98 European Journal of Operational Research 48 (1990) 98-104 North-Holland

Physics as a decision theory

Thomas L. Saaty University of Pittsburgh, Pittsburgh, PA 15260, USA

Abstract: We show that there is a close mathematical relationship between physics and the Analytic Hierarchy Process. We argue that numerical scales used in physics must be interpreted in terms which the scientist understands through experience and through theories advocated by experts in the field. In physics there are pr imary variables and secondary variables defined in terms of them, all measured on ratio scales. We show that hierarchic composition in the AHP works in a similar way to physics and we illustrate this with an example.

Keywords: Decision, physics, scaling, measurement, optics

1. The measure of all things - Human judgment

All measurement data, whether in physics, en- gineering or sociology must be interpreted to be understood. Such numbers describe the degree of a property an object or event possesses; how fast, how long, how poor. Numbers tell us how much more of the property an object has one day than another; or how much more it has it than another object, or how much more or less than a certain standard.

Our ability to assess the meaningfulness of measurement is limited. For example, beyond cer- tain cold temperatures we observe in everyday experience, we have no idea how much colder - 160 o C is than - 140 ° C. One is a greater or smaller refinement than the other, but we have no feeling for it. On the other hand, a difference of 20 ° in our range of sensation has much more meaning to us. The temperature reading of 50 ° is more comfortable than 30 °, and 100 o is much less comfortable than 80 °. Even what is a comfor- table temperature depends on whether we are ac- customed to the weather in New York, Siberia or Kenya. Understanding measurements depends on our experience and perception acquired through living, learning and training. The significance of measurements on different scales is a phenomenon

Received November 1989

cultivated in us through conditioning. It has no significance in itself. Our conclusion is that we always interpret the meaning of data subjectively, as we interpret other stimuli with our senses--such as how bright light is to the eye or how soft velvet is to the touch. The basic problem is to create a scientific framework for interpreting data.

The purpose of this paper is to demonstrate the close analytical relationship between the Analytic Hierarchy Process (AHP) and physics. The AHP is a decision theory that directly interprets data and information by forming judgments and per- forming ratio scale measurement on them within a prescribed hierarchical framework. We illustrate this relationship with the following experiment.

2. Illustrating the connect ion - The inverse square law

Four identical chairs were placed on a line from a light source at the distances of 9, 15, 21, and 28 yards. The purpose was to see if one could stand by the light and look at the brightness of the chairs and compare their relative brightness in pairs, fill in a judgment matrix and obtain a relationship between the brightness of the chairs and their distance from the light source. The ex- periment was performed twice with different judges whose judgment matrices are given below.

0377-2217/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)

T.L. Saaty / Physics as a decision theory 99

Relative visual

cl 1 5 6

c2 1 4

1 1 G ~ _~ 1

1 1 1

Relative visual

C1 C2 Ca

G 1 4 6

c2 k 1 3

1 1 c3 ~ 7 1

1 1 l cn ~ z 7

brightness (1st trial):

c4

brightness (2nd trial):

c4

i) The judges of the first matrix were the author's

young children, ages 5 and 7 at the time, who gave their judgments qualitatively. The judge of the second matrix was the author's wife, who was not present during the children's judgment process. The synthesis of the judgments obtained in the two trials are given by the principal right eigenvec- tors of the following matrices.

Relative brightness eigenvector (1st trial):

0.61

0.24 Xma × = 4.39, 0.10 ' 0.05

CI = 0.13, CR = 0.14.

Relative brightness eigenvector (2nd trial):

0.62 0.22 0.10 ' Xma X =4 .1 ,

0.06

CI = 0.03, CR = 0.03.

The first and second trial eigenvectors should be compared with the last column of the inverse square law table (Table 1) calculated from the inverse square law in optics. It is interesting to observe that the judgments have captured a natu- ral law. Note the sensitivity of the results as the object is very close to the source, for then it absorbs most of the value of the relative index and a small error in its distance from the source yields great error in the values. What is noteworthy from this sensory experiment is the observation or hy- pothesis that the observed intensity of illumina- tion varies (approximately) inversely with the square of the distance. The more carefully design- ed the experiment, the better the results obtained from the visual observations. A variety of similar measurements have been developed to ,give greater validity to this approach.

The paired comparisons are based on forming the ratio of the values for the two objects being compared, with the smaller or less dominant ob- ject serving as the unit and the larger or more dominant one as a multiple of that unit.

3. Multidimensional and unidimensional scales

Among the many possible ways to combine measurements belonging to several ratio scales, two have relevance for this disussion. One is to raise the measurements based on each ratio scale to an appropriate power, and then multiply the resulting weighted scales, thus obtaining a multidi- mensional scale. This is what is done in physics, by using the power law. For example in F = ma, the measurement of force is obtained by directly multiplying the measurements of mass and accel- eration whose units are L / T 2, a ratio of length

Table 1 The inverse square law table

Distance Normalized Square of distance normalized

distance

Reciprocal Normalized Rounding of previous reciprocal off column

9 0,123 0.015129 13 0,205 0.042025 21 0,288 0.082944 28 0,384 0.147456

66.09 0.6079 0.61 23.79 0.2188 0.22 12.05 0.1108 0.11

6.78 0.0623 0.06

100 T.L. Saaty / Physics as a decision theory

and time raised to the second power. Another example is E = m c 2.

The other way of combining measurements first interprets and rescales the magnitudes of measure- ments according to priority or importance with respect to each of a set of criteria, which in turn receive priorities with respect to a single overall goal. These priorities are then multiplied by the measurements with respect to the criteria. Finally, corresponding measurements with respect to each criterion are summed to obtain a unidimensional scale. This is what is done in the AHP approach to multicriteria decision-making.

Physics uses instruments to perform measure- ment. What the measurements signify must be interpreted according to a theory that is agreed upon by experts. Decision theory also transforms information and data to judgments within a frame of reference, a hierarchy of criteria and goals, of an individual or a group. We will show that the two ways of combining measurements are related in both structure and numerical representation.

Combining measurements in the Analytic Hierarchy Process gives rise to multilinear forms and to tensors. They are the basic formal interface between hierarchic measurement and physics where they play an important role in Maxwell's equations, in the General Theory of Relativity and in Differential Geometry. They are the abstract generalization of expressions which are obtained by multiplying and adding numbers from different scales in the process of synthesizing measure- ments.

4. Representing measurement: Multilinear forms and tensors Ill

There is an important formal area of mathe- matics where hierarchic decision-making and physics meet. It has to do with the representation of ratio scales as products of vectors in the various levels of the hierarchy. These transformations give rise to multilinear forms which correspond to mul- tiplication (weighting) and summation (composi- tion) of measurements on different attributes. Op- erating on measurement can be the process of weighting and adding to obtain a single number as one obtains in the AHP or in a physics formula.

A multilinear or a p-linear form on p vector spaces E 1 . . . . . Ep, defined over the same field K,

is a mapping f of E 1 . . . . . Ep into K such that if we fix all vectors except those of Ei, we obtain a linear form on Ei. When the vector spaces E 1 . . . . . Ep are identical (e.g., when the measure- ments belong to the same basic scale), we can form the tensor product f ® g of a mapping f of a p- form and a mapping g of a q-form. It is the ( p + q)-form defined by

( f ® g ) ( v 1 . . . . . Vp, /)p+l . . . . . Vp+q)

= f ( v , . . . . . Vp ) " g ( op + 1 . . . . . Up+q).

This product is associative and distributive with respect to the sum and also associative with re- spect to exterior multiplication. The tensor prod- uct of p forms that are linear is a p-linear form:

( f l ® " ' " ®fp ) (v t . . . . . op) = f , ( v , ) . . . f p ( V p ) .

Consider a p-linear form on E of dimension n related to a basis e~, and let f ( e , , . . . . . e~p) = t~, ... , .

i i Let v k = E ~ x k e k. Linearity enables us to write

f ( v l , . . . , V p ) = Y'~ xli,x2i2 . . . xipp ti 1 . . . tie. i l , . . . , ip

Consider a change of basis defined by uj = Zia~e~. Let

Tj, . . . . . Tip = f ( uj , , . . . . ujp)

= E a ~ a , ' " a ~ ; t i , ' " t , ; i l , . . . , ip

This transformation is a generalization of the for- mula for a linear form called a covariant vector and justifies the name of a covariant tensor of order p of the p-form. Here we have a natural way to see that the tensor calculus on a finite dimensional vector space can be characterized as a multilinear form on a product of identical spaces. A polynomial in one or more variables can be written as a multilinear form in which a variable raised to a power is expressed as a product of that variable as many times as its power. In this sense, multilinear forms are more general and have wider applicability than polynomials. It is known that polynomials can be used to approximate any con- tinuous function on a closed interval as closely as desired. Multilinear forms generalize on this con- cept to many dimensional spaces for approximat- ing functions that may not be continuous such as in the measurement and composition of ca- tastrophic events with respect to several criteria.

T.L. Saaty / Physics as a decision theory 101

5. The measurement link between hierarchies and physics [2,7,8]

Physics uses absolute measurement. The for- mula

M = a log s + b

describes the psychophysical law of Weber - Fechner concerned with scaling sensation as a function of stimulus. Here, a and b are constants, a ~ 0, s is the magnitude of a measurable stimulus and M is a quantity for scaling sensation. For example if I denotes the physical intensity of loudness, then the subjective impression of loud- ness as a result of experienced sensation is given by

L = a l o g I + b .

The constants a and b are fixed by requiring that at a frequency of 1000 Hz (cycles per second) the lowest intensity that can be heard is I 0 = 10 -12 W a t t / m 2 and one requires that the following be satisfied:

L = 10(log I - log I0) = 10 log I / I o.

For tones of frequency differing from 1000 Hz this formula cannot be applied because the sensi- tivity of the ear is not the same for tones of different frequencies. Here a subjective method of scaling is required when the tone deviates from 1000 Hz, or if there are mixed tones. This points to the need for a hierarchical structure to deal with measurement at varying thresholds.

As another readily understandable interpreta- tion, we note that despite precise subdivisions producing accurate measurement, the sensation of length also differs among people; the numerical values for length can be the same as measured by different people but their perception and interpre- tation can be considerably different; thus, re- sponse to the length stimulus depends on people's ability to sense and perceive it.

Let us now examine absolute measurement in hierarchical structures. The absolute measure- ments in one level B are multiplied (weighted) by the relative measurements in level A immediately above and the results are then summed for each element in B (a state variable) for which measure- ments are being carried out with respect to ele- ments (properties) in A.

If the sensations are given first, stimuli intensi- ties are obtained from them by exponentiation, the inverse of the logarithmic function. Thus, if xl, x 2 . . . . , x n are the measurements or ratings of each state variable with respect to the criteria, and if al, a 2 . . . . . a n are the relative measurement prior- ity of the ratings, then the overall measurement of each state variable with respect to the criteria is given by

a 1 log x 1 q-- a 2 log x 2 + • • • + a n log x n

log x i x i . i = 1 i = 1

Because this composite outcome is a logarithmic function, it represents the measurement of a com- posite sensation. It is sufficient to exponentiate

n at obtaining l-li=lx ~ for the actual intensities of the stimuli. This is precisely the kind of answer one obtains in physics.

One can show through dimensional analysis that the exponents a i, i = 1 , . . . , n, of the funda- mental variables are rationals. This is done by equating the powers of the fundamental variables appearing on both sides of the equation. Thus, in carrying out pairwise comparisons, the exponents are principal eigenvector coefficients arising from a consistent matrix. In principle, each exponent could be multiplied by the same normalizing con- stant to obtain integer values for all the expo- nents. When a variable appears in a formula with negative powers, it may be conceptually repre- sented in the hierarchy as the reciprocal of the original variable.

6. An example

An abstract hierarchy which corresponds to a physics problem consists of three levels. It is an incomplete hierarchy. The focus or goal is the new concept or composite variable defined by or de- composed into the second level elements. These are the secondary variables that are composites of primary variables. They are like the clusters or criteria of a decision hierarchy. Each secondary variable is defined in terms of either a primary variable or the reciprocal of a primary variable. In addition, a secondary variable can itself be a primary variable. The primary variables and their reciprocals are in the third level of the hierarchy.

102 T.L. Saaty / Physics as a decision theory

The hierarchy would only show those primary variables needed to define the secondary variables in the second level. They are measured on an absolute scale that is homogeneous, usually ex- tending from zero to infinity. Let us now consider how hierarchic composition gives the desired out- come. We must caution the reader that the hierarchical formulation will mimic physics knowledge. It cannot set down arbitrary judg- ments and magically lead to new physics formulas. We can also learn how to create structural links in domains whose relations and data are external to our feelings. Our purpose here is to illustrate the parallel in structure and calculation.

Since the power of a variable is a measure of the dominance of that variable with respect to other variables, we assume that it is meaningful to say that one variable is more important (has a higher priority) than another. The parallel be- tween physics and the AHP suggests that such importance is reflected in the power of the primary variable.

The period of a pendulum is given by

t = 2~Nrh/g.

Our object is to show that this relation can be derived hierarchically using knowledge from physics instead of subjective judgments. We first conjecture that t depends on the mass m of the bob of the pendulum, the height h of the pendu- lum, acceleration due to gravity g and the angle O. This leads to the hierarchy in Figure 1.

The composite concept or goal is the period t. It needs to be expressed as a composite function of the primary variables and only after that, these are grouped as secondary variables. Hierarchi- cally, the unknown priorities ka, kb, kc, kd, where k = 1/ (a + b + c + d), give rise to the con- sistency matrix of paired comparisons, as shown

Goal

Secondary Variables

Primary Variables

Composite Concept t x

m g h 0

M L 1/T

Figure 1. Hierarchy of physics variables

Table2

t m h g 0

m 1 a / b a / c a / d h b /a 1 b /c b /d g c /a c /b 1 c /d 0 d / a d / b d / c 1

in Table 2; ka, kb, kc, and kd are 'priorities' of the secondary variables, mass, gravity, and so on.

What does it mean to compare mass versus gravity for importance as we are asked to do in an AHP formulation? We believe it means this: Which is more important in determining the period (or time) of a pendulum, gravity or mass? And what does 'more important ' imply? If, for example, a 20% increase in mass affects t more than a 20% increase in gravity, then mass is a more important factor.

Because the matrix of paired comparisons is consistent, any column of this matrix gives the relative weights. Let us use the first column. If we divide each entry in a column by the sum of the column, we have for the relative values

ka, kb, kc, kd.

Next we determine the priorities of the primary variables in the third level under each secondary variable. These are absolute numbers known from the definition in physics of each secondary varia- ble and are shown in Figure 1. To clarify the ideas we list in a table all primary variables and their reciprocals and assign to each secondary variable the priority of its corresponding primary variables. For example, by definition, g = L / T 2 and in Ta- ble 3, we enter under g, 1 in the L row as L appears to the first power and 2 in the 1 / T row which appears to the second power and 0 elsewhere.

On noting that 1 / x = x - i , we can reduce this array to one involving the primary variables by

Table 3

m g h 0

M 1 0 0 0 1 / M 0 0 0 0 L 0 1 1 0 1 /L 0 0 0 0 T 0 0 0 0 1 / T 0 2 0 0

T.L. Saaty / Physics as a decision theo O' 103

Table 4

w g h 0

M 1 0 0 0

L 0 1 1 0

T 0 - 2 0 0

subtracting the rows with the reciprocal variable from the row above it. Clearly, the negative num- bers that appear indicate that the reciprocal varia- ble is used. See Table 4.

Finally, Fechner's formula says we perceive intensities of M, L and T, as the logarithms of the actual values. We then apply the composition principle of the AHP to these perceived values of the absolute measurements of M, L and T. We have, as one can surmise from the diagram,

k a . l . log M + k b . l . log L + k c . l . l og L

+ k b . 2 . log l / T = log M ~a L k(b+C) T -2kh .

To recover the actual measurement period t we take the exponential of this result. However, be- cause the actual measurement of t belongs to a ratio scale, we multiply by a constant K to write the following equality:

t = K ( M a L ( b + ' ) T - 2 h ) k

which is the most important result of this analysis. Here the parameters a, b and c are still unknown and must be determined. We do not involve the intensities of M, L and T in that process since their priority is defined by their power which is what we must find. We use knowledge from physics to equate powers of primary variables. We write

1 00 01(i/ = 0 1 1 0 • 0 - 2 0 0

We have on the left the dimensions of the period of the pendulum which are simply time. On the right we have the hierarchic composition of the priorities in the third and second levels, respec- tively. Dimensional equivalence (which in AHP jargon we call priority equivalence) of the two sides for each primary variable leads to the three equations

k a = O, k b + k c = O, - 2 k b = l ,

and their solution is b = - 1 / 2 k and c = 1 / 2 k . In dimensional terms, d = 0 because the angle 0 is

the ratio of the radius to the circumference and is dimensionless.

Having obtained the powers of the secondary variables, we can use them to relate the basic concept which is the period t of the pendulum to the product of the secondary variables because of the logarithmic relation. The final result is t = K h ~ . The value K is determined experimen- tally. Our final expression is t = 2,~ hv/~- ~ .

7. Conclusion

We have shown how the Analytic Hierarchy Process along with Fechner's formula can be used to generate expressions known in physics. This creates an opportunity for extending the basic approach of physics to include procedures to in- terpret the outcome of measurement. It can be done by extending the basic 'hard ' hierarchy to include judgmental criteria. Conversely, it has been well established in applications that whenever the AHP is used in scientific problems, there must be a level or levels which incorporate scientific data from the problem domain. These data cannot be simply replaced by the individual's judgments.

A fundamental difference in the structure of physics and of multicriteria processes is the Cos- mological Principle of physics which assumes that phenomena of any magnitude, at any point in space and time, can be directly compared using the same homogeneous scale of measurement. Were this principle to be rejected, a problem would have to be structured in greater detail with more levels in the hierarchy. In multicriteria decisions a hierarchic structure is needed to decompose com- plexity into levels of homogeneous entities such that magnitudes in one level differ by only one order from the magnitudes in the levels above or below it. Given that all learning takes place through gradual comparisons, it is difficult to accept the unrestricted use of homogeneous scales to measure all objects sharing an attribute. The question remains as to the wide disparity between how we can satisfactorily handle complex prob- lems at our level of understanding and how physics scales a gigantic universe without concern for de- composition and comparisons to link widely dis- parate entities as we find to be essential for any framework to interpret the meaning of measure- ment.

104 T.L. Saaty / Physics as a decision theory

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