Fuzzy Sets and Systems 1 (1978) 57-68. © North-Holland Pubi,,,hing Company

E X P L O R I N G T H E I N T E R F A C E B E T W E E N H I E R A R C H I E S , M U L T I P L E O B J E C T I V E S A N D F U Z Z Y SETS

Thomas L. SAATY Wharton School, University of Pennsylvania, Philadelphia 19174, U.S.A.

Received January 1977 Revised June 1977

The complexity of experience acqvired through our senses and as interpreted by our mind, is fuzzy and must remain so as long as the meaning of things change as they are embedded in larger or different contexts to relate them to new ideas and new experiences. Here we give a method for measuring the relativity of fuzziness by structuring the functions of a system hierarchically in a multiple objective framework. The method is based on computing the principal eigenvector of a positive matrix with reciprocal entries (i.e., aji = l/ao). The eigenvector provides an estimate for an (assumed) underlying ratio scale. For a single property the scale provides a measure of the grade of membersltip of elements in a fuzzy set according to that property and the departure of the eigenvalue from the dimension of the matrix serves as a measure of departure from consistency. For a number of properties, the principle of hierarchical composition enables us to compose the eigenvectors into a priority vector which measures the fuzziness of the elements in the lowest level of the hierarchy with respect to the relative dominance of the purposes or properties represented in the hierarchy.

1. Introduction

The French essayist Montaigne warned "There never were in the world two opinions alike".

By being democratic to accord other people's opinion a place beside ours we add to the existing fuzziness. The result of this is to keep a many handed point of view. This kind of ambiguity or fuzziness we regard as good because it encourages diversity. Our problem is to find out what we understand by fuzziness and learn to deal with it to our advantage.

Two types of fuzziness concern us here. They are fuzziness in perception and fuzziness in meaning. I will address both types and show that we have a powerful methodology today for coping with fuzziness. It is the theory of prioritized hierarchies.

Fuzziness is a property arising out of complexity and out ofour limited capacity to deal with complexity. Every system has a structure and a function both of which have fuzziness associated with them. We can reduce functional fuzziness more easily than we can structural fuzziness. We understand things fuzzily because they do not have a sharp meaning by themselves but through interaction with other things. We cannot apprehend at once the totality of objects and interactions but need to decompose them. When we decompose them they appear fuzzy because they have different meanings

57

58 T.L. Saaty

according to the context of the decomposition. An individual is a fuzzy idea because he cannot be grasped at once. As we focus on him we see different significances. They only relate well if the objectives are considered together.

Any object we talk about genetically is a fuzzy concept. When we say cow, it could be a large cow or a small one, a healthy cow or a thin cow, a pregnant cow or an old cow. Because of our fertile imagination we are bombarded simultaneously with many ideas and properties of the same object.-Even when we speak of the same cow, we have it in mind yesterday or today, in the barn or out grazing. If we were to specify all the conditions we have in mind when we mean "that" cow, we would run out of time and serve no good purpose. Even then we can never speak of the same thing at all times since time changes and the object may occupy different positions in space. In sum, we always need a way to get at this elasticity in meaning, tightening and increasing its specificity to uniformize it across experience, and operationalize it for practical usage.

The problem is compounded with the use of a highly limited language in which the same word is used for many purposes whose meaning, no matter how standardized, evokes different feelings and hence different meaning and interpretation from individual to individual and sometimes in the same individual.

The boundaries of properties and even objects appear fuzzy (where fuzziness really concerns us) because they are a part of a continuum whose boundaries merge into one another. Why don't we just discard the idea of boundaries? We have to keep it because we need to decompose systems into parts so we can understand what flows into them and out of them at what we call boundaries. It is this analytic process of discretization and decomposition that creates problems and difficulties for us. if we wish to cope with '.,i,- type of fuzziness we must treat reality as a whole in systems terminology and minimize decomposing it into parts.

The problem of boundaries has to do with the degree of belongingncss to sets. Thus the fuzziness of objects as we probe deeper and deeper into o u fuzzy perceptions of them turns into a question of meaning.

For each element of a set the question is not whether it has or ',:oes not have a given property (e.g., a rose being red or pink) but how strongly it has ::he property. A fuzzy pcoperty such for example as beauty is an aggregation, or more sp::cifically a hierarchy, of various properties of different importance.

We may distinguish between two types of boundary propertie~: those amenable to direct quantitative measurement and those that are qualitative and require a framework of comparison with other properties in order to measure their relative standing with regard to a given property. When dealing with a measurable property such as height it should be possible by direct measurement to determine the grade of membership or relative height of an individual member of a set with respect to all other elements. This type ofapproach sounds very much like relative frequency measurement and hence also like probability measurement. There is no fuzziness about height (despite the possible error) because in the final analysis it is defined in precise measurement terms. Now let us look at beauty. Unlike height or length, beauty is not an elementary but a composite concept. In addition there are practically no universal elementary measurable properties in terms of which beauty is defined. Even if such elementary properties did exist, a measurable functional relationship relating them to beauty as the dependent variable has been unavailable. Yet we not only can tell what is

Exploring the interface between hierarchies 59

beautiful but we can also say that certain things are more beautiful than others. Beauty is a fuzzy concept which is not amenable to the type of measurement that would enable us to construct a meaningful relative frequency curve which would serve as a representative for our subjective probability estimates of the grade of membership [2].

Objects and ideas have no meaning in themselves apart from how we experience and interpret them. It appears that meaning is tied to what function objects perform in the fulfillment of different purposes. The meaning is clearer when the purpose is unique. But most of ~he materials of experience can be viewed in different ways as they fulfill different purposes. An experience (object or idea} looks fuzzy by itself bu t less fuzzy in terms of each purpose taken separately. Now purposes as part of experience are themselves fuzzy except when they are part of still higher purposes.

It follows that fuzziness is a basic quality of understanding and canno! be done away with. What we need is to recognize this relativism of meaning in the context of increasing purposes.

Next we note that despite dedication to purpose our feelings themselves arc imprecise and flexible. That this is a good thing which we need to carefully refine may be explained as follows. Our system of feelings has to adapt itself to interpret the experiences with which it is faced. It would be too bad if our minds were precise measuring instruments because they would limit the way we view most of the materials ihat we have io work with. Our judgment and feelings which discriminate between experiences have flexible ranges which often allow instantaneous identification of a pt~rp6se for what we experience or reinterpret the function of objects to fit the current nceds of experience. It is likely that the greatest asset we have is this adaptive ability. If our mind were a precise measuring instrumepk it could not become unstuck fast enough from one experience and adapt its framework for a new one happening side by side wilh the first but having nothing to do with it.

What really matters to us is how understanding and knowledge can be used to achieve whatever purposes we wish to pursue. Man can use a fuzzy world to serve his well-defined and unfuzzy needs. When he is hungry he eats and is satisfied; when ill with some diseases he takes antibiotics which cure him; when he has difficulty, he sometimes uses his mind to surmount it successfully and he is happy. What counts is that we perceive and interpret experience within out limited framework to serve our needs. Although we may never know the world as it really is we can know it to the extent that it fulfills our needs and helps to expand our experience through interaction and adaptation. This tells us that to decrease fuzziness in functional interpretation of reality we need clear definitions of our purposes and how they relate to the real world.

We will now discuss a method to give meaning to all this fuzziness. Any system can be functionally represented in terms of a hierarchy of purposes. By means of a hierarchy we can view the functions of the components of a system according to priority with regard to multiple purposes or objectives. The levels of the hierarchy define the priority classes. Within each level these priorities may differ but more significantly, they differ from level to level by increasing orders of magnitude.

The experience which we wish to evaluate usually occupies the lowc~;, level and we seek a measure of its impact on this hierarchy of purposes. In this manner we give it meaning within that context. We never can interpret it universally because there are an infinity of other hierarchies in which it may Conceivably fit. However, we should

60 T.L. Saa ty

concentrate on the hierarchy which we consider tile most relevant in our view for that experience•

Thus for us a fuzzy property i~ a system of properties which are aggregated in levels with higher levels dominating lower ones. A fuzzy property is a hierarchical system of properties. The lowest level of the hierarchy is reached when we run out of both imagination and experience. That is the hierarchy with which we would work.

Because of our limited discrimination capacity, even decomposition is complex unless we are dealing with a situation which does not have too many elements. Therefore the hierarchical structure is very important as it allows us to focus on a few elements at a time in each level.

The way to handle fuzziness so that one can determine a course of action in spite of it is to structure a hierarchy. We need a brief discussion of a new theory for deriving ratio scales and of the numerical scale to be used in representing judgments. An example to illustrate the idea is given. It is then followed by some relevant formalism about hierarchies. Finally all these ideas are applied to two examples of fuzziness within a hierarchical setting.

2. Ratio scales from reciprocal pairwise comparison matrices

Suppose we wish to compare a set of n objects in pairs according to their relative weights. Denote the objects by A 1,..., A, and their weights by w~,...,w,. The pair-wise comparisons may be represented by a matrix as follows:

At

A2

A.

At A2 ... A.

W t W t W I

W I W2 W n

w2 w2 w2 • ,

WI W 2 W n

W n W n W n

W t W 2 W n

This matrix has positive entries everywhere and satisfies the reciprocal property aji = I/a~. It is called a reciprocal matrix. We note that if we multiply this matrix by the transpose of the vector w r = (wt,..., w, ) we obtain the vector nw.

Our problem takes the form

a w -- ~w.

We started out with the assumption that w was given, But if we only had A and wanted to recover w we would have to solve the system (A - n l ) w = O in the unknown w. This

Exploring the interface between hierarchies 61

has a nonzero solution if and only if n is an eigenvalue of A, i.e., it is a root of the characteristic equation of A. But A has unit rank since every row is a constant multiple of the first row. Thus all the eigenvalues 2~, i = 1,..., n of A are zero except one. Also it is known that

!1

2~ = tr(A)=sum of the diagonal elements = n. i=!

Therefore only one of 2i, we call it 2m.~, equals n, and

2 i - -0 , "~'i ~: 2max"

The solution w of this problem is any column of A. These solutions differ by a multiplicative constant. However, it is desirable to have this solution normalized so that its components sum to unity. The result is a unique solution no matter which column is used. We have recovered the scale from the matrix of ratios.

The matrix A satisfies the cardinal consistency property aoatk = aik and is called consistent. For example if we are given any row of A, we can determine the rest of the entries from this relation. This also holds for any set of n entries whose graph is a spanning cycle of the graph of the matrix.

Now suppose that we are dealing with a situation in which the scale is not known but we have estimates of the ratios in the matrix. In this case the combined consistency relation (elementwise dominance) above need not hold, nor need an ordinal relation of the form: Ai> A t, At> Ak imply A,> A k hold

As a realistic representation of the situation in preference comparisons, we wish to account for inconsistency in judgments because, despite their best efforts, people's feelings and preferences remain inconsistent.

We know that in any matrix, small perturbations in the coefficients imply small perturbations in the eigenvalues. Thus the problem Aw=nw becomes A'w'=2,~axW'. We also know from the Perron-Frobenius theorem that a matrix of positive entrie.~; has a unique real positive eigenvalue whose modulus exceeds those of all other eigenvalues. Some of the remaim~g eigenvalues may be complex. The corresponding eigenvector solution has nonnegative entries and when normalized is unique.

Suppose then that we have a reciprocal matrix. What can we say about an overall estimate of inconsistency for both small and large perturbations of its entries ? In other words how close is 2max to n and w' to w ? If they are not close we may either revise the estimates in the matrix or take several matrices from which the solution vector w' may be improved. Note that improving consistency does not mean getting an answer closer to the "real" life solution. It only means that the ratio estimates in the matrix, as a sample collection, are closer* to being logically related than to being randomly chosen.

From here on we shall use A={ao) for the estimating matrix and w for the eigenvector. There should be no confusion in dropping the primes.

It turns out that a reciprocal matrix A with positive entries is consistent if and only if /~max =rl . With inconsistency 2ma~>n always. One can also show that ordinal consistency is preserved, i.e., if Ai>A~ (or aik>ajk, k = 1,...,!1) then w~ > w~. Finally the

62 ZL. Saaty

departure of 2m~ from n can be shown to be a measure of increased inconsistency which calls for revising judgments in the matrix.

As outlined in my paper on fuzziness [2!, the scale which I have been using (to derive ratio scales from matrices of pairwise comparisons) ranges from 1 to 9 to represent judgment entries as follows:

1: equally important 3: weakly more important 5: strongly more important 7: demonstratedly more important 9: absolutely more important

2, 4, 6, 8 are values for compromise in judgment of importance between 1 and 3, 3 and 5, 5 and 7, 7 and 9 respectively. Given a o, we enter the reciprocal value aii= l/a o. Thus also a , = 1 always.

2 he question to ask in a matrix of pairwise comparison is: when the element on the left side of the matrix is compared with an element on top, how much more strongly does it have the property in question.

This scale has been compared with dozens of other scales. It seems to come the closest to representing our judgment about reality when compared with actual measures of reality already known. The subject will be explored extensively in a forthcoming book ~McGraw-Hill~. See also [3].

2.1. Illumination intensizy and the inverse square law The rate at which a source emits light energy evaluated in terms of its visual effects is

spoken of as light flux. The illumination of a surface is defined as the amount of light flux it receives per unit area.

The following experiment was conducted in search of a relationship between the illumination received by four identical objects (placed on a line at known distances from a light source) and of the distance from the source. The comparison of illumination intensity was performed visually and independently by two sets of people. The objects were placed at the following distances measured in yards from the light source: 9, 15, 21 and 28. In normalized form these distances are: 0.123, 0.205, 0.288, 0.384.

The two matrices of pairwise comparisons of the brightness of the objects labelled in increasing order according to their near:less to the source where the judges were located are:

Relative visual brightness (lst Trial)

Relative visual brightness (2nd Trial)

Cl

C2

C3

C,

C~ C2 C3 C4 1 5 6 7 C~

1/5 1 4 6 C_,

1/6 1/4. 1 4 C3

1/7 1/6 1/4 1 c,

C1 C2 Ca C4 1 4 6 7

1/4 1 3 4 1/6 1/3 1 2

1/7 1/4 1/2 1

Exploring the interface between hierarchies 03

Relative brightness Relative brightness eigenvector eigenvector (lst Trial) (2nd Trial) 0.62 0.63 0.23 0.22 0.10 0.09 0.05 0.06 2 =4.39 2 =4.1

Normalized reciprocal distance square

0.61 0.22 0.11 0.06.

The first two columns should be compared with the last column calculated from the inverse square law in optics

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 hypothesis that the observed intensity of illumination varies (approximately) inversely with the square of the distance. The more carefully designed the experiment the better results obtained from the visual observation. This example shows that judgment can in fact capture natural law. Thus our proposed method of measuring fuzziness gives answers which relate somewhat objectively to the situation.

The normalized antipriority vector of the brightness of the chairs is obtained by solving the left hand eigenvalue problem. It is given by:

0.05 0.14 0.29 0.52.

This is the dual vector to the dominance priority vector. It provides a measure of how strongly each chair is relatively dominated in brightness.

In reference [3-1 we have given a mathematical definition of a hierarchy. It is essentially a formalization in terms of partially ordered sets of our intuitive understandin~ of the idea. It has levels; the top level consists of a single element and each element of a given level dominates or covers (serves as a property or a purpose for) some or all c,f th~ ~, elements in the level immediately below. The pairwise comparison matrix approach is then applied to compare elements in a single level with respect to a purpose from the adjacent higher level. The process is repeated up the hierarchy and the problem is to compose the resulting eigenvectors in such a way as to obtain one overall priority vector of the impact of the lowest level elements on the top element of the hierarchy by successive weighting and composition. We now simply show how this is done between two adjacent levels and the reader should have no problem in generalizing it to the entire hierarchy. In the same reference we have proved that this process of hierarchical composition preserves ordinal preferences among the elements in a level, thus it appears to be a very good way of composing the eigenvectors. We use the symbol Lk to represent the kth level of a hierarchy of h levels. The elements of this level cover those of L k + 1 and are covered by the el~ements of Lk- 1.

64 TL. Saaty

Assume that Y = {yl,.. . ,ymk}eLk and that X = {Xl , . . . ,Xmk+~}eLk+ 1" Also assume that there is an element z e Lk-1 such that Y is covered by z. Then we consider the priority functions

w_,'Y~[O, 1] and wr:X~[O, 1 ] j= l , . . . ,mk .

We construct the "priority function of the elements in X with respect to z," denoted w, w:X ~[0 , 1], by

m k

j = l i - " 1 , . . . , ink+ 1.

It is obvious that this is no more than the process of weighting the influence of the element Yi on the priority of xi by multiplying it with the importance of yi with respect to z.

The algorithms involved will be simplified if one combines the wyj(x~) into a matrix B by setting bo=wrj(xi). If we further set W~=w(xi) and W~=wz(yj), then the above formula becomes

ra k

Wi= Z boWj j=t

i - 1,..., mk+ 1.

Thus, we may speak of the priority vector W and indeed, of the priority matrix B; this gives the final formulation W = BW'.

A hierarchy is complete if all x e L k are dominated by every element in Lk-1, k = 2,..., h. The following is easy to prove"

Theorem 2.1. Let H be a complete hierarchy with largest element b and h levels. Let Bk be the priority matrix of the kth level, k = 1,..., h. I f W' is the priority vector of the pth level with respect to some element z in the ( p - 1 )st level, then the priority vector w of the qth level (p < q) with respect to z is given by

• • W t. W=BqB~_I "Bp+l

Thus, the priority vector of the lowest level with respect to the element b is given by:

W =BhBh_ 1 " ' " B2W'.

The following observation holds for a complete hierarchy but it is also useful in general. The priority of an element in a level is the sum of its priorities in each of the subsets to which it belongs, each weighted by the fraction of elements of the level which belong to that subset and by the priority of that subset. The resulting set of priorities of the elements in the level is then normalized by dividing by its sum. The priority of a subset in a level is equal to the priority of the dominating elea~ent in the next level.

Exploring the interface between hierarchies 65

2.2. School selection

Three highschools--A, B, C--were analyzed from the standpoint of a candidate according to their desirability. Six characteristics were selected for the comparison. They were: learning, friends, school life, vocational training, college preparation and music classes. The pairwise judgment matrices are given below:

Comparison of characteristics with respect to overall satisfaction with school

Learning Friends School Vocational College Music life training preparation classes

Learning 1 4 3 1 3 4 Friends 1/4 1 7 3 1/5 1 School life 1/3 1/7 1 !/5 1/5 1/6 Vocational

training 1 1/3 5 1 1 1/3 College

preparation 1/3 5 5 1 1 3 Music classes 1/4 1 6 1/3 1/3 1

The pairwise judgment matrices of comparison of schools with respect to the six characteristics are given in matrices (iHvi).

Learning A B C

a 1 1/3 1/2

3 1 3 1 1/3 1

(i)

A

B

C

Friends A B C

1 1 1 A

1 1 i B

1 1 1 c (ii)

School life A B C

1 5 1

1/5 1 1/5 1 5 i (iii)

Vocational training A B C

1 9 7 A

1/9 1 1/5 a 1/7 5 1 c

(iv)

College preparation

A B C

1 1/2 1 A

2 1 2 B 1 1/2 l C Iv)

Music classes

A B C

1 6 4

1/6 1 1/3 1/4 3 1

(vi)

The eigenvector of the first matrix is given by:

(0.32, 0.14, 0.03, 0.13, 0.24, 0.14)

and its corresponding eigenvalue is 2 .--- 7.49 which is far from the consistent value 6. No revision of the matrix was made. Normally such inconsistency would indicate that we should reconsider the array elements. The eigenvalues and eigenvectors of the other six

66 T.L. Saaty

matrices are"

2 = 3.05 2 = 3 2 = 3 2 = 3.21 2 = 3.00 2 = 3.05

Learning Friends School life

0.16 0.33 0.45 0.59 0.33 0.09 0.25 0.33 0.46

Vocational College Music training preparation

0.77 0.25 0.69 0.05 0.50 0.09 0.17 0.25 0.22

To obtain the overall ranking of the schools, we multiply the last matrix on the right by the transpose of the vector of weights of the characteristics. This yields:

A =0.37 B =0.38 C =0.25.

The individual went to school A because it had almost the same rank as school B, yet school B was a private school charging close to $1600 a year and school A was flee. This is an example where we were able to bring in a lower priority item e.g. the cost of the schooi to add to the argument that A is favored by the candidate. The actual hierarchy is shown in Fig. 1.

2.3. Choosing a job

A student who had just received his Ph.D. was interviewed for three jobs. His criteria for selecting the jobs and their pairwise comparison matrix are given on the next page.

SATISFACTION WITH SCHOOL

A B C

Fig. 1. The school satisfaction hierarchy.

Exploring the interface between hierarchies 67

Overall satisfaction with job

Research Growth Benefits Colleagues Location Reputation

Research 1 1 1 4 1 1/2 Growth 1 1 2 4 1 1/2 Benefits 1 1/2 1 5 3 1 / 2 Colleagues 1/4 1/4 1/5 1 1/3 1/3 Location 1 1 1/3 3 1 1 Reputation 2 2 2 3 3 1

The pairwise comparison matrices of the jobs with respect to each criterion are given in matrices {vii)-lxii).

A

B

C

Research A B

1 1/4 4 1 2 1/3

{vii)

C

1/2 a

3 B 1 C

Growth A B C

A 1 1/4 1/5

B 4 1 1/2

C 5 2 1 (viii)

Benefits A B

1 3

1/3 l 3 1 (ix)

C

1/3 1

1

Colleagues A B

1 1/3

3 1 1/5 1/7

ix)

C

5 A

7 B 1 c

Location A B

1 1

1 1

1/7 1/7 (xi)

C

7 A 7 B 1 C

Reputation A B

1 7

1/7 1 1/9 1/5

lxii)

The eigenvalue and eigenvector of the first matrix are, respectively"

2max=6.35 0.16 0.19 0.19 0.05 0.12 0.30.

The remaining eigenvalues and eigenvectors are given by:

m a x - "

Research Growth Benefits Colleagues Location Reputation 3.02 3.02 3.56 3.06 3 3.21 0.14 0.10 0.32 0.28 0.47 0.77 0.63 0.33 0.22 0.65 0.47 0.17 0.24 0.57 0.46 0.07 0.07 0.05.

68 T.L. Saaty

The composite vector for the jobs is given by

A =0.40 B =0.34 C =0.26.

The differences were sufficiently large for the candidate to accept the offer to job A.

3. Conclusion

We conclude by again drawing attention to the fact that our estimate of an underlying ratio scale together with our measure of consistency investigated in depth elsewhere [3], offer a meaningful way for measuring fuzziness. The unidimensionality of the final scale vector through the use of a hierarchical structure describing fuzzy phenomena is of considerable use. We have a good way to represent fuzziness which reveals properties of consistency, stability and pareto optimality. It also offers an adaptive framework for collective bargaining both to structure a problem and provide numerical judgments. Many applications of the theory have been made among which are those in the design of a transport system for the Sudan in 1985 [6], by establishing priorities for the projects with respect to economic social and political factors. Another application has been made to the allocation of electric energy to industries according to their contributions to the economy, health, national defense, etc. Yet another example is in the area of conflict resolution i, Northern Ireland, and the solution was communicated to various leaders of the conflicting parties [1]. Finally, two more applications of the theory have been made to study world influence of nations and the future of higher education in the U.S. in 1985 [4, 5].

References

[ 1 ] J.M. Alexander and T.L. Saaty, The forward and backward processes of conflict analysis, Behavioral Science (March 1977).

[2] T.L. Saaty, Measuring the fuzziness of sets, J. Cybet net. 4 (1974) 53-61. [3] T.L. Saaty, A scaling method for priorities in hierarchical structures, J. Mathematical Psychology I June

1977). [4] T.L. Saaty and M. Khouja, A measure of world influence, Peace Science (June 1976). [5] T.L. Saaty and P. Rogers, The future of higher education in the United States in 1985, Socio-Economic

Planning Sciences (December 1976). [6] T.L. Saaty, The Sudan transport study, Transportation Research (to appear).