What is the age of replacement?

Question

The following problem is from 'Replacements of models'.

The cost of a new machine is \$5000. The maintenance cost of the $n$th year is given by $C_n=500(n-1)$, $n=1, 2, \dots$. Assuming that the dollar value will not change in time, after how many years will it be economical to replace the machine by new one?

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To do solve this kind of problems, one needs to make a tabular where the entries are years, scrap value (resale value), maintenance cost, cumulative maintenance cost, total cost, anaual average cost.

In this case what is the scrap value and how many years do we need to calculate (because years are not mentioned)?

Please help me.

Answer 1

In this we have to assume that the scrap value does not exist, which mean that it is zero. The criteria is the average annual cost. When it is minimum at year $n$ you are supposed to replace the machine at year $n+1$.

Year n  Cost (C)    Scrap v. (S)  Mainten. Cost  Depreciation  Cumulative  Total Cost   Average Cost  
                                      M (n)        C – S      Mainten. Cost   (TC)      per Year ATC

 (1)       (2)          (3)         (4)            (5)           (6)           (7)           (8) 
                                              = (2) – (3)       = ∑ (4)   = (5) + (6)      = (7)/(1)  
1         5000          0         0                5000           0        5000              5000      
2         5000          0         500              5000         500        5500              2750        
3         5000          0         1000             5000        1500        6000              2000         
4         5000          0         1500             5000        3000        8000              2000       
5         5000          0         2000             5000        5000       10000              2000         
6         5000          0         2500             5000        7500       12500              2083,33         
7         5000          0         3000             5000       10500       15500              2214,29         
8         5000          0         3500             5000       14000       19000              2375    

In this case I would say that the machine has to replaced after 5 years, since in year 6 the ATC starts to increase.