higher education in the united states (1985–2000): scenario construction using a hierarchical...

Sow-Econ. Plan. Sci., Vol. IO, pp. 251-263. Pergamon Press 1976. Printed m Great Britain

HIGHER EDUCATION IN THE UNITED STATES (1985-2000)

SCENARIO CONSTRUCTION USING A HIERARCHICAL FRAMEWORK WITH EIGENVECTOR WEIGHTING

THOMAS L. SAATY

Department of Social System Sciences, The Wharton School, University of Pennsylvania, Philadelphia, PA 19174, U.S.A.

and

PAUL C. ROGERS

Department of Mathematics, University of Maine, Portland, ME 04103, U.S.A.

(Received 2 hly 1976)

Abstract-The object of this paper is to illustrate an application of hierarchies and eigenvalues developed by the first author to the area of planning in higher education, We start with a brief discussion of the role of scenario construction in planning and follow it with a discussion of the role of hierarchies in system planning. We then summarize the eigenvalue procedure in measurement and use it to study impacts in a hierarchy. We illustrate its use in elementary examples by way of validation. Finally, we use all the fore-going ideas to construct a composite and likely future for higher education in the United States around 1985.

I. INTRODUCTION

A useful tool rather widely employed in planning today is the method of scenario construction. The method of scenarios is a synthetic approach which stimulates coherently and plausibly, in a step-by-step fashion a sequence of events with emphasis on casual relation which direct a system to a future outcome among a set of possible outcomes. Scenario construction is a young art.

A scenario is a portrayal of the future with strong focusing on the particular idea or subject being emphas- ized (e.g. a transport system) with an “adequate” account of its interaction with environmental, social, political, technological and economic factors. It follows that a faithful scenario analysis must examine, in considerable depth, projections of all these factors in order to arrive at a convincing description of the state of the particular subject under various possible assumptions. In some of these approaches to scenario construction, one must guard against free use of uninhibited or undisciplined imagination and avoid falling into science fiction type of prognostication. There are two general types of scenarios.

A. The exploratory scenario The idea here is to explore in a set of trend-seeking

scenarios alternative futures, examining events that are logically necessary for a possible future by paramateriz- ing the principal components of the system under study. Its starting point is the present. Limiting scenarios are constructed in conjunction with trend-seeking scenarios to constrain the possible futures through paramatric variations and by careful examination of the hypotheses of evolution from the present. The exploratory scenario is often used as a technique to force the imagination, stimulate discussion and attract the attention of decision- makers to specific issues. The trend-seeking scenario does not make use of references to theory and methodology. Its practitioners, although they take its conclusions with a grain of salt, argue that so far as making errors in

predicting the future they are in good company with all the other methods.

B. The anticipatory scenario This approach is concerned with the conceptualization

of feasible and desirable futures. Unlike the exploratory scenario which proceeds from the present to the future, anticipatory scenarios follow the inverse path by starting with the future and working backwards to the present to discover what alternatives and actions (trajectory correc- tions) are necessary to attain these futures. There are two types of anticipatory scenarios: the normative scenario, which determines at the start a set of given objectives to be realized and defines a path for their realization (one version is to idealize the objectives and find technologi- cally feasible means with viable descriptions to reach them); and a contrast scenario which is characterized by a sketch of the desirable and feasible future which is on the boundary of the anticipatory scenario. Each contrast scenario emphasizes sharply a particular range of assumptions whose totality comprises the convex hull of the possible futures. Normative or contrast scenarios are synthesized into a composite scenario which retains the properties of each of the scenarios with appropriate mix of emphases. Since the future is shaped by a variety of forces or interests each seeking the fulfillment of its particular objectives, the synthesis of a wide ranging set of scenarios into a composite scenario must take into consideration the actors who influence the future, their objectives and the particular policies they will pursue in each scenario to fulfill their objectives. Thus, the normative process of constructing the composite scenario must reflect the priority of the actors according to importance to bring about a certain degree of fulfillment of the building blocks of each scenario. Thus, a major technical problem in scenario construction is how to construct a composite scenario from a large set of scenarios which defines the “cone” of the future.

251

252 T. L. SAATY and P. C. ROGERS

Some of the highlights of scenario construction proceed along the following lines:

Definition of the general system and of external and internal constraints and identification of subsystems; hierarchical structuring of subsystems and identification of regulating components; definitions of the states of the system and modelling its historical development; scenario treatment of the historical analysis high-lighting the system’s evolution and its impacts on characteristics of the society together with examination of the internal dynamics of the model; definition of the objectives of the scenario with a discussion of their values; choice of the types of scenario to be used; development of a data base of past, present and future information; identification of the structural components, factors which offset equilib- rium, evolutionary tendencies of the system; description of the tensions inherent in the functional mechanisms, analysis of the regulators of the system and of its coherence; critique and revision of the previous analysis, refinement of the scenario by examining constraints, disequilibrium, tensions, forces, contradiction, interven- tion of the regulators and put forward the contradictions which affect the survival of the system; produce an improved scenario.

Frequently the Delphi method of eliciting judgment from knowledgeable people is used to make a systematic identification of the elements of a scenario. The advan- tages and shortcomings of the method are discussed at length in Ref. [l] and will not be repeated here. The method is considered as an indispensible tool.

Probably, the best answer to the question of validation of the scenario approach is that it is a unique aid in forecasting the future. Its conclusions should be amenable to reasonable interpretation. The results derived from it for implementation should be categorized into urgency classes and only the most urgent projects implemented first and after a period, the planning process is then revised or iterated.

2. THE CONCEFKIAL FRAMEWORK

For the study of causal relations, a hierarchical structure is the single most powerful idea associated with systems. Hierarchic organization is crucial to the synthesis and survival of large complex systems. In addition, it is a source of isomorphism among the widest possible range of natural and man-made systems. Hierarchic systems have common properties that are independent of their specific content.

Any system is a large matrix of interactions between its components in which most of the entries are (close to) zero. Ordering those entries according to their orders of magnitude, a distinct hierarchic structure is discerned. In fact, this arrangement of the elements of a system in an incidence type matrix can be used to identify the levels of a hierarchy but what is needed is a theory for the measurement of interactions within the hierarchy. The stability, or functional efficacy of the higher level structures, can be made relatively independent of the detail of their microscopic components by virtue of hierarchic structure. Such a structure also provides explanation as to why elements in any given level can preserve a measure of independence to adapt their structure and function to the environment without destroying their usefulness to the system.

The laws characterizing different levels of a hierarchy are generally different. The levels differ both in structure

and function. The proper functioning of a higher level depends on the proper functioning of the lower levels. The basic problem with a hierarchy is to seek understand- ing at the highest levels from interactions of the various levels of the hierarchy rather than directly from the elements of the levels. At this stage of development of the theory the choice of levels in a hierarchy generally depends on the knowledge and interpretation of the observer. Rigorous methods for structuring systems into hierarchies are gradually emerging in the many areas of the natural and social sciences and, in particular, in general systems theory as it relates to the planning and design of social systems.

A hierarchy is a non-empty set H with a partition of H into disjoint subsets (called the levels of the hierarchy) such that H is a sup-lattice that is a chain with respect to a simple order on its levels and such that each element of each level which is not the sup is dominated by at least one element (called the boss) in the immediately higher level and each element of a higher level dominates at least one element of the immediately following level. Each level is partitioned into (not necessarily disjoint) subsets each consisting of all those elements dominated by each element of the immediately preceeding level. On each of these sets we have a total order (relating to the dominating element of the higher level) that is reflexive and anti-symmetric (but not necessarily transitive). A hierarchy is complete if each element of a given level is dominated by every element of the next higher level.

We shall return to hierarchies after we summarize a useful method of measuring the interaction between elements of a hierarchy level and then examine the measurement of the impact of each level on the hierarchy[2].

3.THE MODEL AND THE SCALE

In performing numerical pairwise comparisons between complex activities, ordinarily one has difficulty in translating feeling and experience to numbers which say exactly how much more one activity impacts on a given objective than another. The business of assigning numbers seems arbitrary, and the numerical approach to prioritization acquires the semblance of artificiality. It will be clear later that when comparison is equal, a unit value should be assigned. All other feelings should be equally spread starting from unity. Scales from 1 to 100 do not reflect this property if improperly used. Thus, 10 is ten times one, whereas 20 is two times ten and there is bias at the lower end of the scale. The numbers should also be such that one can distinguish between their reciprocals if those comparisons were to be made first. It is practically impossible to have the reciprocals of large numbers serve to distinguish between separate judgments.

In constructing a scale of importance, one customarily asks a decision-maker to state (a) which of two activities, in his opinion, is more important, and (b) his perception of this intensity of difference in importance, expressed as a rank number on a given numerical scale. Only direct effects of the activities on the objectives are considered in the judgment process. One method for dealing with indirect effects is to consider input-output type of relations between the activities. We do this elsewhere when considering, for example, the allocation of energy to interdependent industries according to priority.

As a prelimary step towards the construction of an intensity scale of importance for activities, we have broken down the importance ranks as follows:

Higher education in the United States (1985-2000) 253

Intensity of Importance Definition Explanation

2,468

Reciprocals of above non- zero numbers

Rationals

Equal importance

Weak importance of one over another

Essential or strong importance

Demonstrated importance

Absolute importance

Intermediate values between the two adjacent judgments If activity i has one of the above non- zero numbers assigned to it when compared with activity j, then j has the reciprocal value when compared with i

Ratios arising from the scale

Two activities contribute equally to the objective The judgment is to favor one activity over another, but it is not conclusive The judgment is to strongly favor one activity over another Conclusive judgment as to the importance of one activity over another The judgment in favor of one activity over another is of the highest possible order of affirmation When compromise is needed

See below

In the consistent case (see consistency discussion for these numbers)

To explain the various numbers note that the integers assigned attempt to estimate a scale of values (w,, . . ., IV,) of the activities. Each value a,, assigned in the matrix of pairwise comparison A may be regarded as an estimate of the ratio wi / w, taken to the nearest integer after appropriate scaling to make values between the (unknown) underlying scale and the approximation scale appropriately correspond through ratios.

the first row than a, = ari/aLi (a rational assuming, of course, that a,i # 0 for all i). The use of zero is reserved for the general case to indicate that a pair is not comparable.

If we assume that the values are estimated precisely, i.e. a, = w,/w,, it is then sufficient to require consistency of the judgment matrix to obtain such equality. Consis- tency means aiiaa = aik from which we have for the main diagonal entries aii = 1 and the reciprocal relations a, = l/a,. In general, we do not expect “cardinal” consistency to hold everywhere in the matrix because people’s feelings do not conform to an exact formula such as the one just given. Nor do we expect “ordinal” consistency as people may not want to behave that way (see below). However, to improve consistency in the numerical judgments, when people become accustomed to the method and provide numbers instead of qualitative judgements, it is recommended to them that whatever value a, they may assign in comparing the ith activity with the jth one, they should consider assigning the reciprocal value to aji thus putting u,~ = (l/~,~). It follows that aii = 1. Roughly speaking, if one activity is judged to be a times stronger than another, then the latter is only l/a times as strong as the former. It can be easily seen that when we have consistency, the matrix has unit rank and it is sufficient to know one row of the matrix to construct the remaining entries. For example, if we know

We do not insist that judgments be consistent and, hence, they need not be transitive; i.e. if the relative importance of C, is greater than that of C3, then the relation of importance of C, need not be greater than that of C,, a common occurrence in human judgments. An interesting illustration is afforded by tournaments regarding inconsistency or lack of transitivity of preferences. A team C, may lose again another team C, which has lost to a third team C,, yet C, may have won against C,. Thus, team behavior is inconsistent-a fact which has to be accepted in the formulation, and nothing can be done about it. I

The treatment of the easier and more elegant case of consistency is a special case. The point is that one must develop rational means for making decisions in spite of inconsistency. We shall assume that our judgmental inputs are all provided by a single individual expert or are the collective view of several individuals as the case may be if no single individual knows enough to supply all the judgments by himse1f.t A major difficulty lies in the large number of questions an individual must answer in order to obtain the n(n - I)/2 judgments for each objective, where n is the number of activities, and reciprocals are used, There is room for improvement in facilitating the process of supplying these judgments.

With inconsistency we no longer have aii(wj/w,) = 1, i, j = 1,. , n. We seek a condition on each row of the

matrix. We note that with consistency i aii(wi/wi) = n, j-l

tSee Ref. [13] for a discussion of the problem of judgmental i = 1,. . . , n, where the right side is the same constant n, consensus. the largest eigenvalue of A, whose other eigenvalues are


all zero since the rank of A is unity and the sum of the eigenvalues is equal to the trace Ba,, = n.

It is possible to argue for the general case that the destred scale w = (w,, . . . , wn) must satisfy the eigenvalue problem Aw = h,,,w where A,,, is the largest eigenvalue of A, which by Perron Forbenius’ theory, turns out to have an essentially unique non-negative solution w if A is non-negative and irreducible.

To see how one arrives at this eigenvalue formulation for the general problem, note first that in the consistent case if we take a typical row a,,, ai2,. . , ain and multiply a,, by WI, ai2 by WZ,. . , a, by w, we would obtain wi, . . . , wi. In this case all these values are exact and therefore when we perform the operation Aw we obtain the vector nw and hence, as just stated, we solve the problem Aw = nw to obtain the vector w which we

normalize by dividing each of its entries by 2 wi to

obtain the desired scale. i=l

In the general case the process of multiplying the ith row as above does not yield exact values wi, . . , wi but deviations about them which amount to perturbations of the exact values. It is known in matrix theory that the eigenvalues of a matrix are continuous functions of the coefficients. If we perturb the coefficients of a consistent matrix, the largest eigenvalue would remain near n and the remaining ones near zero. Thus, our problem becomes: find w which satisfied Aw = A,,,w and the results would be the more valid the closer A,,, is to n as indicated in an index of consistency. It is for this reason that use of the relation uL = l/a,, should be encouraged to improve consistency. The deviation of A,,, from n could be used as a measure of the consistency and hence usefulness (in some sense) of the results. A,,, is known to be a monotone increasing function of a,. Thus an error in aZi reflected in A,,, is compensated for by a, = l/a,,

It is worth noting that we have a ratio scale (invariant under positive similarity transformations) as it easily follows from the fact that the scale is derived from the solution to the (linear) eigenvalue problem.

Once ratio judgments are elicitated from the subjects to form one row or column, the use of paired comparisons can make further contributions to locating the stimuli by possibly enhancing stability where the data are nearly but not quite perfectly consistent. More importantly, requir- ing each subject to rate each pair of stimuli separately permits tests of dimensionality to be undertaken.

The expression for (A,,, - n)/n provides an index for testing the efficiency of various psychophysical laws e.g. S. S. Steven’s power law and the Weber-Fechner law. It is easy to prove that A,,, 2 n. Although it is easy to prove that the method leads to a ratio scale, it is not so easy to justify (although the many applications confirm it) that it is the ratio scale which applies to the real world; hence, the value of interfacing with the psychophysical axioma- tic work of Krantz[4] on ratio scaling.

To determine the rank of an element of a subset of a hierarchy level defined by the dominance of all its members by a single element of the adjacent higher hierarchy level, a matrix of pairwise comparisons is constructed, each of whose entries indicates the degree of (relative) dominance of a member of a pair over the other member in its contribution to the status of the dominating element in this adjacent upper level. For example, if the elements of the subset are industries and the dominance relation is their contribution to health which is an element of the next hierarchy level, the food industry

would be ranked numerically to indicate that it is absolutely dominant over the tobacco industry. The tobacco industry would then have the reciprocal numerical value when compared with the food industry in regard to contribution to health.

Given a subset of elements, x = (xi,. . . , x,) of a hierarchy level dominated by the set y = (y,, . , y,,,) if w, is the priority of x,, i = 1,. . . . , n in x and w: is the priority of y,, j = 1,. . . , tn in y, then we assume that the following set of linear relationships holds:

w~=u,,w:+~~~+~,,w~ i=l,..., n.

To generalize this relation to a complete hierarchy of h levels with nt, k = 1,. . . , h elements in the kth level, where obviously n, = 1 we write

w,, = qi,.lwl,., + a,,,2w2,., +.’ . + a,.,., ik = 1, . . . , nk.

In a complete hierarchy H let wli be the vector of priorities of the kIh level and let V~ denote the corresponding matrix of coefficients expressing linear dependence on wli_,. We state without proof:

Theorem:

We = vI-vk_, .v,+,w, k > m.

In particular we have w,, = vh . .vzwI. The following observation holds for a complete

hierarchy but is intended to be useful in general. The priority order 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 order of that subset. The resulting set of priorities is then normalized by dividing by its sum.

The priority of a subset in a level is equal to the sum of the priorities of the dominating elements in the next level

The consistency index of a hierarchy is the difference between the sums of the products of the maximum eigenvalues corresponding to the matrices on each path from the top level to the bottom level (that is used to construct the composite eigenvector of the bottom level) and the products of the dimensions of these matrices along these paths with the result (of this difference) divided by the second term in the difference.

For example, in a three level hierarchy level the dimensions of the levels be L, = 1, L, = 3, and L3 = 5 and assume that the hierarchy is complete. Let the eigenvalues for the second and third levels be A,,, Aj2, A33, A34 and A,,. Then the consistency index is given by

hz(A31 + Ag2 + A3, t A34 t A35) - 3(5 + 5 t 5 t 5 t 5) > o 3(5 + 5 t 5 t 5 + 5)

since A,,, 3 n

always. It can also serve as a measure of the stability of the hierarchy with respect to perturbations in the interactions between the elements in the levels.

4. VALIDATION OF THE SCALE AND THE METHOD

We will present three examples, one physical, one economic and one social, to illustrate the use of the method. Over two dozen other applications are available (see Section 6).

Higher edu~atinn in the United States (l98S-20~) 2.55

Example 1. The rate at which a source emits light illumination varies (approximately) inversely with the energy evaluated in terms of its visual effects is spoken of square of the distance. The more carefully designed the as a light flux. The illumination of a surface is defined as experiment the better results obtained from the visual the amount of light flux it receives per unit area. observation.

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 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.

Example 2. A number of people have studied the problem of measuring world influence of nations. We have briefly examined this concept within the framework of our model. We assumed that influence is a function of several factors. We considered five such factors: (1) human resources; (2) wealth; (3) trade; (4) technology; and (5) military power. Culture and ideology, as well as potential natural resources (such as oil) were not included.

The two matrices of pairwise comparisons of the brightness of the objects iabelied in increasing order according to their nearness to the source are, along with the eigenvector of the maximum eigenvalue:

Seven countries were selected for this analysis. They are the U.S., U.S.S.R., China, France, U.K., Japan and West Germany. It was felt that these nations as a group comprised a dominant class of influential nations. It was desired to compare them among themselves as to their overall influence in international relations. We realize that

Relative visual brightness Relative visual brightness

C, C, c, c, Ev. C, C* c, C, Ev.

C, 1 5 6 7 0.62 Cl 1 4 6 7 0.63 C, l/S I 4 6 0.23 C, t/4 I 3 4 0.22 C, 116 t/4 1 4 0.10 C1 116 l/3 1 2 0.99 C, 117 116 l/4 1 0.05 C4 117 114 r/2 1 0.06

h = 4.39 h =4.1

Thus for example, object C, appeared definitely brighter than C, (7 in row I, column 4) on each trial,

When we compare the components of the eigenvectors with the following reciprocals of distance squares we see good agreement.

Reciprocal of corresponding normalized distance square

Normalized reciprocal distance square

0.67 0.61 0.24 0.22 0.12 0.11 0.07 0.06

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

Table 1. Wealth

what we have is a very rough estimate-mainly intended to serve as an interesting example of an application of our approach to priorities. We will only iitustrate the method with respect to the single factor of wealth. The problem is studied in a separate paper.

In Table 1, we give a matrix indicating the pairwise comparisons of the se; en countries with respect to wealth. For example, the value 4 in the first row indicates that wealth is between weak and strong importance in favor of the U.S. over the U.S.S.R. The reciprocal of 4 appears in the symmetric position, indicating the inverse relation of relative strength of the wealth of the U.S.S.R. compared to the U.S.

Note that the comparisons are not consistent. This basic inconsistency cannot be eradicated merely by changing scale. Moreover, there are also apparent numerical (or scale) inconsistencies. For example, the U.S.: China = 9 (not 28) despite the fact that U.S.:U.S.S.R. = 4 and U.S.S.R. : China = 7.

Nevertheless, when the requisite computations are performed, we obtain relative weights of 43.3 and 21.7 for the U.S. and Russia, and these weights are in striking

U.S. U.S.S.R. China France U.K. Japan W. Germany

U.S. 1 4 9 6 6 5 5 U.S.S.R. Ii4 1 7 s 5 3 4 China Ii9 117 1 US 115 117 115 France t/6 r/s 5 1 1 l/3 113 U.K. l/6 1.5 5 1 1 l/3 l/3 Japan l/5 l/3 7 3 3 I 2 W. Germany l/5 1.4 5 3 3 l/2 1


Table 2. Normalized wealth eigenvector Cluster analysis can be used to show that China probably should not be in the group.

Normalized Actual Fraction of eigenvector GNPt (1972) GNP total

Example 3. Three high schools were analyzed from the standpoint of a candidate according to their desirabil-

U.S. 0.429 U.S.S.R. 0.231 China 0.021 France 0.053 U.K. 0.053 Japan 0.119 W. Germany 0.095

tBillions of dollars.

Total

1,167 635 120 196 154 294 251

2,823

0.413 0.225 0.043 0.069 0.055 0.104 0.091

ity. Six characteristics were selected for the comparison. They are: learning, friends, school life, vocational training, college preparation and music classes. The following dominance matrix shows the pairwise relative importance of each pair of characteristics to the person(s) making the study. The column labelled Ev. at the right is the normalized eigenvector of the resulting matrix. This shows the relative importance of each characteristic. The eigenvalue of 7.49 is slightly far from the consistent value of 6.

Which characteristic is more important when selecting a school?

Learning Friends School

life Vocational

training College

preparation Music

classes

Learning Friends School Vocational College Music

life training preparation classes

1 4 3 1 3 4 l/4 1 7 3 l/5 1

l/3 l/7 1 l/3 l/5 l/6

1 113 5 1 1 l/3

113 5 5 1 1 3

114 1 6 113 113 1

Ev.

0.32 0.14

0.04

0.13

0.24

0.14

A = 7.49

Each of the following six matrices gives the pairwise relative ratings of the schools vis-a-vis the indicated characteristic.

Learning Friends School life

A B C Ev. A B C Ev. A B C EV.

A 1 113 l/2 0.16 A I I 1 0.33 A I 5 I 0.45 B 3 1 3 0.59 B I I I 0.33 B l/S I l/5 0.09 c 2 l/3 I 0.2s c I 1 I 0.33 c I 5 I 0.46

A = 3.05 A =3 A=3

Vocational training College preparation Music classes

A B C Ev. A B C Ev. A B ( Ev.

A 19 7 0.77 A I I/? I 0.25 A I 6 4 0.69 B l/9 I l/S 0.05 B 2 I 2 0.50 B l/6 I I/! 0.09 c l/7 5 I 0.17 c I l/2 I 0.25 C l/4 3 I 0.22

A = 3.21 A=3 A = 3.0s

agreement with the corresponding GNP’s as percentages To obtain the overall ranking of the schools we form the of the total GNP (see Table 2). Thus, despite the apparent 6 x 3 matrix whose columns are the six eigenvectors just arbitrarines of the scale, the irregularities disappear and obtained and multiply on the right by the eigenvector of the numbers occur in good accord with observed the characteristics. This yields the relative weights tif the data. schools:

Compare the normalized eigenvector column derived by using the matrix bf judgments in Table 1 with the actual GNP fraction given in the last column. The two are very close in their values. Estimates of the actual GNP of China range from 74 to 128 billion.

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. He had been going to school B before the analysis was made.


KSCENARIO CONSTRUCTION, A CASE STUDY

This description is based on an experiment conducted by twenty-eight college level teachers, mostly from the mathematical sciences, under the leadership of T. L. Saaty at an NSF Chautauqua type course in operation Research and the Systems Approach in February 1976. The problem was to construct seven weighted scenarios and a composite scenario which would describe the attitude of the group about the future of higher education in the United States during the period 1985-2000.

Figure 1 presents the hierarchical structure of the factors, actors and their motivating objectives which the group saw as chain of influences which would affect the form that higher education will take between 1985 and 2000. No strict definitions of the various terms will be given although during the development (which took approx. 9 hr) comments were made on some of the intended meanings.

Seven scenarios are offered.

(1) U’ROJ) 1985-Projection of the present status quo (slight perturbation of present)

(2) (VOTEC) Vocational-Technical Oriented (skill orientation)

(3) (ALL) Education for All (subsidized education)

(4) (ELITE) Elitism (for those with money or exceptional talent)

(5) (APUB) All Public (government owned) (6) (TECH) Technology Based (little use of

classroom-use of media, computers) (7) (P.T.) Part-Time Teaching-no research

orientation.

The characteristics which were considered and which were considered and which were calibrated so as to give profiles of the various scenarios are given in Table 3. The calibration numbers are integers between - 5 and 5. These measurements were arrived at by consensus.

Zero (0) represents things as they now are (in the group’s opinion). Positive integers represent the various degrees of “increasingness” or “more than now”. Negative integers represent various degrees of “decreas- ingness” or r“less than now”. For example under Institution-Governance Structure we see a 5 for Scenario

6. this means that the group thought that there would be a very large measure of administrative control (relative to the state of things as present) in a technology based higher education system in 1985 and after. On the other hand, if Scenario 2 (Education for All) were to prevail, then the value of a degree (Education-Value of a Degree) would diminish considerably (- 2) compared to how it is valued today. The row “Scenario Weight” and the column “Composite Weight” are empty. Filling them is the object of this study. Table 4 at the end of the study is Table 3 with these columns filled.

In the matrices listed below the group’s consensus as to how much and which factor dominated which factor relative to the criterion being considered are presented in the light of the question asked. In all cases the list on the left is compared with the factor listed on the top. Thus an integer indicates “left factor dominance over top factor”; whereas a fraction indicated top factor dominance over left factor. It is always true that a,, = l/a,, i # j and aii = 1. Only the a, with i 5 j or with i 2 j, whichever gave more integers is recorded.

The Roman numbers refer to the levels as indicated in Fig. 1. Thus, for example, in the matrix I-I the group believed that the Economy would have strong dominance over Technology vis-a-vis their impact on higher education in the United States in 1985. Thus a 5 is recorded in row 1 column 4. A l/5 is understood to be in the row 4 column 1 position of the matrix. The Ev. column to the right is the normalized eigenvector associated with the maximum eigenvalue (A) listed below it.

There is only one matrix of pairwise dominance measures at the I-level since each pair is compared relative to its impact on higher education.

I-I. Which factor has the greater impact on higher education?

Higher

y

A = 4.06

A hierarchy of influences on higher education

Level I II 12 I3 I4

I Primary Factors

The primary factors are affected by the

Economic Political Social Technological

II Actors III Students

II? Faculty

113 Administration

II4 Government

115 Private sector

116 Industry

The Actors are motivated by

III Objectives I Voc. Trng 2 Self-dev. 3 Social

status

I

Jobs Prof’l. growth Promo. of knowledge Power

I Fig. I.

1 Control of social change

~ 2 Knowledge 3 Culture 4 Vested

interests

I Manpower 2 Technology 3 Profit 4 Perpet. and

Dower


Table 3. Seven Scenarios and the calibration of their characteristics (Scale: - 5 @ + 5)

Scenario weights Characteristics PROJ VOTEC ALL ELITE APUB TECH P.T. COMP

Students 1. Number 2. Type (I.Q.) 3. Functiont 4. Jobs

Faculty 1. Number 2. Type (Ph.D.) 3. Function (role on

campus) 4. Job security 5. Acad. freedom

Institution 1. Number 2. Type (acad./non-acad.) 3. Governancet 4. Efficiency§ 5. Accessibility 6. Culture-entertain. 7. Avail. $ and other

resources

Education 1. Curriculum (life-long

learning) 2. Length of study 3. Value of a degree 4. Cost per student 5. Research by faculty

-2 -1 t1 +1

-2 +1

-2 -2

0

-1 -1 +2 t2

0 0

-1

1 0

-1 t3 +I

+2 -2 -1 +4

+2 0

-3 +1 -2

t2 -4 t4 t3 +2 -2

t2

-2 -3

0 +3 -1

+4 -3

0 -3

t4 -2

-2 t2

0

t2 -3 +1 -2 t5 t3

+2

t2 +2 -2 +3 -1

-3 +3 +1 +4

-3 t3

fl -3 t2

-3 +3 -2 t4 -3 +3

-2

+3 0

+4 t4 +3

-1 -1

0 +1

-1 +1

-2 -1 -1

-1 -1 t2 -1 t2 t1

0

t1 t1 -1 t2 +1

t2 -2 -2 -2

-5 +2

-5 -4 -4

-4 -3

5 -1 t4 -3

-1

to t2 -2 -1 -3

-2 -1 t2 t1

-4 -3

-5 -4 -5

-1 -3

5 0

t1 -1

-3

-1 0

-2 -1 -4

tIf more role in social change in and out of school. *If administration in control. PIf student attrition is less (survival rate more).

From the I-I matrix we see, for instance, that the group considered the economic factor somewhat dominant (4) over the political factor in its impact on higher education.

Now each pair of items (actors) at level II is compared

with respect to which of the pair has more impact on the way education affects each of the factors on level I. There are four dominance matrices as shown.

II-II. Who (what) has more impact on the way education affects 11-13. Who (what) has more impact on the way education affects the economy of the United States? the social issues in the United Statts?

;;

A = 6.67

1 A = 6.59

11-12. Who (what) has more impact on the way education affects 11-14. Who (what) has more impact on the way education affects the political situation in the United States? the technology of the United States?

~

A = 6.93

Tech.

stu. Fat. Adm. Gov. Pri. Ind.

I 1

8 3 5 1 A = 6.67


From these matrices and eigenvectors we observe, for example, that the group thought that the Industrial sector is the dominant force on higher education impact on the technology of the United States (11-14).

The objectives of each of the six actors are compared (1111-111 through 1116-116) pairwise for each of the actors resulting in an eigenvector which essentially orders and weights the objectives.

1116-116. Which objective has more impact on industry uis-a-uis its set of objectives?

1111-111. Which objective has more impact on the student vis-a-G education?

~‘:

A = 3.12

1112-112. Whichobjective has more impact on the faculty vis-a-vis education?

ti

A = 2.00

1113-113. Which objective has more impact on the administration vis-a-vis education?

Admin.

Perpetuation (tradition) Financial security

P F.S. Ev.

1 0.250 3 1 0.750

A = 2.00

1114-114. Which objective has more impact on the government uis- a -uis its set of objectives?

Gov. P C.O. M RIP T OPI: Ev.

Prosperity Civ. order Manpower Rel. int’l

power Technology Creat. oppor.

I l/5 3 3 5 6 1 5 7 8 8

1 l/2 3 5

I 3 5 1 4

I

0.203 0.516 0.092

0.110 0.05 I 0.027

A = b.56

1115-115. Which objective has more impact on the priz;ate sector vis-a-k its set of objectives?

& A = 4.31

Ind. ) M T P P&P I/ Ev.

Manpower Technology Profit Perpetuation and power

1 0.040 4 I 0.084 9 7 1 0.331

7 1 3 I 0.546 A = 4.40

The next step was to find the importance of the actors relative to their impact on the factors which affect higher education. This is done by multiplying the matrix of eigenvectors of the actors with respect to each factor in level I on the right by the eigenvector obtained for level I.

Econ. Pal. Sot. Tech.

S 0.04 0.04 0.10 0.02 0.55 E 0.05 S F 0.02 0.04 0.07 0. IO 0.04 F A 0.06 0.03 0.04 0.03 0.11 P=O.O5 A G 0.47 0.49 0.41 0.23 x 0.24 s 0.44 G P 0.12 0.12 0.16 0.13 P I 0.28 0.29 0.26 0.44 0.21 T 0.30 !

Since Government and Industry account for 74% ( = 0.44 + 0.30) of the impact on the four primary factors which affect higher education, it was decided to use only these two actors to obtain the weights for the scenarios. Should one decide to use more actors, the computations follow the same procedure shown below but the amount of work is increased.

Now we want to find the important objectives of the two actors; government and industry. To do this. we multiply the eigenvector for objectives by the respective actor weight which was just calculated.

0.20 0.09 Prosperity 0.52 0.?3Civ.

For 0.44 0.09

government: JBManpowel 0.1 I 0.05 R.I.P. 0.05 0.02 Technology 0.03 0.01 Create oppor.

0.04 0.01 Manpower

For industry: 0.30 ;:;; = 0.02 Technology

0.10 Profit -__ 0.55 0.17 Peroet. & Dower

From this we see that the most influential objectives are prosperity and civil order for Government and profit and perpetuation and power for Industry. Using these four objectives and normalizing their weights we get the weight vector.

0.15 Prosperity 0.39 Civ. order 0.17 Profit 0.29 Perpet. & power

This vector will be used to get our scenario weights. The final step necessary to get our scenario weights is to

construct the dominance matrices for the seven scenarios with respect to each of the objectives (four in our case).


6. DOMINANCE MATRICES FOR SCENARIOS AND ACTORS ORJECTIVES

11141-Scenarios. Which scenario has more impact on the prosper- 11142-Scenarios. Which scenario has more impact on the ci~~~~r~e~ ity of the United States? of the United States?

Prose.

STAT. QUO VOC. TECH. ED. ALL ELITE ALL PUB TECH. BASED PART-TIME

SQ VT EA E AP TB PT Ev.

I l/S If3 5 1 , 5 5 0.129 1 3 7 1 5 5 0.329

1 7 5 5 5 0.275 I l/5 3 1 0.041

1 3 5. 0.149 1 l/3 0.032

1 0.045 A = 7.96

Civ. ord. PROJ VT EA E AP TB PT Ev.

PROJ 1 l/3 I/S 5 1 3 3 0.125 VOC. TECH. 1 113 5 1 3 3 0.180 ED. ALL 1 5 3 5 5 0.369 ELITE I l/5 l/3 l/2 0.033 ALL PUB 1 5 5 0.177 TECH. BASED 1 l/3 0.050 PART-TIME 1 0.065

A = 7.60

III63-Scenarios. Which scenario has more impact on ~ra~t-u6i/~t~?

Profit PROJ VT ALL ELITE APUB TECH PT Ev.

PROJ 1 0.067 VOC. TECH. 5 I 0.309 ED. ALL f/4 I17 1 0.028 ELITE 5 1 8 1 0.33 1 ALL PUB l/3 l/3 3 l/6 1 0.048 TECH. 3 l/S 3 l/5 4 I 0.129 PART-TIME 3 115 3 115 3 l/3 1 0.089

A = 7.79

III64-Scenarios. Which scenario has more impact on perpetuating industrial methods and power?

P&P PROJ VT ALL ELITE APUB TECH PT Ev.

PROJ 1 0.062 VOC. TECH. 7 1 0.306 ED. ALL 117 115 1 0.026 ELITE 5 I 8 1 0.330 ALL PUB 1 l/5 5 l/6 1 0.085 TECH. 3 l/3 3 l/5 l/3 1 0.075 PART-TIME 4 l/S 4 l/S 2 2 1 0.115

A = 7.94

To obtain the scenario weights, we multiply the matrix I.Q.) and who will be a little less active in influencing of eigenvectors just obtained by weight vector for the the institution, but they will have no problem in getting four most influential factors (prosperity, civil order, profit jobs upon graduation. and perpetuation and power). This product yields the -There will be more faculty of about the same scenario weights. Table 4 is Table 3 with these weights intellectual level as today, but they will have considerably placed over the scenario names. less to say about the governing of the university. Their job

PROS C. ORD. PROF. P&P SC.

I 0.129 0.125 0.067 0.062 0.15 0.097 1 2 0.329 0.180 0.309 0.306 0.261 2

Scenario 3 0.275 0.369 0.028 0.026 0.39 0.197 3

4

=

0.041 0.033 0.331 0.330 0.171 4 5 0.149 0.177 0.048 0.08s 0.17 0.124 5 6 0.032 0.050 0.128 0.075 0.068 7 0.045 0.065 0.089 0.11s 0.29 0.081 7”

We note that the second scenario has the greatest security will be a little better than it is now, but there will weight (0.261). This can be interpreted as the scenario be Iess academic freedom. As for the institutions, there most heavily favored by the group. A description of this will be more of them, but with much less academic scenario could be: orientation. The administration will control things to a

-Higher education in the United States in 1985 and much greater degree and the efficiency (less student beyond will be vocational-technical oriented. There will attrition) will be considerably higher. The schools will be be more students who will be less bright (as measured by more accessible, but their cultural and entertainment roles

Higher education in the United States (1985-2~) 261

Table 4. Seven scenarios and the calibration of their characteristics (Scale: - 5 ++ + 5)

0.097 0.261 0.197 0.171 0.124 0.068 0.081 Scenario weights 1 2 3 4 5 6 I Characteristics PRQJ VOTEC ALL ELITE APUB TECH P.T. COMP

Students 1. Number 2. Type (I.Q.) 3. Functiont 4. Jobs

Faculty 1. Number 2. Type (PhD.) 3. Function (role on

campus) 4. Job security 5. Acad. freedom

Institution 1. Number 2. Type ~acad,~no~-acad.) 3. Governances 4. Efficiency8 5. Accessibility 6. Culture-entertain. 7. Avail. $ and other

-2 -1 +1 +1

-2 4-l

-2 -2 0

-I -1 i-2 +2

0 0

-1

Education 1. Curriculum (life-long

learning) 2. Length of study 3. Value of a degree 4. Cost per student 5. Research by faculty

1 0

-1 +3 +1

+2 -2 -1 +4

+2 0

-3 +t -2

+2 -4 +4 +3 +2 -2

+2

-2 -3

0 +3 -1

+4 -3

0 -3

+4 -2

-2 +2

0

+2 -3 +1 -2 t5 t3

12

-l”2 +2 -2 +3 -1

-3 +3 +1 +4

-3 +3

+1 -3 t2

-3 +3 -2 t4 -3 t3

-2

+3 0

+4 +4 t3

-I -I 0

t1

-1 t-1

-2 -1 -1

-t -1 +2 -1 +2 1-t

0

+1 t-t -1 +2 if

t2 -2 -2 -2

-5 +2

-5 -4 -4

-4 -3

5 -1 t4 -3

-1

0 +2 -2 -1 -3

-2 0.45 -1 -1.04 +2 - 0.05 i-1 1.30

-4 -0.18 -3 +0.23

-5 -2.19 -4 - 0.77 _. 5 -0.98

-1 -0.17 -3 - 1.79

5 2.08 0 1.07

+I 1.60 -1 0.42

-3 0.17

-1 0.53

-! -0.13 - 0.23 -I 2.45 -4 -0.25

iif more role in social change in and out of school. $If administration in control. §If student attrition is less (survival rate more).

will decrease somewhat. The av~labiIity of dollar and other resources will be greater than at present.

-Finally, the type of curriculum will be more vocation~iy (skill) oriented with less of the learning experience which benefits one for a lifetime. The length of time it takes to complete a degree program will be considerably less and the value of a degree will not be any more or less than at present. The per student cost will rise quite a bit. There will be a little less research going on.

We now obtain the composite scenario: a single scenario obtained by finding composite scale measurement for each of the characteristics. The composite scale measurement for a characteristic is obtained by forming the sum of the products of scenario weight by the corresponding characteristic measurement. For example, for the number of students we have:

(- 2)(0.~7) f (2)(0.261~ t (4)(0.197) + (- 3)(0.171)

( - l)(O. 124) + (2)(0.~8) + ( - 2)(0.081) = 0.454.

This measurement is found in Table 4 in the last column on the right. Similarly for the other characteristics. An interpretation of the composite scenario might be:

-Higher Education in the United States in 1985 and beyond will witness not much, if any, increase in total enrollment. The student will exhibit slightly lower

performance levels as measured by the type of standardized tests we have today. Students will play about the same role as they do today in setting university educational policy. Their chances for jobs upon ~aduation will be a little better than at present.

-The faculty characteristics will be about the same as today regarding numbers, Ph.D. holders and job security. However, faculty will play considerably less of a role in campus affairs while possessing a little less academic freedom.

-The number of institutions of higher education will not change much, if at all. They will be definitely less academically oriented with the administration exhibiting more control. There will be some increase in efficiency (less student attrition). Accessibility will be greater, but their cultural and entertainment roles will be about the same as today. There will be practically no increase in dollar resources.

-The life-long learning qualities of the cu~iculum will not undergo much change, nor will the length of study, the value of a degree. Costs will continue to increase significantly. The amount of faculty research will be at a much lower level.

It was suggested that different results might be obtained had we not had level I. Then we would need the eigenvector for the 6 x 6 dominance matrix for the actors (II-II). The dominance matrix was constructed yielding:


II-II. Who will have more impact on education in the United States in 1985?

~

x = 6.50

When this vector is compared with the actors’ weights obtained previously, we see close agreement.

It was further suggested that another primary factor, Ideology, be added at level I. A new dominance matrix was formed for I-I [same as original with an added row and column). It follows:

~

h =5.06

Comparing this eigenvector with the original one for I-I we again see close agreement.

7. COMMENTS AND CONCLUSIONS

Care must be taken that the actors do not exaggerate the influence of their objectives on the set of scenarios. Actors

pursue their objectives according to a “policy” and this policy should be checked for compatibility with the actors’ past behavior, this check was not done in the case study just described.

More than two dozen applications have been made of the method particularly in the area of impact analysis and in planning. Examples include:

(1) Allocation of energy to industries in case of short fall (within the framework of input-output analysis).

(2) Design of future scenarios in conjunction with a plan for a transport system for the Sudan by 1985 [5]. The method was used to determine priorities of projects and to determine cost effectiveness. Implementation is in prog- ress.

(3) The role of actors in an energy environment game[6]. Here the macro approach of hierarchies was combined with micro economic models so that each actor could see the outcome of his policies and those of the other actors.

(4) Planning the future of a corporation and measuring the impact of env~onmental factors on its development.

(5) Selecting a job. (6) Analyzing the fuzziness of sets [7]. (7) The candidacy problem for the Democratic Party in

1976[8]. (8) Group evaluation of paper presentations. The results of the interactions are usually very

satisfying to the participants. Policy change can only take place through interaction and through revising assumptions after trying them out in paractice. The process is both helped and expedited by well designed sessions in which interaction and evaluation occur. Note that for

planning purposes, one may trace back to the judgments which gave rise to a composite scenario result and design a way to change that judgment.

Because of the development of the method for deriving ratio scales (with intrensic consistency tests) which have valid appIications in the physical sciences, modeling complex problems with qualitative attributes in a hierarchical framework for impact analysis is becoming an established practice. It offers a way for analyzing the beliefs of scientists in the conceptualization stage of problem solving and underlines the priority factors which should receive the greatest emphasis in the micro- modeling process. In planning, it can be used to devise new policies in the solution of the two point boundary value problem: given the present and given a desired future, determine feasible policies and actions which carry a system from the present to the desired future. In the above example we have shown how to solve the one point boundary problem, (the forward problem), from the present to the future, from the s~ndpoint of ail actors simul~neously. For the corporate problem just men- tioned, we have solved the two point problem. The prospects for this methodology are exciting and educational.

REFERENCES

1. P.-A. Julien. P. Lamonde and D. Latouche, La M~thode des Sc~nurjos. Groupe de Recherches sur ie Futur UniversitC du Qubbec (Nov. 1974).

2. T. L. Saaty, Hierarchies and priorities-eigenvalue analysis. J: Math. Psycho/. In press.

3. T. L. Saaty, An Eigenvalue Allocation Model in contingency Planning. University of Pennsylvania (1972).

4. D. H. Xrantz, R. D. Lute, P. Suppes and A. Tversky, Fo~~dutions of Measnrenle~r. Academic Press, New York (1971). .

5. T. L. Saaty (Project Director), The Sudan Transport Study, 5 Volumes. The Democratic Republic of the Sudan in association with the Kuwait Fund for Arab Economic Development (1975).

6. T. L. Saaty, F. Ma and P. Blair, Hierarchical Theory and Operational Gaming for Energy Policy Analysis. Study done for ERDA (1975).

7. T. L. Saaty. Measurihg the fuzziness of sets. J. Cy~e~efjcs 4(J), 53-61 (1974).

8. T. L. Saaty and J. Bennett, A hierarchical approach to political candidacy. Submitted to Public Opinion J. (1976).

FURTHERREADING

Beals, R., Krantz, D. H. and Tversky, A., Foundations of multidimensional scaling. Physichol. Reu. 75(2), 127-142 (1968).

Berge, C., Theory of Grup~s and Its App~icatio~s. John Wiley, New York (1962).

Bogart, K. P., Preference structures-i: Distances between transitive ureference relations. J. Math. So&l. 3.49-67 (1973).

Bogart, K. ‘P., Preference structure--II. Distances between &transitive preference relations. SIAMJ. Appl. Math. In press.

Busacker, R. and Saaty, T. L., Finite Graphs and Networks. McGraw-Hill, New York (1965).

Cant, G.. An X-rav analvsis of doctors’ bills, Money (Aug. 1973). Census if M&factu~es, 1967. Bureau of the Cen& U.S.

Department of Commerce. Chenery, H. and Clark, P., Interindustry Economics. John Wiley,

New York (1959). Eckenrode, R. T., Weighting multiple criteria. Mgmt Sci. 12(3),

180-192 (l%S). The Quality of Life Concept: A Potential New Tool for

recision-augers. Environmental Protection Agency, Office of Research and Monitoring (1973).

Fechner, G., Elements of Psychophysics (Translated by Helmut E. Adler), Vol. 2. Holt,.Rinehart & Winston, New York (1966).

Franklin, J. N., Matrix Theory. Prentice-Hall, Englewood Cliffs, N. J. (1968).


Frazer, R. A., Duncan, W. J. and Collar, A. R., Elementary Martices. Cambridge University Press, Cambridge, Mass. (1963).

Gantmacher, F. R., The Theory ofMatrices, Vol. II, pp. 53 and 63. Chelsea Publishing, New York (1960).

Gillett, J. R., The football league eigenvector. Eureka (Oct. 1970). Herstein, I. N., Topics in Algebra. Blaisdell, New York (1964). Kemeny, J. G. and Snell J. L., Mathematical Models in the Social

Sciences. Blaisdell, New York (1962). Klee, A. J., The role of decision models in the evaluation of

competing environmental health alternatives. Mgmt Sci. 18(2), 53-67 (1971).

Krantz, D. H., A theory of magnitude estimation and cross modality matchings. J. Math. Psychol. (9). 168-199 (1972).

Landsberg, H. H. and Schurr, S. H., Energy in the United States: Sources, Uses and Policy Issues. Random House, New York, for Resources for the Future, Inc. (1963).

Identification of War Reserve Stock, Task 72-04, Logistics Management Institute, Washington, D.C. (1972).

Mariano, R. et al., Application of the eigenuector prioritization model to the beverage container problem, Energy Management & Power Memorandum, University of Pennsylvania (Sept. 1973).

Mason, E. S. and the staff on the NPA project on the economic aspects of the productive uses of nuclear energy, energy requirements and economic growth, Washington, National Planning Association (1955).

Moreno. J. L., Fondements de la Sociomettie (trans. Lesage- Maucorps). Presses Universitaires, Paris (1954).

Pattee, H. H. (Ed.), Hierarchy Theory, The Challenge of Complex Systems. George Braziller, New York (1973).

Pfanzagl. J.. Theory of Measurement. John Wilev, New York (1968). . -

Saaty, T. L. and Khouja M., A measure of world influence. Peace Science (June 1976).

Savage, C. W., Introspectionist and behaviorist interpretations of ratio scales of perceptual magnitudes. In Psychological Monographs: General and Applied, Vol. 80, No. 19, Whole No. 627 (1966).

Scott, D., Measurement structures and linear inequalities. J. Math. Psychol. (l), 233-247 (1964).

Shepard, R. N., A taxonomy of some principal types of data and of multidimensional methods for their analysis. In Multidimen- sional Scaling: Theory and Applications in the Behavioral Sciences (Edited by R. N. Shepard, A. K. Romney and S. B. Nerlove), Vol. 1, pp. 21-47. Seminar Press, New York (1972).

Stevens, S. S., On the psychophysical law. Psychol. Rev. 64, 153-181 (1957).

Stevens, S. S., To honor Fechner and repeal his law. Science 13 (13 Jan. 1961).

Suppes, P. and Zinnes J. L., Handbook of Mathematical Psychology, Vol. I. John Wiley, New York (1963).

Thurston, L. L., A law of comparative judgment. Psychol. Rev. 34, 273-286 (1927).

Torgerson, W. S., Theory and Methods of Scaling. John Wiley, New York.

Wei, T. H., The algebraic foundations of ranking theory, Thesis, Cambridge University, Mass. (1952).

Weiss, P. A., Hierarchically Organized Systems in Theory and Practice. Hafner, New York (1971).

Whyte, L. L., Wilson, A. G. and Wilson D., (Eds.). Hierarchical Structures. American Elsevier, New York (1969).

Wielandt, H., Unzerlegbare, nicht negative Matrizen. Math. Z. 52, 642-648 (1950).

Wilf, H. S., Mathematics for the Physical Sciences. John Wiley, New York (1962).

Wilkinson, J. H., The Algebraic Eigenvalue Problem, Chap. 2. Clarendon Press, Oxford (1965).

Woodall, D. R., A criticism of the football league eigenvector. Eureka (Oct. 1971).

higher education in the united states (1985–2000): scenario construction using a hierarchical...

Documents