The Scale

2. The Scale

In the paired comparison approach of the AHP one estimates ratios by using a fundamental scale of absolute numbers. In comparing two alternatives with respect to an attribute, one uses the smaller or lesser one as the unit for that attribute. To estimate the larger one as a multiple of that unit, assign to it an absolute number from the fundamental scale. This process is done for every pair. Thus, instead of assigning two numbers w i and w j and forming the ratio w i /w j we assign a single number drawn from the fundamental 1-9 scale to represent the ratio (w i /w j )/1. The absolute number from the scale is an approximation to the ratio w i /w j . The derived scale tells us what the w i and w j are. This is a central observation about the relative measurement approach of the AHP and the need for a fundamental scale.

The names of Ernest Heinrich Weber (17951878) and Gustav Theodor Fechner (180187) stand out as one considers the subject of stimulus, response, and ratio scales. In 1846 Weber formulated his law regarding a stimulus of measurable magnitudes. He found, for example, that people while holding in their hand different weights, could distinguish between a weight of 20 g and a weight of 21 g, but not if the second weight is only 20.5 g. On the other hand, while they could not distinguish between 40 g and

41 g, they could between the former weight and 42 g, and so on at higher levels. We need to increase a stimulus s by a minimum amount ∆ s to reach a point where our senses can first discriminate between s and s + ∆ s . ∆ s is called the just noticeable difference (jnd). The ratio r = ∆ s / s does not depend on s . Weber's law states that change in sensation is noticed when the stimulus is increased by a constant percentage of the stimulus itself. This law holds in ranges where ∆ s is small when compared with s , and hence in practice it fails to hold when s is either too small or too large. Aggregating or decomposing stimuli as needed into clusters or hierarchy levels is an effective way for extending the uses of this law.

In 1860 Fechner considered a sequence of just noticeable increasing stimuli. He denotes the first one by s 0 . The next just noticeable stimulus [1] is given by

having used Weber's law. Similarly

In general

Thus stimuli of noticeable differences follow sequentially in a geometric progression. Fechner noted that the corresponding sensations should follow each other in an arithmetic sequence at the discrete points at which just noticeable differences occur. But the latter are obtained when we solve for n. We have

( log s n - log s 0 )

log α

and sensation is a linear function of the logarithm of the stimulus. Thus if M denotes the sensation and s the stimulus, the psychophysical law of WeberFechner is given by

M = a log s + b, a ≠ 0

We assume that the stimuli arise in making pairwise comparisons of relatively comparable activities. We are interested in responses whose numerical values are in the form of ratios. Thus b = 0, from which we must have log s 0 = 0 or s 0 = 1, which is possible by calibrating a unit stimulus. This is done by comparing one activity with itself. The next noticeable response is due to the stimulus

This yields a response log α/log α = 1. The next stimulus is

which yields a response of 2. In this manner we obtain the sequence 1, 2, 3,... . Our ability to make qualitative distinctions is well represented by five intensities: equal, moderate, strong, very strong, and extreme. We can make compromises between adjacent intensities when greater precision is needed. Thus we require nine values which, according to the previous discussion, should be consecutive. The following refinement of the above is also possible: equal, tad, weak, moderate, moderate plus, strong, strong plus, very strong, very very strong, and extreme. The resulting scale would then be validated in practice.

We use the scale of absolute values shown in Table 3.1 to make the comparisons. We deal with widely varying measurements of alternatives by grouping those of similar magnitude in clusters and then uniformizing the measurement with a pivotal alternative that appears in two adjacent clusters. We do this because in using judgments, people are usually unable to accurately compare the very small with the very large. But they can make the transition gradually from clusters of smaller elements to clusters of larger ones. This approach is the valid way to extend the 1-9 scale as far out as one wants. In any case one does not need to go too far out on the scale to set priorities against one's personal goals. Thus the problem is to find the right numbers to represent comparisons of objects that are close on some property. The 1-9 scale is a simple scale that serves well.

If the need arises for a judgment larger than 9, the larger element could be placed in another homogeneous set of comparisons and the 1-9 scale is also applied to that set. Clustering is used to combine different homogeneous groups. When we have a situation like a i preferred to a j by 3 and a j preferred to a k by 5, implying that a i is preferred to a k by 15, we need not conclude that there is a need for a wider scale but that there is inconsistency in the judgments already given, because we know from the homogeneity requirement that there is no need for a number outside the scale. No matter what finite scale one chooses to represent the outer limit of perception, 1 to 9 or 1 to α n , inconsistency might seemingly require an even larger number. Instead of extending the scale one should look for better understanding of the inconsistency of judgments.

Table 3.1 The Fundamental Scale

Intensity of Definition

Explanation

Importance 1 Equal Importance

Two activities contribute equally to the objective

2 Weak

3 Moderate

Experience and judgment

importance

slightly favor one activity over another

4 Moderate plus

5 Strong importance

Experience and judgment strongly favor one activity over another

6 Strong plus

7 Very strong or

An activity is favored very

demonstrated

strongly over another; its

importance

dominance demonstrated in practice

8 Very, very strong

9 Extreme importance The evidence favoring one

activity over another is of the highest possible order of affirmation

Reciprocals If activity i has one

A reasonable assumption

of above of the above nonzero numbers assigned to it when compared with activity j , then j has the reciprocal value when compared with i

Rationals Ratios arising from

If consistency were to be forced

the scale

by obtaining n numerical values to span the matrix

There are many situations where elements are close or tied in measurement and the comparison must be made not to determine how many times one is larger than the other, but what fraction it is larger than the other. In other words there are comparisons to be made between 1 and 2, and what we want is to estimate verbally the values such as 1.1, 1.2, ..., 1.9. There is no problem in making the comparisons by directly estimating the numbers. Our proposal is to continue the verbal scale to make these distinctions so that 1.1 is a "tad", 1.3 indicates moderately more, 1.5 strongly more, 1.7 very strongly more and 1.9 extremely more. This type of refinement can be used in any of the intervals from 1 to 9 and for further refinements if one needs them, for example, between 1.1 and 1.2 and so on.

Let us note that we are unable to distinguish between object sizes as they become very small or very large. Sometimes we can with the aid of an instrument like the microscope or the telescope which bring things to our range of abilities, but we must then relate the new magnitudes to those we know best in our daily experience. The idea of a logarithmic scale arises from saturation of the ability to make distinctions which happens at both ends of the scale. What we have done with the 1-9 scale, which we are compelled to use, as we can use no other scale with clear understanding of magnitudes, is to piecewise linearize the logarithmic idea in a limited operational range of magnitudes.

Naturally if one uses actual measurements to form the ratios, one gets them back by solving for the derived scale because in the consistent case one gets back exactly what one puts in. In fact Expert Choice, the implementation software package of AHP, uses fractional values. They allow one to put very close approximations to whatever number one may think of between 1 and 9. But what arbitrary values would one assign to feelings. It is better to have a well tested protocol to go from comparisons to words to numbers validated to work in known situations, than to guess at widely disparate numbers assigned to firm judgments associated with perception.

There are undoubtedly a few situations with known scales whose numbers can be used by an experienced person to form ratios and one need not use the scale 1-9. But AHP was developed to set There are undoubtedly a few situations with known scales whose numbers can be used by an experienced person to form ratios and one need not use the scale 1-9. But AHP was developed to set

he is well equipped to make unambiguous distinctions.

Figure 3-1 Pairwise c ompare five areas for size

Figure 3-1 gives five areas to which the paired comparison process and the scale can be tested. One may approximate the outcome by adding the rows of the matrix and dividing by the total. Compare the answer with A = .471, B = .050, C = .234, D = .149, E = .096. We have considered the use of all kinds of scales other than the 1-9 scale. One particular

0 instance is the power scale, 1 α α , ..., , α n .

There are several problems in using such a scale. Clearly

2 3 one can choose the numbers 1, 2, 2 2 , 2 , or 1, 3, 3 , or similar subsets that are already in the 1-9 scale.

A rule is needed to identify the value of α, the base in the geometric scale, to be associated with a verbal expression. Note that if we find one counterexample that produces a poor result with such a scale, for whatever α one may choose, then we would no longer be tempted to try a power scale. It is difficult to see how a power scale would be as natural a representation of the semantic scale when comparing homogeneous alternatives.

If we extend the 1-9 scale in an extreme case to 100, the scale 1, α, ... , α n with n = 100 can give astronomically large values even for small values of α near one. We not only have difficulty estimating

such values, but lower down on the scale we could have a problem distinguishing between some of the small values.

The last value is beyond our ability to compare it semantically, say with 2 10/2 . If instead of α = 2 we use α = 1.009 for example, it is difficult to distinguish between the scale values for very small n.

When the use of one of the two scales gives rise to a consistent matrix, the use of the other would make it inconsistent. There are situations where both matrices are inconsistent and there is reversal in the ranks of the first and second alternatives as represented by the two eigenvectors as in the following example

Scale Matrix Eigenvector

1-9 1/2 1 6 1 .2939 1/3 1/6 1 1/3 .0808

C.R. = .062

Scale Matrix Eigenvector

C.R. = .065