611*A + 0.567*S ... (1)

For Used Vehicle:

Initial cost (IC) = $75,000, Salvage value (S) = $20,000

Labor hours required = 8 hours, work days in month = 20 days

Thus, monthly hours of labor = 8*20 = 160

With 12 months in an year, annual labor hours = 160*12 = 1,920, and annual labor costs, A = 1920*w = 1920*50 = $96,000

From (1), PW = - 75000 - 3.611*96000 + 0.567*20000 = -410,316

For New System:

Initial cost (IC) = $150,000, Salvage value (S) = $40,000

Labor hours required = 20% less than that under used vehicle

So, labor hours required = (1 - 0.2)*8 = 6.4 hours, work days in month = 20 days

Thus, monthly hours of labor = 6.4*20 = 128

With 12 months in an year, annual labor hours = 128*12 = 1,536, and annual labor costs = 1536*w = 1536*50 = $76,800

From (1), PW = - 150,000 - 3.611*76800 + 0.567*40000 = -404,644.8

Clearly, present worth is higher in case of new system than for used vehicle. Thus, city should purchase the new system.