Economic Analysis
- Comparing Alternatives

Present Worth and Annual Worth Methods

When several alternatives are being considered, each will have different cash flows over time. We assign the notation as the investment and and to indicate the income and the cost for alternative k at time t. is the life of alternative k and K is the number of alternatives. In the typical problem, the alternatives are mutually exclusive; only one may be chosen.

Alternatives may be compared with the present worth method or the annual worth method. For the present worth method, the NPW of the cash flows for alternative k is computed using the MARR.

We use the argument to emphasize that the present worth is computed over periods.

When all alternatives have the same life, the one with the greatest NPW is selected as the best.

When the alternatives have different lives, we must compare their present worths over a common time horizon called the study period. There are two ways to establish a study period. The simplest is to truncate the longer lived alternatives at the termination of the alternative with the smallest life. We must then include a salvage value for the truncated alternatives for the study period. The second method is to use a study period equal to the least common multiple of the lives. We use the latter method because it does not require estimates of the salvage values for time periods other than the project life. When we have two or more alternatives let n be the least common multiple of the lives.

One way to calculate the NPW over the study period is to first compute the NAW of each alternative and then compute the present worth for n periods.

The best alternative is the one with the greatest value of . An equivalent and sometimes easier method is to chose the alternative with the greatest value of .

These concepts are illustrated with the two alternatives below. The least common multiple of the two lives (6 and 9) is 18 years. Comparing the two on the basis of a single life using the NPW values in I2 and I17 respectively indicates that Machine A should be selected, but this is incorrect. Comparing their NAW values in cells I3 and I18 we see that Machine B should be selected. The NPW values over the study period in cells I4 and I19 also indicate that Machine B is best. This is the correct decision.


Incremental Analysis


Alternatives may also be compared using incremental analysis. The process starts with the alternative with the smallest investment. Then each alternative with successively higher investment is tested to determine if the added investment yields a return at least equal to the MARR. This results in a series of pairwise comparisons that selects an alternative at each step and is left with the winning alternative when the process is complete.

Incremental Analysis:

  1. Rank the alternatives in order of investment.
  2. Select as the incumbent the alternative with the smallest investment.
  3. Let the challenger be the alternative with the next higher investment.
  4. Accept or reject the challenger on the basis of the return on the extra investment of the challenger over the incumbent. The winner becomes the incumbent.
  5. If the highest level of investment has been reached, stop. The last incumbent is the best alternative. Otherwise return to step 3.

For the example above with two alternatives, Machine A with an investment of $9000 is the incumbent. Machine B with an investment of $16000 is the challenger. The increment to be tested is the extra investment of Machine B over Machine A. The extra investment is $7000. The cash flow for the increment over the 18 year study period is shown below.


Analysis of this cash flow yields the results below. The NPW (B over A) is positive, so the increment of investment is justified. The IRR of the incremental cash flow is almost 19%, much greater than the MARR.

This example is considered again on the next page where we illustrate the use of the Economics add-in to perform economic comparisons.


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Operations Management / Industrial Engineering
by Paul A. Jensen
Copyright 2004 - All rights reserved

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