# When Capital or Operating Costs Are Non Monotonic

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Sometimes operating costs do not increase smoothly and monotonically over time. The same can happen to capital costs. Y\Ten the operating or capital costs are not smooth and monotonic, the one year principle does not apply. The reason that the principle does not apply is that there may be periodic or one-time costs that occur over the course of the next year (as in the case where periodic overhauls are required). These costs may make the cost of keeping the defender for one more year greater than the cost of installing and using the challenger over its economic life. However, there may be a life longer than one year over which the cost of using the defender is less than the cost of installing and using a challenger. Consider this example concerning the potential replacement of a generator.

### EXAMPLE 7.9

The Colossal Construction Company uses a generator to produce power at remote sites. The existing generator is now three years old. It cost Sll 000 when purchased. Its current salvage value of S2400 is expected to fall to S1400 next year and S980 the year after, and to continue declining at 30% of current value per year. Its ordinary operating and maintenance costs are now Si000 per year and are expected to rise by S500 per year. There is also a requirement to do an overhaul costing Si000 this year and every third year thereafter.

New fuel-efficient generators have been developed, and Colossal is thinking of replacing its existing generator. It is expected that the new generator technology will be the b est available for the foreseeable future. The new generator sells for \$9500. Installation costs are negligible. Other data for the new generator are summarized in Table 7.4.

 End of Year Salvage Value Operating Cost 0 S9500 1 8000 \$1000 2 7000 1000 3 6000 1200 4 5000 1500 5 4000 2000 6 3000 2000 2000 2000 8 1000 3000

Should Colossal replace the existing generator with the new type? The MARR is 12%.

We first determine the economic life for the challenger. The calculations are shown in Table 7.5.

Sample calculations for the EAC of keeping the challenger for one, two, and three years are as follows:

 End of Year Salvage Value Operating Costs EAC 1 S8000 SI 000 S3640.00 2 7000 1000 3319.25 3 6000 1200 3236.50 4 5000 1500 3233.07* 5 4000 2000 3290.81 6 3000 2000 3314.16 2000 2000 3318.68 8 1000 3000 3393.52

Lowest equivalent annual cost.

Lowest equivalent annual cost.

EAC(3 years) = (P-S)(A/P, 12%,3) + Si + 1000 + 200(a/F,12%,3) = (9500-6000) (0.41635) + 6000(0.12) + 1000

As the number of years increases, this approach for calculating the EAC becomes more difficult, especially since in this case the operating costs are neither a standard annuity nor an arithmetic gradient. An alternative is to calculate the present worths of the operating costs for each year. The EAC of the operating costs can be found by applying the capital recovery factor to the sum of the present worths for the particular service period considered. This approach is particularly handy when using spreadsheets.

By either calculation, we see in Table 7.5 that the economic life of the generator is four years.

Next, to see if and when the defender should be replaced, we calculate the costs of keeping the defender for one more year. Using the capital recovery formula:

EAC (keep defender 1 more year)

= EAC(capital costs) + EAQoperating costs) = (2400- 1400)(A/P,12%,1) + 1400(0.12) + 2000 = 3288

The equivalent annual cost of using the defender one more year is S3288. This is more than the yearly cost of installing and using the challenger over its economic life. Since the operating costs are not smoothly increasing, we need to see if there is a longer life for the defender for which its costs are lower than for the challenger. This can be done with a spreadsheet, as shown in Table 7.6.

We see that, for an additional life of three years, the defender has a lower cost per year than the challenger, when the challenger is kept over its economic life. Therefore, the defender should not be replaced at this time. Next year a new evaluation should be performed.

 Additional Life in Years Salvage Value Operating Costs EAC 0 S2400 1 1400 S2000 S3288 7 980 1500 2722 5 686 2000 2630 4 480 3500 2872 5 336 3000 2924 6 235 3500 3013