003%Ol2lj86 53.00 +o.oo Pergamon Journals Ltd

Socio-Econ. Plann. Sci. Vol. 20, No. 6, pp. 397-398, 1986 Printed in Great Britain

A NOTE ON THE AHP AND EXPECTED VALUE THEORY

THOMAS L. SAATY University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.

Abstract-It is shown here that one cannot simply take columns of numbers, normalize them and add to obtain results corresponding to operations in the AHP. This is what traditional expected value theory using a single scale would lead one to do. Care needs to be exercised. What one must do is to interpret the data represented by each column according to relative importance to a decision maker so that the alternatives under each criterion are pairwise compared according to the fundamental scale used to represent judgments. This procedure then leads to a set of vectors which belong to the same ratio scale and they can now be combined by using the weights of the criteria.

Consider the following example of a standard method used in decision theory to make a best choice based on expected value theory. It illustrates how to use expected return to invest money in three time periods. The investments will require varying sums of money in each of the three years, but the total amount to be invested is $65 for each of the three investment plans considered here. We have:

Yl Y2 Y3 Pl: 40 10 15 P2: 30 15 20 P3: 30 25 10

We will assume that the interest rate for the three years is 5% per year. Which plan should we choose?

In effect we will be using the reciprocals of 1.05, its square and cube or multiply by 0.95, 0.91 and 0.87 respectively, as the weighting factors for the three years.

The expected value approach yields:

E(P1) = 40 x 0.95 + 10 x 0.91

+ 15 x 0.87 = 60.15

E(P2) = 30 x 0.95 + 15 x 0.91

+ 20 x 0.87 = 59.55

E(P3) = 30 x 0.95 + 25 x 0.91

+ 10 x 0.87 = 59.95

The first alternative would be the best choice. Suppose that instead of doing it this way, we wish

to see how the expected value idea works if we use ratios instead of absolute numbers. We do this be- cause the AHP uses ratios (although in a different way) and we wish to find the relationship between the two approaches.

Our first matrix of paired comparisons would use the first column to construct the paired comparisons. We have:

40140

(

40130 40130

30140 30130 30130

30140 30130 1 30130

and the scale of relative values derived from this

consistent reciprocal matrix is obtained by nor- malizing any of its columns. We have:

40/100

30/100

30/100

which is simply the original first column normalized. Similarly for the second and third columns, yield-

ing:

Yl Y2 Y3 Pl: 40/100 10/50 15145

P2: 30/100 15/50 20145

P3: 30/100 25150 1 o/45

Notice, for example, that previously the (P3,Yl) element was 30 and the (P3, Y2) element was 25 and the latter was smaller than the former, whereas now we have 30/100 and 25/50, respectively, and the opposite is true. In other words, we no longer can calculate expected values in the same way from the normalized numbers as we did prior to normaliz- ation. Why did this happen? Because it violated the stringent conditions that the weights of the altema- tives be measured in the same dollar unit for all the years and column normalization in this special case of using the same scale throughout destroys the relationship between row elements.

What must one then do to choose the best altema- tive if he were to apply normalization? After one has normalized the columns to unity, the criteria must be resealed to preserve the dollar relationship between the row elements as follows:

E(P1) = 40/100 x 100 x 0.95

+ lo/so x 50 x 0.91

+ 15/45 x 45 x 0.87 = 60.15

E(P2) = 30/100 x 100 x 0.95

+ 15/50 x 50 x 0.91

+ 20145 x 45 x 0.87 = 59.55

397

398 THOMAS L. SAATY

E(P3) = 30/100 x 100 x 0.95

+ 25/50 x 50 x 0.91 + 10145 x 45 x 0.87 = 59.95

Note that these operations do not alter the final answer. We continue by dividing each term by the sum of the column sums, 100 + 50 + 45 = 195, which is the sum of all the elements in the matrix. We obtain:

E(P1) = 40/100 (lOOj195 x 0.95) + lo/so (59/195 x 0.91)

+ 15/45 (45/195 x 0.87) = 60.15/195

E(P2) = 30/100 (loo/195 x 0.95)

+ l5/50 (50/195 x 0.91)

+ 20/45 (45/195 x 0.87) = 59.55/195

E(P3) = 30/100 (loo/195 x 0.95)

+ 25/50 (50/195 x 0.91)

+ lo/45 (45/195 x 0.87) = 59.95/195

It is clear that all the operations we have carried out so far are legitimate and give the same answer as calculating expected values on unnormalized entries. In addition we have learned something important. With normalization, when the same unit of mea- surement is used throughout, the criteria weights must be resealed proportionately by multiplying each weight by the ratio of the column sum under it to the total sum of the elements in the matrix. Thus, even if the criteria are considered to be independent from the alternatives, their weights depend on the measure- ments of the alternatives given under them. We call this structural dependence.

With the AHP when one assumes that the criteria are independent from the alternatives, one does not then assume that there is scale dependence between the criteria as we have seen above. When such linkage exists the criteria are said to depend on the alterna- tives and the network of supermatrix approach of the AHP is used.

Obviously we cannot measure everything with one monolithic scale. Not all people subscribe to expected value theory as used in the money example, nor to the idea that a single scale such as money is a good one to apply to all problems. Which should come first, an individual’s subjective preference among alternatives, or a scale used by others to indicate their value? Clearly, the first. In general with the AHP we do not have an underlying scale but derive one. Further- more, we are dealing not with one scale but with several different scales. Even if two criteria were somehow known to be measured with the same scale, we do not ordinarily use this scale, but interpret it and derive our own scales through the AHP. There are people who do not believe that the value assigned by society or even by scientific measurement to things in terms of dollars or other units reflects their own idea of the worth or significance of those things. Rather, they prefer to think in terms of overall goals and purposes and the criteria which serve their purposes in a particular decision. They then pursue their own evaluation of what is most important in terms of these criteria.

Assume that we have the identical setting given in the single scale numerical example, but let the in- vestments be people and the years be criteria: fame (measured by how many people know an individual), fortune (measured by money) and intelligence (mea- sured by I.Q.). The relative importance of the criteria is 0.95, 0.91 and 0.87. We want to find an overall ranking of the people. As pointed out above, this problem must first be reduced to a paired comparison framework to determine the extent to which one agrees with the ratios formed from these scales before one can carry out weighting and composition. For example, an individual with an 1.Q. of 180 is not merely twice as intelligent as another with an I.Q. of 90, but signifies the difference between a genius on one side and a low-average individual on the other. The point is not whether the I.Q. scale is a ratio scale subject to certain laws or not, but to be wary of numbers and of their ratios. They need filtering through one’s judgments to find out what sense they make. In the AHP the comparisons are made accord- ing to a particular fundamental scale which allows one to compare things that are close with respect to a property. A good policy is not to use prescribed scales, but to interpret the information of such scales in terms of this fundamental scale and in terms of hierarchic decomposition and clustering.

Scales are derived for the alternatives under each criterion and no scale dependence is assumed to exist between the elements of the rows, although the criteria may depend on the alternatives according to function. Composite weights are obtained directly by using the normalized column weights and the criteria weights without resealing the criteria. Only after normalization do all the column priorities under the criteria belong to the same ratio scale. Numbers which belong to different ratio scales can be multi- plied giving rise to new numbers which also belong to a ratio scale, but they can only be added if they belong to the same ratio scale. Hence the need for a unique way to fix the multiplicative constant of all the different ratio scales derived in the AHP through normalization.

Let us summarize. If we have a matrix of data using the same scale in the background and are given the normalized columns of that matrix, we would be unable to reconstruct the highly interrelated matrix of coefficients without knowledge of resealing factors. If on the other hand there is no scale relationship between the columns of the matrix it would not make sense to simply divide the elements in each column by some number such as the sum of the elements in that column and then obtain the weighted sums of each row. The procedure would be quite arbitrary. What one must do is to interpret the data represented by each column according to relative importance to a decision maker so that the alternatives under each criterion are pairwise compared according to some fundamental scale. This procedure leads to a set of vectors which belong to the same ratio scale. Only in that case can they be combined by using the weights of the criteria. There is no need to rescale the criteria because there is no scale relationship that one would be required to preserve as an initial constraint of the problem.