To illustrate this process, we take the case of Judy Grim, who is looking for a new computer system for her small business. She has determined that the most important overall factors are hardware, software, and vendor support. Furthermore, Judy has narrowed down her alternatives to three possible computer systems. She has labeled these SYSTEM-1, SYSTEM-2, and SYSTEM-3. To begin, Judy has placed these factors and alternatives into a decision hierarchy (see Figure M1.1).

The decision hierarchy for the computer selection has three different levels. The top level describes the overall decision. As you can see in Figure M1.1, this overall decision is to select the best computer system. The middle level in the hierarchy describes the factors that are to be considered: hardware, software, and vendor support. Judy could decide to use a number of additional factors, but for this example, we keep our factors to only three to show you the types of calculations that are to be performed using the AHP. The bottom level of the decision hierarchy reveals the alternatives. (Alternatives have also been called items or systems.) As you can see, the alternatives include the three different computer systems.

The key to using the AHP is pairwise comparisons. The decision maker, Judy Grim, needs to compare two different alternatives using a scale that ranges from equally preferred to extremely preferred.

We use the following for pairwise comparison:

• 1—Equally preferred

• 2—Equally to moderately preferred

• 3—Moderately preferred

• 4—Moderately to strongly preferred

• 5—Strongly preferred

• 6—Strongly to very strongly preferred

• 7—Very strongly preferred

• 8—Very to extremely strongly preferred

• 9—Extremely preferred

Judy begins by looking at the hardware factor and by comparing computer SYSTEM-1 with computer SYSTEM-2. Using the preceding scale, Judy determines that the hardware for computer SYSTEM-1 is moderately preferred to computer SYSTEM-2. Thus, Judy uses the number 3, representing moderately preferred. Next, Judy compares the hardware for SYSTEM-1 with SYSTEM-3. She believes that the hardware for computer SYSTEM-1 is extremely preferred to computer SYSTEM-3. This is a numerical score of 9. Finally, Judy considers the only other pairwise comparison, which is the hardware for computer SYSTEM-2 compared with the hardware for computer SYSTEM-3. She believes that the hardware for computer SYSTEM-2 is strongly to very strongly preferred to the hardware for computer SYSTEM-3, a score of 6. With these pairwise comparisons, Judy constructs a pairwise comparison matrix for hardware. This is shown in the following table:

This pairwise comparison matrix reveals Judy’s preferences for hardware concerning the three computer systems. From this information, using the AHP, we can determine the evaluation factors for hardware for the three computer systems.

Look at the upper-left corner of the pairwise comparison matrix. This upper-left corner compares computer SYSTEM-1 with itself for hardware. When comparing anything to itself, the evaluation scale must be 1, representing equally preferred. Thus, we can place the number 1 in the upper-left corner (see the next table) to compare SYSTEM-1 with itself. The same can be said for comparing SYSTEM-2 with itself and comparing SYSTEM-3 with itself. Each of these must also get a score of 1, which represents equally preferred.

In general, for any pairwise comparison matrix, we will place 1s down the diagonal from the upper-left corner to the lower-right corner. To finish such a table, we make the observation that if alternative A is twice as preferred as alternative B, we can conclude that alternative B is preferred only one-half as much as alternative A. Thus, if alternative A receives a score of 2 relative to alternative B, then alternative B should receive a score of ${}^1{/}_2$ when compared with alternative A. We can use this same logic to complete the lower-left side of the matrix of pairwise comparisons:

Look at this newest matrix of pairwise comparisons. You will see that there are 1s down the diagonal from the upper-left corner to the lower-right corner. Then look at the lower left part of the table. In the second row and first column of this table, you can see that SYSTEM-2 received a score of ${}^1{/}_3$ compared with SYSTEM-1. This is because SYSTEM-1 received a score of 3 over SYSTEM-2 from the original assessment. Now look at the third row. The same has been done. SYSTEM-3 compared with SYSTEM-1, in row 3 and column 1 of the table, received a score of ${}^1{/}_9.$ This is because SYSTEM-1 compared with SYSTEM-3 received a score of 9 in the original pairwise comparison. In a similar fashion, SYSTEM-3 compared with SYSTEM-2 received a score of ${}^1{/}_6$ in the third row and second column of the table. This is because when comparing SYSTEM-2 with SYSTEM-3 in the original pairwise comparison, the score of 6 was given.

Now that we have completed the matrix of pairwise comparisons, we can start to compute the evaluations for hardware. We start by converting the numbers in the matrix of pairwise comparisons to decimals to make them easier to work with. We then get column totals:

Once the column totals have been determined, the numbers in the matrix are divided by their respective column totals to produce the normalized matrix that follows:

To determine the priorities for hardware for the three computer systems, we simply find the average of the various rows from the matrix of numbers, as follows:

The results are displayed in Table M1.6. As you can see, the factor evaluation for SYSTEM-1 is 0.6583. For SYSTEM-2 and SYSTEM-3, the factor evaluations are 0.2819 and 0.0598, respectively. The same procedure is used to get the factor evaluations for all other factors—software and vendor support, in this case. But before we do this, we need to determine whether our responses are consistent by determining a consistency ratio.

To arrive at the consistency ratio, we begin by determining the weighted sum vector. This is done by multiplying the factor evaluation number for the first system by the first column of the original pairwise comparison matrix. We multiply the second factor evaluation by the second column and the third factor by the third column of the original matrix of pairwise comparisons. Then we sum these values over the rows:

The next step is to determine the consistency vector. This is done by dividing the weighted sum vector by the factor evaluation values determined previously:

So far, we have determined the factor evaluations for hardware for the three different computer systems along with a consistency ratio for these evaluations. Now, we can make the same calculations for other factors—namely, software and vendor support. As before, we start with the matrix of pairwise comparisons. We perform the same calculations and end up with the various factor evaluations for both software and vendor support. We begin by presenting the matrix of pairwise comparisons for both software and vendor support:

With the matrices shown, we can perform the same types of calculations to determine the factor evaluations for both software and vendor support for the three computer systems. The data for the three different systems are summarized in Table M1.7. We also need to determine the consistency ratios for both software and support. As it turns out, both consistency ratios are under 0.10, meaning that the responses to the pairwise comparison are acceptably consistent.

Note that the factor evaluations for the three factors and three different computer systems shown in Table M1.7 are similar to the factor evaluations in Table M1.2 for the job selection problem. The major difference is that we had to use the AHP to determine these factor evaluations using pairwise comparisons because we were not comfortable with our abilities to assess these factors subjectively without some assistance.

Next, we need to determine the factor weights. When we used the MFEP, it was assumed that we could simply determine these values subjectively. Another approach is to use the AHP and pairwise comparisons to determine the factor weights for hardware, software, and vendor support.

In comparing the three factors, we determine that software is the most important. Software is very to extremely strongly preferred over hardware (number 8). Software is moderately preferred over vendor support (number 3). In comparing vendor support to hardware, we decide that vendor support is more important. Vendor support is moderately preferred to hardware (number 3). With these values, we can construct the pairwise comparison matrix and then compute the weights for hardware, software, and support. We also need to compute a consistency ratio to make sure that our responses are consistent. As with software and vendor support, the actual calculations for determining the factor weights are left for you to make on your own. After making the appropriate calculations, the factor weights for hardware, software, and vendor support are shown in Table M1.8.

After the factor weights have been determined, we can multiply the factor evaluations in Table M1.7 by the factor weights in Table M1.8. This is the same procedure that we used for the job selection decision in Section M1.2. It will give us the overall ranking for the three computer systems, which is shown in Table M1.9. As you can see, SYSTEM-3 received the highest final ranking and is selected as the best computer system.

As you can see from the previous pages, solving AHP problems can involve a large number of calculations. Fortunately, computer programs are available to make the AHP easier. Appendix M1.1 demonstrates how Excel can be used for the AHP calculations shown in this module. A commercial package called Expert Choice for Windows can also be used to solve the types of AHP problems discussed in this module. In addition, it is possible to use the AHP with group decision making. Team Expert Choice helps groups brainstorm ideas, structure their decisions, and evaluate alternatives.