Why Barzilai's Criticisms of the AHP are Incorrect

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					Why Barzilai’s Criticisms of the AHP are Incorrect: Validation

                                  Rozann Whitaker
                            Creative Decisions Foundation
                               4922 Ellsworth Avenue
                                Pittsburgh, PA USA
                                rozann@creativedecisions.net

Abstract
AHP has been claimed to have flaws on the basis of examples that have been offered.
We show that such an AHP model was incorrectly formulated and offer many examples
of AHP giving correct answers.



1. Introduction
Jonathan Barzilai has made several criticisms of the AHP (Analytic Hierarchy Process) in
various papers he has written. One such paper, brought to our attention by a Canadian
friend who asked us to write a commentary on it, is “On the decomposition of value
functions,” Operations Research Letters 22 (1998) 159-170. The flaws he thinks he has
found are due solely to the incorrect way he formulates his AHP models. We use his
same examples to show they give the right results when modeled according to AHP
principles.

The main thrust of Barzilai’s paper is that AHP gives different results for different
decision hierarchy structures, and that therefore it must be flawed. But his different
structures give the same results when they are set up properly. In the AHP the criteria
inescapably depend on the alternatives when data are known for the alternatives. One of
Barzilai’s mistakes is that he assigned equal default values to the criteria. This is not
correct when the criteria depend on the outcomes or alternatives because they have
known data, as they do in his marketing example.

Truth often takes time to surface and it is no different in this case. Perhaps there are
critics of the AHP because the AHP offers such new ideas that people are unable to
reconcile them with their traditionally acquired knowledge, and mixing old and new
knowledge does not work.

An important point in the AHP is that there can be decisions where the criteria weights
depend on the alternatives and decisions where they do not. In Section 2 we give a small
instructional example to show how to determine the priorities of the criteria and sub-
criteria when data are given for the alternatives. Then we turn to Barzilai’s marketing
example and apply the same process to his example to get the right weights for his
criteria and subcriteria. When the weights are right all his structures give the same
outcome, and it is the outcome he himself expects: that the two marketing strategies
result in the same overall income and are equally preferred, regardless of the hierarchical
structure.


                                             1
The same marketing problem can be formulated as an ANP (Analytic Network Process)
network with feedback that leads to the right priorities for the alternatives. We show how
to structure the ANP network for our simple example of Section 2 and for Barzilai’s
marketing example. The limit supermatrix of the ANP automatically produces the correct
weights for both the criteria and alternatives when the data are entered as priority vectors
in the supermatrix. It is simpler in a way because no thinking is required on the part of
the user. Just plug in the known data and the supermatrix will crank out the priorities for
all the nodes in the model.

Criteria may depend on alternatives even for intangibles. If one is dealing with
intangibles, the priorities for the criteria are determined by using judgments to pairwise
compare the criteria, but judgments on the criteria should be made only after carefully
studying the alternatives. For example, in a car purchase decision where one of the
criteria is style and another is cost, one might have a pre-conceived idea that cost is the
most important criterion. But after actually visiting the auto showroom and seeing the
stylish cars side-by-side with the cheap cars, it is surprising how many people revise the
importance of their criteria and walk out with a beautiful but costly car. This is a case of
careful study of the properties exhibited by the alternatives influencing the weights of the
criteria. The Analytic Network Process (ANP) makes it clear how to determine the
priorities of criteria, tangible and intangible, that depend on alternatives.


2. Determining Priorities for the Criteria when Alternatives have known Outcomes
expressed as Data
Priorities like probabilities are relative numbers. Numbers from a scale can be
transformed to relative priorities, but not conversely, unless there is some link to the
actual values. Consider the case of two objects A and B worth 25 dollars and 50 dollars
respectively. One converts from dollars to relative numbers by normalizing as follows:

                        Object A = 25/75 = 1/3 of the total dollars
                        Object B = 50/75 = 2/3 of the total dollars

If we are told only that A = 1/3 and B = 2/3, we can say B is worth twice A, but we cannot
say whether A and B originally were 25 dollars and 50 dollars or 1000 dollars and 2000
dollars. In passing we note that the AHP with its homogeneity requirement requires
elements of different orders of magnitude to be grouped together and appropriately linked
for comparison purposes. One would not directly pairwise compare $25 and $2000 using
judgments from the Fundamental Scale of the AHP because the largest number should be
no more than 10 times the smallest number. When using data it is all right to lump them
together. Though the ratio of 2000/25 is 80, is a much bigger number than the maximum
9 of the Fundamental Scale, when using direct data for the entire comparisons set it is
okay.

The following example will make clear how one determines the criteria weights. The
investments A and B, for the same period of time, have both interest and capital gains



                                             2
returns as shown in Table 1. The capital gains returns are assumed to be known as well
as the interest returns (certainly not true in real life!). The first criterion C1 is the interest
return and the second C2 is the capital gains return. Note that the capital gains returns are
much heftier than the interest returns.

We can see from Table 1 that investment A’s gain of 13 would be 0.382 of the total
dollars ($34) that might be brought in by the two together while investment B’s would be
0.618.


 Table 1 Investments with Returns Measured in Dollars using Ordinary Arithmetic
                            to give the Right Answer
         Values (in dollars)
                               C1         C2                Total  Normalized
                               Interest   Capital Gains     Return Total
                               Return     Return                   Return
         Investment A          3          10                13     13/34 = 0.382
         Investment B          6          15                21     21/34 = 0.618
         Totals                9          25                34

This is, however, not the result we get if we normalize first before adding. In Table 2 we
first converted the dollars in each column to relative values by dividing by the total, and
then summed the row. The numbers in the “Sum of Normalized Return” column do not
sum to 1, so they must be normalized to obtain the final column, and it does not
correspond to the final column of Table 1.

        Table 2 The Wrong Outcome is Obtained by Normalizing then Adding
                                     Relative Values
                      (normalize each column by dividing by its sum)
                       C1              C2                    Sum of               Normalized
                       Interest        Capital Gains         Normalized           Sum
                       Return          Return                Return
                       (normalized)    (normalized)
   Investment A        3/9             10/25                 11/15                11/30=.367
   Investment B        6/9             15/25                 19/15                19/30=.633
   Totals              1               1                     2

To make the values in Table 2 correspond to those in Table 1, the correct relative values,
we need to determine weights for the criteria. The weight of each criterion is obtained by
dividing the sum of the values in its column in Table 1 by the total value of 34 for both
criteria. The priority of C1 is 9/34, or about 0.265, and that of C2 is 25/34 or 0.735. We
then multiply the normalized returns of the alternatives in each column in Table 2 by
these criteria weights and add. This yields Table 3 whose final column is identical with
the final column of Table 1. Thus we see here that the criteria in a sense “inherit” their


                                                3
priorities from the measurements of the alternatives under them and cannot be arbitrarily
assigned. This is always the process in multi-criteria decisions when one has
measurements – in the same scale – that need to be converted to relative values.
            Table 3 Relative Values Weighted by the Criteria then Added
                               Relative Values (weighted)
                   C1                C2                   Total
                   (9/34)=0 .264706   (25/34)=0.735294
             A     3/9×9/34=3/34 10/25×25/34=10/34 13/34 =0 .382
             B     6/9×9/34=6/34 15/25×25/34=15/34 21/34 = 0.618

ANP Solution to the Investment Problem
The same problem can be solved by structuring it as an ANP network with feedback and
dependence with two clusters: Criteria and Alternatives as shown in Figure 1 from the
SuperDecisions ANP software. The Criteria cluster contains the elements Interest and
Capital Gains and the Alternatives cluster contains the elements Investment A and
Investment B. Compare both alternatives with respect to Interest and Capital Gains, then
invert the process and compare both criteria with respect to Alternative A and Alternative
B. The question to be answered in the latter case is, for example: “Which criterion is
more dominant for Alternative A? For Alternative B? By how much?” Since we have
data, for Investment A, for example, this question is answered by normalizing the Capital
Gains and Interest return, obtaining 10/13 = 0.769 for Capital Gains, and 3/13 = 0.231 for
Interest Return. This normalized data is entered in the column headed Investment A in
the supermatrix, shown in Table 5 .



              Criteria
              Cluster
                                  Interest         Capital Gains
                                  Return           Return




              Alternatives
              Cluster
                              Alternative A          Alternative B




                     Figure 1The ANP Model for the Investment Example




                                              4
We start with the raw data shown in Table 4 below. We then normalize the data so it
sums to 1 for each cluster in each column as shown in Table 5. We raise this matrix to
powers until it converges to obtain the limit supermatrix which has all its columns the
same, as shown in Table 6. Finally, normalize for each cluster in the column to obtain
the overall priorities as shown in the rightmost column of Table 6. Note that we now
have overall or synthesized priorities for all the nodes in the entire structure.




                                   Table 4 Data for the Investment Problem

                                                                 Criteria                      Alternatives
                                                         C1             C2                A                B
                                                      Interest        Capital            First          Second
                                                                      Gains           Investment      Investment
           1Criteria       C1 - Interest                   0                0              3              6
                           C2 - Capital Gains              0                0             10              15
           2Alternatives A - First inv.                    3                10             0              0
                           B - Second inv.                 6                15             0              0



                                 Table 5 Supermatrix with Data Normalized

                                                                 Criteria                      Alternatives
                                                         C1             C2                A                B
                                                      Interest        Capital            First          Second
                                                                      Gains           Investment      Investment
           1Criteria       C1 - Interest                   0                0           0.230769          0.285714
                           C2 - Capital Gains              0                0           0.769231          0.714286
           2Alternatives A - First inv.              0.333333           0.400012           0              0
                           B - Second inv.           0.666667           0.599988           0              0



              Table 6 Limit Supermatrix Yields Priorities of Alternatives and Criteria
                                                                                                         OVERALL
                                                      Criteria                         Alternatives     PRIORITIES
                                                   C1              C2                A             B   (Normalize to
                                                Interest         Capital            First       Second    1.0 for
                                                                 Gains           Investment Investment each cluster)
       1Criteria       C1 - Interest             0.132353        0.132353          0.132353    0.132353        0.264706
                       C2 - Capital Gains        0.367647        0.367647          0.367647    0.367647        0.735294
       2Alternatives A - First inv.              0.191181        0.191181          0.191181    0.191181        0.382362
                       B - Second inv.           0.308819        0.308819          0.308819    0.308819        0.617638


The overall priorities of the nodes in the model are given in the rightmost column in
Table 6. They are correct and are what we computed they should be in Table 3 using
ordinary arithmetic:

A1 (Investment A) = 0.382
B1 (Investment B) = 0.618


                                                               5
C1 (Interest Criterion) = 0.265
C1 (Capital Gains Criterion) = 0.735

The final priorities fall out of the ANP model in a very natural way by normalizing the
data to get priorities and entering the priorities in the ANP supermatrix. There is no need
to do any of the side calculations that were necessary to produce the required priorities
for the criteria to get the correct answer with the AHP model. The ANP model produces
the correct priorities for all the nodes in one fell swoop. This is a simple case involving
an ANP framework, but ANP models generally deal with far greater complexity and
interdependence than this trivial example shows.

3. Barzilai’s Marketing Example
Consider the following problem. Barzilai uses it to illustrate his proposition that the AHP
gives different results depending on the structure of the decision hierarchy. He uses data
in his exercise so that it will not be clouded by the issue of deriving priorities from
pairwise comparisons.

“The president and three vice-presidents of a company are analyzing their marketing
options. The company produces and sells a single product for a fixed price through five
stores in the city. Stores 1 and 2 are in the city’s West Side, store 3 is at City Centre, and
stores 4 and 5 are in the East Side. They all agree to define the company’s value function
v( x)  v( x1 ,..., x5 ) as its total annual revenue where xi represents annual sales in millions
of dollars in store i, i = 1,…,5. The company needs to choose between marketing
strategies A and B. These strategies will result in annual revenue of P = (3, 3, 1, 1, 1)(i.e.,
 x1  3, x2  3, x3  1, etc.) if strategy A is chosen and Q = (1, 1, 1, 3, 3) if strategy B is
implemented. In our terminology, the criteria are xi , i  1,...,5 , the alternatives are
marketing strategies A and B and the coordinates of the alternatives in the evaluation
space are given by the points P and Q.

The four executives agree that the criteria are identical: xi differs from x j only in its
index. Therefore, the president concludes that
                                 v( x)  x1  x2  x3  x4  x5
and since v( P)  v(Q)  9, the two alternatives are equally preferred.”

Barzilai is somewhat confused in the above paragraph when he says: “The four
executives agree that the criteria are identical: xi differs from x j only in its index.”
Earlier he had defined the xi as: “ xi represents annual sales in millions of dollars in store
i, i = 1,…,5.” Perhaps this was a simple grammatical error that makes it seem as though
he is referring to the xi as criteria when what he really means is that the stores
aggregated into territories, si in Figure 2, are criteria, but it may have contributed to his
misunderstanding.




                                               6
In any case, the four executives cannot simply agree that “ xi differs from x j only in its
index.” They are the annual sales from the stores and their values are either 1 or 3
depending on which of the two marketing strategies is employed. This confusion of
using the same symbol to denote both a node and a value for the node leaves his whole
work mathematically inaccurate and highly questionable. Even when one tries with the
best will to interpret it in the most positive light, one is able to see much confusion in
Barzilai’s logic about working with value functions and the way he assigns values to the
variables.

Barzilai’s value function v( x)  x1  x2  x3  x4  x5 is a common sense arithmetic
solution to the problem that everyone would agree gives the correct answer. This value
function uses data, the annual sales for each store, in terms of dollars and the answer
happens to be the same, 9 million dollars, whether one applies it to strategy A or strategy
B. He then structures three different hierarchies for this decision problem and obtains a
different value function for each hierarchy, on page 168. But the value functions this
time are weights, not dollars, though he compares the results with his original value
function of 9 million dollars. He incorrectly concludes that the AHP is flawed, rather
than that the flaw is in how he went about calculating his value functions.

In fact, all three hierarchical structures do give the same result when the priorities of the
nodes in the three structures are correctly calculated along the lines of the measurement
example of Section 2 above. His basic reference is a value function, shown above, which
expresses the total annual sales in dollars as a function of the xi . But writing a single
value function is incorrect. The 9M dollars is computed as the outcome of a strategy and
there should be two value functions, one for each strategy. It is a coincidence that his
original value function happens to give the same result, with annual sales totaling 9
million dollars, for both strategies:
vA ( x)  x1  x2  x3  x4  x5  3  3  1  1  1  9M , and
vB ( x)  x1  x2  x3  x4  x5  1  1  1  3  3  9M
According to Barzilai the three vice-presidents use the AHP to decompose the problem as
follows. The first, whose territory is City Centre, decomposes the criteria into the tree in
Figure 2. The variables xi are defined as above with s1 representing total revenue from
sales in stores 1 and 2, s2 sales in store 3 and s3 sales in stores 4 and 5, that is, this vice
president groups the stores the same way they are grouped into sales territories. As
before, he assumes that equal weights must be attached to the edges at each node
(essentially a criterion) of the tree “Therefore, according to the AHP procedure, in an
obvious notation, the intermediate weights (“local” priorities in AHP terminology) are
given by ws1  ws2  ws3  1/ 3 , wx1  wx2  1/ 2 , wx3  1 , and wx4  wx5  1/ 2 .”     This
casual and unjustifiable assumption is why Barzilai’s AHP formulations do not give the
correct result. The priorities of the criteria inescapably depend on data when it is given
for the alternatives. One would have to calculate what those priorities must be in order
for the model to result in the given data. It is a case of dependence not independence that
Barzilai assumes. Usually in an AHP model the priorities of the alternatives are derived,
not given, but Barzilai takes this approach in an effort to prove the AHP is faulty.


                                              7
                                v1



           s1               s2                  s3



x1                   x2     x3            x4          x5


      Figure 2 Hierarchy by VP for “Center”


                            v2



                t1                              t2



     x1               x2        x3         x4         x5


          Figure 3 Hierarchy by VP for “East”




                           v3



           u1                                  u2



x1                   x2              x3   x4         x5




                            8
                             Figure 4 Hierarchy by VP for “West”


Starting with these three structures and arbitrarily assuming default equal priorities for
the criteria at the same time that he assumes data for the alternatives that contradict the
assumed priorities Barzilai ends up with three different answers. Furthermore, he
computes a value function on the structures that is in terms of priorities, but he then
compares it to his original value function that was in terms of dollars. The value
functions for the three structures, his measure of whether the AHP composition works
correctly or not, are computed without any input concerning the known annual sales data
of the two strategies for the stores. How could a value function applied to some
hierarchical structure with arbitrary priorities be expected to give back known answers
when there is no information supplied to the function about the assumed data? It puts one
in mind of giving a Chinese student without knowledge of English a question written in
English and when he does not reply properly concluding that he does not know the
correct answer.

The central issue in this example, and the wrong thinking that may be at the root of many
of Barzilai’s criticisms, is that he mixes up how he is deals with data and with priorities
and furthermore he does not know that though it is possible to go from data to priorities it
is not possible to go the other way without outside information to link the priorities to the
data.

One would rarely if ever set up a practical decision problem in the AHP the way he does.
If there are two criteria with outcomes in the same scale, one would usually first combine
them into a single criterion using whatever arithmetic is used in practice to combine the
outcomes. One would then use that criterion in the AHP problem. In any case, the AHP
always gives the correct answer if the model is set up correctly.


4. Barzilai’s Wrong Assumption
To formulate his problem correctly with AHP, Barzilai should have followed his two
strategies, A and B, with corresponding outcomes (alternatives) Pi and Qi, for the ith store.

Let us show that when his hierarchies are set up properly for the data that are given, the
AHP composition is correct and gives the same results, in relative terms, for v A ( x) and
 vB ( x) . This problem can also be formulated as an ANP network with dependence and it
gives the correct answer as well but we demonstrate it here only for the first of Barzilai’s
three cases.


Correct AHP Model for “Center” VP’s Structure
The first vice president, whose territory is the center of the city comes up with the
structure shown in Figure 5. In Barzilai’s notation the strategies P and Q for the
alternatives are given in terms of millions of dollars for each store, with
P=(3M,3M,1M,1M,1M) the result of applying strategy A and Q=(1M,1M,1M,3M,3M)



                                             9
the result of applying strategy B, having aggregated the data upward through the
hierarchy for all nodes above the bottom level alternative nodes. The value xi associated
with the ith store Si is the sum of the ith component of the strategies P and Q immediately
below it and is written in the second line of each box. Normalizing the data for the nodes
directly under each parent node one obtains the relative values, the priorities, shown in
the first line of each box.




                                                             V1
                   s1=8/18=.444                              s2=2/18=.111                   s3=8/18=.444
                        8M                                        2M                             8M

     Store S1                 Store S2                Store S3                Store S4                 Store S5
    x1=4/8=.5                x2=4/8=.5                x3=2/2=1               x4=4/8=.5                x5=4/8=.5
       4M                       4M                       2M                     4M                       4M


                P1=3/4=.75               P2=3/4=.75              P3=1/2=.5               P4=1/4=.25               P5=1/4=.25
                   3M                       3M                      1M                      1M                       1M

                Q1=1/4=.25               Q2=1/4=.25              Q3=1/2=.5               Q4=3/4=.75               Q5=3/4=.75
                   1M                       1M                      1M                      3M                       3M


                             Figure 5 Marketing Decision Tree by VP of “Center”
Below the Goal are the territories, below them the stores and below the latter the two
alternative strategies.

v1 ( A) = 0.444×0.5×0.75+0.444×0.5×0.75+
                0.333×1×0.5+0.444×0.5×0.25+0.444×0.5×0.25 = 0.50

v1 ( B ) = 0.444×0.5×0.25+0.444×0.5×0.25+
                0.333×1×0.5+0.444×0.5×0.75+0.444×0.5×0.75 = 0.50

We see that the two value functions, for strategy A and strategy B, have correct equal
priorities of 0.50, and not those Barzilai calculated incorrectly.

The priorities of the criteria need to be calculated from the priorities of the alternatives in
the part of the tree beneath each criterion. Barzilai does not calculate his value function
for the two strategies. Had he done it correctly he would have had to develop priorities
for the criteria above the strategies as shown in Figure 5. All three of his examples would
give the correct answer when done correctly. One cannot simply assign arbitrary weights


                                                                 10
to the criteria and hope that these weights will somehow give the correct answer for value
functions at whatever level in the tree they are computed.

The ANP Formulation of the “Center” VP’s Structure
We will formulate the AHP structure above as an ANP model. Here again we get the
precise answer we obtained before. It too is at variance with what Barzilai obtained. We
start by putting the known data in the supermatrix and showing that the resulting
priorities for each of the alternative strategies are 0.5. The same results would be
obtained if the ANP model were done for the other two structures.




                                  Table 7 The Data* for Center VP’s Structure

                              Districts                                        Stores
                                                                                                               Alternative
                                                                                                               Strategies
                    S1            S2          S3        X1          X2           X3          X4       X5         P           Q
Districts     S1     0.000         0.000      0.000           4           4      0.000       0.000    0.000     0.000    0.000
              S2     0.000         0.000      0.000     0.000       0.000               2    0.000    0.000     0.000    0.000
              S3     0.000         0.000      0.000     0.000       0.000        0.000            4        4    0.000    0.000
Stores        X1         4         0.000      0.000     0.000       0.000        0.000       0.000    0.000          3           1
              X2         4         0.000      0.000     0.000       0.000        0.000       0.000    0.000          3           1
              X3     0.000                2   0.000     0.000       0.000        0.000       0.000    0.000          1           1
              X4     0.000         0.000           4    0.000       0.000        0.000       0.000    0.000          1           3
              X5     0.000         0.000           4    0.000       0.000        0.000       0.000    0.000          1           3
Alternative   P      0.000         0.000      0.000           3           3             1         1        1    0.000    0.000
 Strategies Q        0.000         0.000      0.000           1           1             1         3        3    0.000    0.000

* With data in terms of millions of dollars.
      Table 8 Data Normalized* by Cluster to become the Supermatrix for Center VP’s Structure

                              Districts                                        Stores
                                                                                                               Alternative
                                                                                                               Strategies
                    S1            S2          S3        X1          X2           X3          X4       X5         P           Q
Districts     S1     0.000         0.000      0.000           1           1      0.000       0.000    0.000     0.000    0.000
              S2     0.000         0.000      0.000     0.000       0.000               1    0.000    0.000     0.000    0.000
              S3     0.000         0.000      0.000     0.000       0.000        0.000            1        1    0.000    0.000
Stores        X1         .5        0.000      0.000     0.000       0.000        0.000       0.000    0.000       .333       .111
              X2         .5        0.000      0.000     0.000       0.000        0.000       0.000    0.000       .333       .111
              X3     0.000                1   0.000     0.000       0.000        0.000       0.000    0.000       .111       .111
              X4     0.000         0.000           .5   0.000       0.000        0.000       0.000    0.000       .111       .333
              X5     0.000         0.000           .5   0.000       0.000        0.000       0.000    0.000       .111       .333
Alternative   P      0.000         0.000      0.000          .75         .75            .5     .25      .25     0.000    0.000
 Strategies Q        0.000         0.000      0.000          .25         .25            .5     .75      .75     0.000    0.000
*With normalized data in the form of priorities
                          Table 9 Limit Supermatrix for Center VP’s Structure



                                                               11
                           Districts                                      Stores                     Alternative
                                                                                                     Strategies
                   S1          S2        S3           X1        X2          X3       X4      X5        P           Q
Districts     S1   0.111        0.111    0.111        0.111       0.111     0.111    0.111   0.111    0.111    0.111
              S2   0.028        0.028    0.028        0.028       0.028     0.028    0.028   0.028    0.028    0.028
              S3   0.111        0.111    0.111        0.111       0.111     0.111    0.111   0.111    0.111    0.111
Stores        X1   0.111        0.111    0.111        0.111       0.111     0.111    0.111   0.111    0.111    0.111
              X2   0.111        0.111    0.111        0.111       0.111     0.111    0.111   0.111    0.111    0.111
              X3   0.056        0.056    0.056        0.056       0.056     0.056    0.056   0.056    0.056    0.056
              X4   0.111        0.111    0.111        0.111       0.111     0.111    0.111   0.111    0.111    0.111
              X5   0.111        0.111    0.111        0.111       0.111     0.111    0.111   0.111    0.111    0.111
Alternative   P    0.125        0.125    0.125        0.125       0.125     0.125    0.125   0.125    0.125    0.125
 Strategies Q      0.125        0.125    0.125        0.125       0.125     0.125    0.125   0.125    0.125    0.125


                   Table 10 Final Normalized Priorities for Center VP Structure

                                         Districts            S1             0.444
                                                              S2             0.111
                                                              S3             0.444
                                          Stores              X1             0.222
                                                              X2             0.222
                                                              X3             0.111
                                                              X4             0.222
                                                              X5             0.222
                                        Alternative           P              0.500
                                        Strategies            Q              0.500


Again, as we see, the priorities of P and Q (for strategies A and B) are shown using the
ANP to be 0.5 and 0.5, the correct answer. A side benefit is that we also get the priorities
of all the other nodes in the model. Note that S2 with a priority of 0.111 is the least
desirable district. It has only one store while S1 and S3 have two stores providing
income. This matches our common sense that the less stores in a territory the less the
income for that territory and the less desirable it is regardless of marketing strategy.

Correct AHP Model for “East” VP’s Structure
Let us now create the right AHP model for the second VP’s structure. The second vice
president whose East side territory contains stores 4 and 5 lumps together all sales outside
his territory and uses the structure shown in Figure 6.




                                                           12
                                                                   V2
                                t1=10/18=.555                                                      t2=8/18=.444
                                     10M                                                                8M


   X1=4/10=.40              X2=4/10=.40               X3=2/10=.20                 X4=4/8=.5                 X5=4/8=.5
       4M                       4M                        2M                         4M                        4M

                P1=3/4=.75                P2=3/4=.75                P3=1/2=.50                P4=1/4=.25                P5=1/4=.25
                   3M                        3M                        1M                        1M                        1M

                Q1=1/4=.25                Q2=1/4=.25                Q3=1/2=.50                Q4=3/4=.75                Q5=3/4=.75
                   1M                        1M                        1M                        3M                        3M

                    Figure 6 Marketing Decision Tree by VP of “East” (Stores 4 and 5)

Here we have
v2 ( A) = .555×.40×.75+.555×.40×.75+.555×.20×.5+.444×.5×.25+.444×.5×.25 = .50
v2 ( B) = .555×.40×.25+.555×.40×.25+.555×.20×.5+.444×.5×.75+.444×.5×.75 = .50
and again the two value functions for strategy A and strategy B have equal priority of 0.5.

Correct AHP Model for “West” VP’s Structure
The third vice president whose territory is the West side with stores 1 and 2 and 3 lumps
together all sales outside his territory and arrives at the structure shown in Figure 7.




                                                               V3

                  u1=8/18=.444                                                     u2=10/18=.555
                       10M                                                                8M

                            X2=4/8=.50                X3=2/10=.2                 X4=4/10=.4                X5=4/10=.4
  X1=4/8=.50
     4M                        4M                        2M                         4M                        4M


               P1=3/4=.75                P2=3/4=.75                 P4=1/2=.50                P5=1/4=.25                P5=1/4=.25
                  3M                        3M                         1M                        1M                        1M


               Q1=1/4=.25                Q2=1/4=.25                 Q4=1/2=.50                Q5=3/4=.75                Q5=3/4=.75
                  1M                        1M                         1M                        3M                        3M



                    Figure 7 Marketing Decision Tree by VP of “West” (Stores 1 and 2)
Here we have:
v3 ( A) = .444×.50×.75 + .444×.50×.75+.555×.2×.5+.555×.4×.25+.555×.4×.25 = .50
v3 ( B) = .444×.50×.25 + .444×.50×.25+.555×.2×.5+.555×.4×.75+.555×.4×.75 = .50
and again the value functions for the two strategies are equal.



                                                                13
Finally, we have outside information so in this case we can convert the strategies back
into dollars. We know the total sales are 18M, by adding all the dollars for both
strategies at the bottom of the model. Multiplying the priority value of 0.50 for either
strategy by 18M we obtain 9M. This corresponds to the original value function of the
president of the company, and the actual annual sales from the five stores under both
scenarios.

AHP is known to be particularly appropriate for handling intangibles, so one might ask
how would the criteria be weighted when dealing with intangibles? Criteria, whether
they are tangible or intangible, need to be weighted when they depend on their
alternatives. With intangibles one needs to be very familiar with the alternatives before
pairwise comparing the criteria. In Barzilai’s marketing example, in Figure 5, when
making pairwise comparisons of the importance of the S1 and S2 groupings of stores, but
not directly using the information about the incomes from the stores under the different
strategies, the decision maker would observe that there are two stores in S1 and only one
in S2, and therefore would know that S1 should be about twice as important as S2. But the
annual sales from the store in the middle grouping are known to be less under than the
other districts under either strategy, so the decision maker might raise his or her judgment
from 2 to 3 when comparing the two to take this into account. And as S1 and S2 have the
same number of stores, without precisely calculating the income under the two strategies,
S1 would be judged to be equally as important as S3. The pairwise comparison matrix and
vector of priorities using judgment rather than data might sensibly be:
      Table 11 Pairwise Comparison of Groupings of Stores based on Perceived Importance
                                                                            Priorities
                                                       Priorities
            S1             S2            S3                                 derived earlier
                                                       (from judgment)
                                                                            from data
 S1         1              3             1             .43                  .444
 S2         1/3            1             1/3           .14                  .111
 S3         1              3             1             .43                  .444

The priorities derived from the decision maker’s “soft” judgments about the importance
of the territories are not very different from the values computed using the data. Similar
kinds of analyses can be done throughout the structure, one would see that the priorities
obtained based on judgments, which one would have to use for intangibles, would be
fairly similar to those obtained from data.

5. On Barzilai’s So-Called Linearity Proof
Barzilai assumes what he is trying to prove in his Theorem 2 on page 161. He shows that
the marginal substitution rate of xi with respect to x j for a linear value function is
constant and goes on to enunciate the following theorem on page 162 of his paper,
without proof, that the AHP gives rise to linear value functions. He writes, incorrectly,
that:

“Theorem 3. The value functions generated by the AHP are always linear.”



                                               14
In the AHP with its relative measurement, the priorities of the alternatives are derived
from paired comparisons and the resulting values are not linearly related, therefore
substituting different values of the variables for different alternatives generally produce
nonlinear values for the value function. Later in the appendix we give several validation
examples to show that the priorities in the AHP are non-linear functions of their
attributes. In addition, the AHP is a special case of the ANP with its dependencies in
which composition yields multilinear forms of infinite order whose entries are functions,
not linear variables. The AHP derived function is not a linear Cartesian function
involving orthogonal axes nor is it affine involving non-orthogonal axes, but curvilinear
possibly with locally affine coordinate systems from which it is well known one cannot
read off incremental changes from point to point by simply taking the affine derivatives.
One must consider not only the increments of the components due to the transition from a
point to an infinitesimally close point, but also to the change in the local coordinate
system. Counterexamples are well-known. In other words one must not identify the
components of the increments of a tensor with the increments of its components. Nor are
the coefficients in the AHP arbitrarily determined. When one performs comparisons of
the attributes that depend on the alternatives, one either uses the ANP directly for
accuracy, or approximates with a hierarchy by thinking about what the attributes mean
through the alternatives. Their meaning is not independent of the concrete alternatives
which they characterize. Barzilai, rather haphazardly, although I am certain that he
knows better, concludes that the AHP has a linear value function and proceeds to arrive at
a conclusion having nothing to do with the AHP. In his heart example discussed in the
next section it appears that he thinks that the criteria may depend in their priorities on the
alternatives, and admits that it is a subject for the ANP but then gallantly dismisses the
ANP by, for example, referring to himself and to Kamenetzky, a student of Saaty’s, who
knew the subject of hierarchies well and was correct to observe that hierarchies can only
be used when the criteria are assumed to be independent of the alternatives. At the time
Kamenetzky did not know about the ANP that deals with this type of dependence.

Hierarchic composition is given by:
                                             i i         i
(1)                                        x11 x22     x pp
                                       i1 , ,i p


where the xi do not belong to a linear system of coordinates, but, as we said before, are
generally non-linear functions derived in the AHP through paired comparisons. Recall
that the components of a priority vector are not linear variables, but are readings from
non-linear functions.

The richer the structure of a hierarchy in breadth and depth, for example that shown in
Figure 8, the more complex are the multilinear forms, which of course are non-linear,
derived from it. Let us show that AHP composition gives rise to multilinear forms whose
marginal rate of substitution is not constant. The composite vector for the entire hth level
is shown in (1) above and it has covariant tensorial components. Similarly, the left



                                                   15
eigenvector approach to a hierarchy gives rise to a vector with contravariant tensor
components.

Consider a hierarchy of two criteria, the first with three subcriteria and the second with
two subcriteria and finally with two alternatives as shown in the figure below together
with hierarchic composition and marginal rate of substitution. One can see that neither is
the first linear nor is the second constant contradicting Barzilai’s incorrect claims.




                                           16
Focus:                     At what level should the Dam be kept: Full or Half-Full?

Decision                           Financial                 Political
                                      x11                      x12
Criteria:


Decision        Congress        Dept. of Interior        Courts              State   Lobbies
Makers:           x21                 x22                 x23                 x24      x25




Alternatives:                            Half-Full Dam                   Full Dam
                                               x31                          x32


                v=(x11 x21 x31+ x11 x22 x31+ x11 x23 x31+ x12 x24 x31+ x12 x25x31,
                 x11 x21 x32+ x11 x22 x32+ x11 x23 x32+ x12 x24 x32+ x12 x25 x32)




            Figure 8 A Simple Hierarchy of Three Levels to Demonstrate Hierarchic Composition


   Consider a hierarchy whose goal is to find the relative cost of three alternatives. Let the
   criteria be area (A) and volume (V) and let alternatives be three spheres ( S1 , S2 , S3 ). The
   criteria weights are the relative cost of unit of area versus a unit of volume. In general,
   the area and volume of a sphere with radius r is equal to 4 r 2 and 4  r 3 , respectively.
                                                                              3

   Let the three spheres S1 , S2 and S3 have radii equal to r1 , r2 and r3 , respectively. Thus,
   the priorities of the spheres with respect to their area is given by
                                                                         
                 4 r12                 4 r22                  4 r32 
     f1 (r1 )            , f1 (r2 )            , f1 ( r3 )            
                                                                       2 
                      3                      3                       3

                4  ri 2
                                        4  ri 2
                                                                4  ri 
                   i 1                   i 1                    i 1   

   and their relative volume is given by

                                                                    
                 4
                    r   3            4
                                        r   3            4
                                                            r  3
    f 2 (r1 )  3 3 1 , f 2 (r2 )  3 3 2 , f 2 (r3 )  3 3 3  .
                                                                  3 
                3   ri            3   ri            3   ri 
                4          3        4          3        4
   
                   i 1                i 1                i 1     




                                                             17
Note that the relative areas and volumes are non-linear functions. Let c1 be the cost per
unit area and let c2 be the cost per unit volume. Then, the costs of the spheres are given
by       c1 4 ri2  c2 4  ri3 ,
                        3
                                                            i=1,2,3,              and         their      relative   values    are   given    by
     c1 4 ri  c2 4  ri
                   3
                    2                      3

 3
                                                       , i=1,2,3.
 (c 4 r
i 1
         1              i
                            2
                                 c2 4  ri )
                                     3
                                               3




To obtain the relative costs of the spheres from the relative areas and volumes we would
have to find values 1 and  2 such that

        4 ri 2                            4
                                                ri3                 c1 4 ri 2  c2 4  ri3
1            3
                                 2       3
                                                   3
                                                                3
                                                                                     3
                                                                                                       , i=1,2,3. Clearly, 1 and  2 are given
       4  ri              2          4
                                       3     ri       3
                                                                 (c 4 r
                                                                        1     i
                                                                                  2
                                                                                      c  4
                                                                                        2 3    ri )
                                                                                                 3

             i 1                              i 1             i 1
by
                                      3                               3
                                                                                  
                             c1 4  ri 2                    c2 4   ri3
                                                                 3                
                  1  3            i 1
                                                  , 2  3           i 1
                                                                                  .
                                                                              3 
                 
                 
                         (c1 4 ri  c2 3  ri )
                         i 1
                                   2       4   3
                                                         (c1 4 ri  c2 3  ri ) 
                                                         i 1
                                                                   2      4

                                                                                  
In practice the criteria weights would be estimates of these values but so would also be
the functions f1 and f 2 . We give an example of three spheres with different radii in
Table 12. A sphere costs $1 for each unit of its surface area plus $5 for each unit of its
volume. Think of hot air balloons that have to be purchased by the square unit of area,
but filled with hot gas to fly them according to their volume. Volume and surface area
are not linearly related.




                                   Table 12 Hot Air Balloon Example Showing Non-linear Relationship


                                       f1(x)                         f2(x)


                                           Area                      Volume




                                                                                           18
Spheres     Radius         $1                $5
                                                          Area Cost        Volume Cost      Total Cost        Relative Cost
S1             1        12.56637         4.1887902          $12.57            $20.94         $33.51           0.036036036

S2             2        50.26548         33.510322          $50.27           $167.55         $217.82          0.234234234
                                                                                                              0.72972973
S3             3        113.0973         113.09734         $113.10           $565.49         $678.58


             Total      175.9292         150.79645      $175.93            $753.98                $929.91


                                                               $929.91
          Criteria   $175.93/$929.91 $753.98/$925.91
          Weights
                     =0.189189         =0.8108108


                                                                           Estimated
Spheres    Relative Area               Volume           Final Priorities Final Priorities
S1                      0.071429         0.0277778       0.036036036          0.0351
S2                      0.285714         0.2222222       0.234234234          0.2328
S3                      0.642857            0.75           0.72972973         0.7321


                                                                       1
                            0.166667        0.8333333


Barzilai assumes incorrectly that the AHP approximates hierarchic composition by a
linear value function in his statement on page 165: “It is tempting to assume that when
the components fk of v( x) are replaced by their first-order approximations, these
components may be converted to normalized linear value functions as well. We show in
Section 5.4 that this implicit assumption which is utilized in AHP decompositions is
incorrect.”

In Section 5.4 of his paper Barzilai assumes that the priorities of the alternatives are given
but those of the criteria are arbitrarily determined and from that concludes that his value
functions will be the same under linear transformations but not under affine
transformations. Here again he misses the fundamental point that once the weights of the
alternatives are given, those of the criteria are automatically determined in relative terms
and that composing with these normalized weights of the criteria automatically gives the
correct relative values of the alternatives which he ignores by assuming that the priorities
of the criteria are arbitrarily given. He is correct to conclude that when the weights of the
criteria are independent from those of the alternatives, the final value functions would not
lead to the same relative measurement of the alternatives under affine transformations but
one never does that when one has measurements for the alternatives.

Let Ai , i  1,..., n be the alternatives and let C j , j  1,..., m be the criteria. Let xij be the
                                                                                                       m
actual measurement of alternative Ai with respect to criterion C j then                             x j 1
                                                                                                              ij   is the total
                                                                                       m      n    m
value of alternative Ai with respect to all the criteria C j and                      xij /  xij is its overall
                                                                                     j 1    i 1 j 1




                                                        19
relative value. Let us show now how to obtain this by normalizing the alternatives with
respect to each criterion. With that we have for alternative Ai with respect to criterion C j
                            n    m
the normalized value xij /  xij . Now we assign criterion C j the relative value of the
                           i 1 j 1

sum of the alternative values under it to the sum of the values of all the alternatives with
                                                          n                  n    m
respect to all the criteria. We have                      xij /  xij for the priority of criterion C j .
                                                         i 1               i 1 j 1

Finally we multiply the normalized weight of alternative Ai under each criterion by the
corresponding weight of that criterion and take the sum over all the criteria. We obtain
                                                         n                          m

                                     n     xij         x         ij              x         ij

for the relative value of Ai ,           n
                                                        i 1
                                                       n m
                                                                            
                                                                                   j 1
                                                                                  n m
                                                                                                       which is precisely the overall
                                  j 1
                                          x  x
                                         i 1
                                                 ij
                                                      i 1 j 1
                                                                       ij         x
                                                                                 i 1 j 1
                                                                                                  ij


relative value of alternative Ai obtained above from measurements. It is clear that there is
no other meaningful way to obtain this answer without the particular weights assigned to
the criteria.

Again, we repeat that when given the weights of the alternatives the AHP does how we
just described and not the way Barzilai does it by assigning arbitrary weights to the
criteria. When measurements are not given one either proceeds from the bottom up or the
top down to derive weights for the alternatives in terms of the criteria and the criteria in
terms of higher level criteria. problems are not ever set up this way in practice by starting
with the weights of the alternatives. In the usual AHP problem the alternative weights
are unknown and we need the judgments of a knowledgeable person to derive them. The
many validation examples in the Appendix provide evidence that the AHP correctly
reflects measurable events in the real world.


6. The Heart Transplant Example – A Poorly Drawn Non-conclusion by Barzilai
Not only does Barzilai draw incorrect conclusions and prove false theorems about the
AHP but then also draws misleading attention to examples that he does not show any
fault with. He takes a heart transplant example and says see I have shown with my
examples that the AHP does not work, so how can one trust it with an application like the
heart transplant example. He not only does his mathematics poorly but regrettably
spreads his erroneous thinking and conclusions to groups where he has no justification
whatever to doubt the intelligence and understanding of the medical experts who did the
exercise and who understand the AHP and its fundamentals very well.

In the heart transplant example Barzilai simply shows that hierarchic composition is a
product of matrices of priorities, a multilinear mapping that involves sums of products of
nonlinear variables (functions) linearly. That is, each is raised to the first power. Thus
neither the variables nor the composition are linear. Barzilai incorrectly assumes it to be
linear.



                                                                20
In its simplest hierarchic form with only a goal, criteria and alternatives, the outcome of
the AHP is a convex combination of the weights of the alternatives i fi ,
                                                                                      i

where   i  1 . These priorities ( f i ) are themselves non-linear functions and not
       i
ordinary linear Cartesian variables.


7. Observations and Conclusions

Of course, a person who knows well how the AHP works and has used it in practice
would never actually set up the kind of problem Barzilai does from which he draws
incorrect answers and conclusions to criticize the AHP. If there is more than one
criterion with an existing scale and alternatives under them with actual data, in practice
one would first combine everything into a single criterion using whatever standard
arithmetic is required. These criteria are then usually put alongside intangible criteria and
the usual process of paired comparisons is applied to determine the importance of the
criteria with respect to the goal. Barzilai set up the problem in this way in an attempt to
show that the AHP does not work. But it does work, both on his very artificially
conceived and wrongly developed example, and in general as a sound theory.

In his paper “On MAUT, AHP, PFM” in the Proceedings of the ISAHP meeting in Kobe
in 1999, Barzilai wrote that Belton and Gear and Dyer “have misidentified the problem,
and proposed incorrect revisions and failed to address the fundamental issues” yet he
frequently uses them to support his arguments against the AHP. He also says in that paper
about measurement theory “Basic concepts such as scale type and meaningfulness are not
fully understood… they will be fully resolved in a forthcoming paper.” That was six
years ago, and we await with anticipation a constructive and complete exposition of
Barzilai’s ideas about measurement in multicriteria decision-making and how to use them
in practice to make decisions.

There are numerous validation examples developed by many people using pairwise
comparison matrices, hierarchies and networks for which the answers are already known
that show how robust and accurate the AHP/ANP is.


References

Barzilai, Jonathan, On the Decomposition of Value Functions, Operations Research
Letters 22 (1998) 159-170.

Saaty, Thomas L., Decision-making with the AHP: Why is the principal eigenvector
necessary, European Journal of Operations Research 145 (2003) 85-9.




                                             21
Saaty, Thomas L., Fundamentals of Decision Making and Priority Theory with the
Analytic Hierarchy Process, Vol. VI, AHP Series, RWS Publications, Pittsburgh, PA
2000

Saaty, Thomas L., The Analytic Network Process: Decision Making with Dependence
and Feedback, RWS Publications, Pittsburgh, PA 2001.

                                     APPENDIX
SINGLE MATRIX VALIDATION EXAMPLES
Relative sizes of Areas: Here is a validation exercise we have often used with
individuals when introducing them to the Analytic Hierarchy Process. We ask the
individual to estimate the relative areas of the five geometric figures below using the
AHP. The person makes pairwise comparison judgments as to how much bigger in area
the first figure is than the second as shown in the matrix below. For example, an area of
12 square units is 3 times bigger than an area of 4 square units. So the judgment 3 is
used. Use the numbers 1 to 9 (or their reciprocals if the second of the pair is bigger than
the first) from the Fundamental Scale of the AHP; decimals in between are allowed. The
judgments are entered in the upper triangular half of the matrix. The entire exercise
requires the individual to make n(n-1)/2 judgments, where n is the number of things
being compared, in this case n = 5, so for this exercise 10 judgments are required. Enter
the reciprocals of those judgments in the bottom triangular half of the matrix. The relative
sizes of the figures are then given by the principal eigenvector of the matrix or, in the
absence of a computer, approximately by normalizing each column to one and taking the
average of corresponding entries in the columns. This is the priority vector of relative
values. It is remarkable how close one can come to the actual relative sizes in this way.
Often verbal judgments are made first particularly when dealing with intangibles.




The actual relative values of these areas are approximately A = 0.47, B = 0.05, C = 0.24,
D = 0.14, and E = 0.09. Here A is 47% of the overall area of the five figures, B is 5%



                                            22
and so on. Or, to put it another way, A/B=0.47/0.05=9.4, and A is about 9 times as large
as B, and the relative sizes of the other figures can be determined in the same way.


                              One Individual’s Judgment Matrix
                     Circle    Triangle  Square    Diamond  Rec-          Eigenvector
                                                            tangle          (priority
                                                                            vector of
                                                                             relative
                                                                              sizes)
            Circle     1          9         2.5        3           6          0.488
          Triangle    1/9         1        1/5       1/3.5       1/1.5        0.049
           Square    1/2.5        5          1        1.7          3          0.233
          Diamond     1/3        3.5       1/1.7       1          1.5         0.148
         Rectangle    1/6        1.5        1/3      1/1.5         1          0.082
       *Only the judgments in bold must be made, the others are automatically determined

If there is no computer available, a shortcut for computing the principal eigenvector is to
assume the matrix is consistent, normalize each column and then take the average of the
corresponding entries in the columns to obtain the priority vector. The priority vector
shown in the right column of the matrix is the actual eigenvalue solution, but using the
shortcut calculation gives a very similar result: A=0.486, B=0.049, C=0.233, D=0.148,
E=0.082.

It is easy to make consistent judgments in this exercise because we are dealing with
objects with measures that are quite familiar to most people. If one were to measure the
figures, compute the ratios of their areas exactly and enter these ratios instead of using
judgments, the priority vector would give back the exact relative areas of the figures.
The point here is to show that judgments can give accurate results. Sometimes people try
to measure the figures and use formulas dimly remembered from high school geometry to
verify the results. The errors associated with physically measuring the figures often give
worse results than eyeballing the figures and making judgments.

An interesting thing often happens when the exercise is tried with a group. In the
Analytic Hierarchy Process the judgments of the members of the group on a pair are
combined using the geometric mean. When the combined judgments are used in the
matrix often the answer is closer to the actual relative sizes of the figures than most of the
individuals’ answers are.


Relative Weights of Objects: The matrix below gives the estimated pairwise
comparisons of the weights of the five objects lifted by hand, made by the then President
of the Diners Club, a friend of the author. The two vectors appear to be very close but are
they really close? To determine closeness of two priority vectors one must use the Saaty
Compatibility Index.

                   Table Pairwise Comparisons of the Weights of Five Objects




                                              23
                                                  Large                           Small                Actual
                          Radio        Type-     Attache           Projec-       Attache    Eigen-    Relative
     Weight                            writer     Case               tor          Case      vector    Weights

           Radio           1            1/5        1/3               1/4           4         .09        .10

      Typewriter           5             1            2              2             8         .40        .39
        Large
     Attache Case          3            1/2           1              1/2           4         .18        .20

      Projector            4            1/2           2              1             7         .29        .27
        Small
     Attache Case          1/4          1/8        1/4               1/7           1         .04        .04


Relative Electric Consumption of Household Appliances: In the following table we
have paired comparisons done by students in Electrical Engineering estimating the
consumption of electricity of common household appliances. How compatible are the
derived and actual vectors?
          Table Relative Electricity Consumption (Kilowatt Hours) of Household Appliances



   Annual
                                                                                                        Actual
   Electric        Elec.                         Dish                               Hair     Eigen-
                               Refrig     TV               Iron          Radio                         Relative
                   Range                         Wash                               Dryer    vector
                                                                                                       Weights
   Consumpti
   on
   Electric
   Range             1           2           5    8            7           9           9      .393      .392
   Refrig-
   erator           1/2          1           4    5            5           7           9      .261      .242

   TV               1/5          1/4         1    2            5           6           8      .131      .167
   Dish-
   washer           1/8          1/5     1/2      1            4           9           9      .110      .120
   Iron
                    1/7          1/5     1/5     1/4           1           5           9      .061      .047
   Radio
                    1/9          1/7     1/6     1/9           1/5         1           5      .028      .028
   Hair-dryer
                    1/9          1/9     1/8     1/9           1/9         1/5         1      .016      .003

The hairdryer is of such a small magnitude that it probably should have been left out of
the other homogeneous comparisons.


                                                          24
This exercise was done on an airplane in 1973 by Thomas Saaty and Mohammad Khoja
by simply using their common knowledge about the relative power and standing of these
countries in the world and without referring to any specific economic data related to GNP
values
                    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    1/ 4      1        7         5       5       3          4      
                                                                                   
      China         1/ 9    1/ 7       1        1/ 5    1/ 5   1/ 7       1/ 5     
      France        1/ 6    1/ 5       5         1       1      1/ 3      1/ 3     
                                                                                   
          U .K      1/ 6    1/ 5       5         1       1      1/ 3      1/ 3     
      Japan         1/ 5    1/ 3       7         3       3       1          2      
     
      W .Germany 1/ 5                                                              
                                                                                    
                            1/ 4       5         3       3     1/ 2         1      

                        Normalized               Actual GNP (1972)       Normalized GNP
                        Eigenvector                                      Values
 U.S                    .427                     1,167                   .413
 U.S.S.R                .23                      635                     .225
 China                  .021                     120                     .043
 France                 .052                     196                     .069
 U.K                    .052                     154                     .055
 Japan                  .123                     294                     .104
 W. Germany             .094                     257                     .091


HIERARCHIC VALIDATION EXAMPLES

To make good applications needs expert knowledge of the subject, a structure that
represents the pertinent issues, and a little time to do justice to the subject. In this part we
give three hierarchic examples that gave results close to what the values actually were
and all the works were published in refereed journals.

World Chess Championship Outcome Validation – Karpov-Korchnoi Match

The following criteria and hierarchy were used to predict the outcome of world chess
championship matches using judgments of ten grandmasters in the then Soviet Union
and the United States who responded to questionnaires they were mailed. The predicted
outcomes that included the number of games played, drawn and won by each player
either was exactly as they turned out later or adequately close to predict the winner. The
outcome of this exercise was notarized before the match took place. The notarized
statement was mailed to the editor of the Journal of Behavioral Sciences along with the
paper later. See the coauthored book with Luis Vargas: Prediction, Projection and
Forecasting, Kluwer, 1991. The prediction was that Karpov would win by 6 to 5 games
over Korchnoi, which he did.


                                              25
                                   Table Definitions of Chess Factors
T (1)    Calculation (Q): The ability of a player to evaluate different alternatives or strategies in light of
         prevailing situations.
B (2) Ego (E): The image a player has of himself as to his general abilities and qualification and his
         desire to win.
T (3) Experience (EX): A composite of the versatility of opponents faced before, the strength of the
         tournaments participated in, and the time of exposure to a rich variety of chess players.
B (4) Gamesmanship (G): The capability of a player to influence his opponent's game by destroying
         his concentration and self-confidence.
T (5) Good Health (GH): Physical and mental strength to withstand pressure and provide endurance.
B (6) Good Nerves and Will to Win (GN): The attitude of steadfastness that ensures a player's health
         perspective while the going gets tough. He keeps in mind that the situation involves two people
         and that if he holds out the tide may go in his favor.
T (7) Imagination (IW: Ability to perceive and improvise good tactics and strategies.
T (8) Intuition (IN): Ability to guess the opponent's intentions.
T (9) Game Aggressiveness (GA): The ability to exploit the opponent's weaknesses and mistakes to
         one's advantage. Occasionally referred to as "killer instinct."
T (10) Long Range Planning (LRP): The ability of a player to foresee the outcome of a certain move, set
         up desired situations that are more favorable, and work to alter the outcome.
T (1 1) Memory M: Ability to remember previous games.
B (12) Personality (P): Manners and emotional strength, and their effects on the opponent in playing the
         game and on the player in keeping his wits.
T (13) Preparation (PR): Study and review of previous games and ideas.
T (14) Quickness (Q): The ability of a player to see clearly the heart of a complex problem.
T (15) Relative Youth (RY): The vigor, aggressiveness, and daring to try new ideas and situations, a
         quality usually attributed to young age.
T (16) Seconds (S): The ability of other experts to help one to analyze strategies between games.
B (17) Stamina (ST): Physical and psychological ability of a player to endure fatigue and pressure.
T (18) Technique M: Ability to use and respond to different openings, improvise middle game tactics,
        and steer the game to a familiar ground to one's advantage.




Monetary Exchange Rate – Dollar versus the Yen
In the late 1980’s three economists at the University of Pittsburgh, Professors A. Blair, R.
Nachtmann, and J. Olson, worked with Thomas Saaty on predicting the yen/dollar


                                                     26
exchange rate. The paper was published in Socio-Economic Planning Sciences 31,
6(1987). The predicted value was fairly close to the average value for a considerable
number of months after that.


                                                     Value of Yen/Dollar Exchange
                                                            Rate in 90 Days
               Relative                         Forward                      Official                              Rel. Deg. of                 Size/Direction of Past Behavior of
               Interest                        Exchange                   Exchg. Mkt.                               Confid. in                  US Current Acct. Exchange Rates
                 Rate                          Rate Bias                  Intervention                            the US Econ.                      Balance             .035
                 .423                             .023                        .164                                     .103                           .252

  Federal        Size of       Bank of     Forward Rate      Size of       Con-        erratic        Relative        Relative      Relative      Size of       Antici-     Rele-    Irrele-
Reserve Mon.     Federal    Japan Monet.    Premium/      Forward rate    sistent       .027          Infaltion        Real         Political    Deficit or     pated       vant      vant
   Policy        Deficit       Policy        Discount      Differential    .137                         Rates         Growth        Stability    Surplus       Changes      .004      .031
    .294          .032          .097           .007            .016                                      .019          .008           .075         .032          .221


   Tighter       Contract      Tighter        High           Premium      Strong       Strong          Higher         Higher         More         Large         Decrease    High     High
     .191          .002          .007         .002             .008         .026        .009             .013           .003         .048          .016            .090     .001     .010
   Steady          No          Steady         Medium         Discount     Moderate     Moderate        Equal          Equal          Equal        Small         No Charge   Medium   Med.
     .082        Change          .027           .002           .008           .100        .009          .006           .003           .022         .016             .106      .001    .010
   Easier         .009         Easier         Low                         Weak         Weak            Lower          Lower          Less                       Increase    Low      Low
    .191         Expand         .063          .002                         .011        .009             .001           .003          .006                         .025      .001     .010
                  .021
                                           Probable Impact of Each Fourth Level Fctor

     119.99 and below                           119.99-134.11                        134.11-148.23                       148.23-162.35                        162.35 and above


               Sharp                                Moderate                              No                                     Moderate                            Sharp
               Decline                               Decline                           Change                                    Increase                           Increase
                .1330                                 .2940                             .2640                                      .2280                              .0820



                                    Expected Value is 139.90 yen/$ (in the late 1980’s)


Number of Children in Rural Indian Families
In a hierarchy whose goal is the optimal family size in India (from a study published in
the Journal of Mathematical Sociology, 1983, Vol.9 pp. 181-209, there were four major
criteria of Culture (with subcriteria: Religion, Women Status, Manlihood), Economic
factors (with subcriteria: Cost of child Rearing, Old Age security, Labor, Economic
Improvement, Prestige and Strength),Demographic factors (with subcriteria: Short Life
Expectancy, High Infant Mortality) and the Availability and acceptance of Contraception
(with subcriteria: High Level of Availability and Acceptance of contraception, Medium
level of Availability and Acceptance of contraception, low level of Availability and
Acceptance of contraception. At the bottom three alternatives were considered: Families
with 3 or Less Children, Families with 4 to 7 Children, and Families with 8 or More
Children. The outcome of this example for reasons explained in the research paper had
two projections of 5.6 and 6.5 children per family (due to regional differences.) The
actual value we obtained from the literature after the study was done were 6.8 births per
woman in 1972 and 5.6 in 1978.



                                                                                                 27
  Decision by the US Congress on China Joining the World Trade Organization
  (WTO) in May 2000? Study Done in 1999 (see Socio-Economic Planning Sciences 35
  (2001)

  Briefly, the alternatives of the decision are:
     1. Passage of a clean PNTR bill: Congress grants China Permanent Normal Trade
         Relations status with no conditions attached. This option would allow
         implementation of the November 1999 WTO trade deal between China and the
         Clinton administration. China would also carry out other WTO principles and
         trade conditions.
     2. Amendment of the current NTR status bill: This option would give China the
         same trade position as other countries and disassociate trade from other issues. As
         a supplement, a separate bill may be enacted to address other matters, such as
         human rights, labor rights, and environmental issues.
     3. Annual Extension of NTR status: Congress extends China’s Normal Trade
         Relations status for one more year, and, thus, maintains the status quo.

  Four hierarchies shown in the figure below were considered whose outcomes were
  combined as briefly outlined to derive the final priorities that show how Congress was
  going to vote and in fact China was later admitted to the WTO.
                                                                               Benefits to US (0.25)


           Increased US Exports to China         Improved Rule of Law         China's Promise to Respect    Increased Employment in US   Benefits to Lower Income Consumers
                       0.44                   Intellectual Property Rights,       Anti-Dumping and                     0.07                              0.05
                                           Improved Investment Environment      Section 201 Provisions
                                                           0.26                          0.18


            PNTR:0.59(1)                     PNTR:0.58(1)                     PNTR:0.65(1)                  PNTR:0.54(1)                    PNTR:0.58(1)
            Amend NTR:0.28(0.47)             Amend NTR:0.31(0.53)             Amend NTR:0.23(0.53)          Amend NTR:0.30(0.55)            Amend NTR:0.31(0.53)
            Annual Extension:0.13(0.22)      Annual Extension:0.11(0.19)      Annual Extension:0.12(0.19)   Amend NTR:0.30(0.30)            Annual Extension:0.11(0.19)




  Benefits Synthesis (Ideal): PNTR 1.00 Amend NTR 0.51 Annual Extension 0.21

                                                                 Opportunities for US (0.20)


         Improve                                Promote Democracy                             Improve Environment               Improve Human and Labor Rights
     US-Sino Relations                                0.23                                            0.14                                  0.08
           0.55


PNTR:0.65 (1)                              PNTR:0.57 (1)                                PNTR:0.57 (1)                              PNTR:0.54 (1)
Amend NTR:0.23 (0.35)                      Amend NTR:0.33 (0.58)                        Amend NTR:0.29 (0.51)                      Amend NTR:0.30 (0.44)
Annual Extension:0.12 (0.18)               Annual Extension:0.10 (0.18)                 Annual Extension:0.14 (0.25)               Annual Extension:0.16 (0.20)



  Opportunities Synthesis (Ideal): PNTR 1 Amend NTR 0.43 Annual Extension 0.13




                                                                                      28
                                                     Costs to US (0.31)


                                      Loss of US Access             Workers in Some Sectors
                                       to China's Market         of US Economy May Lose Jobs
                                             0.83                             0.17


                                 PNTR             :0.10 (0.17)    PNTR             :0.57 (1)
                                 Amned NTR        :0.30 (0.5)     Amned NTR        :0.29 (0.50)
                                 Annual Extension :0.60 (1)       Annual Extension :0.14 (0.25)



Costs Synthesis (which is more costly, Ideal): PNTR 0.31 Amend NTR 0.50 Annual
Extension 0.87

                                                           Risks for US (0.24)


         Loss of Trade as                 US-China Conflict           China Violating Regional Stability   China's Reform Retreat
    Leverage over Other Issues                 0.25                                 0.25                            0.07
               0.43


      PNTR            : 0.59            PNTR            : 0.09             PNTR            : 0.09          PNTR            : 0.09
      Amend NTR       : 0.36            Amend NTR       : 0.29             Amend NTR       : 0.28          Amend NTR       : 0.24
      Annual Extension: 0.05            Annual Extension: 0.62             Annual Extension: 0.63          Annual Extension: 0.67




Risks Synthesis (more risky, Ideal): PNTR 0.54 Amend NTR 0.53 Annual Extension
0.58
Figure Hierarchies for Rating Benefits, Costs, Opportunities, and Risks.


                                                Factors for Evaluating
                                                    the Decision


             Economic: 0.56                         Security: 0.32                          Political:0.12
    -Growth (0.33)                        -Regional Security (0.09)                -Domestic Constituencies (0.80)
    -Equity (0.67)                        -Non-Proliferation (0.24)                -American Values (0.20)
                                          -Threat to US (0.67)
.
Figure Prioritizing the Strategic Criteria to be used in Rating the BOCR
How to derive the priority shown next to the goal of each of the four hierarchies shown in
the last figure is outlined in the table below. We rated each of the four merits: benefits,
costs, opportunities and risks of the dominant PNTR alternative, as it happens to be in
this case, in terms of intensities for each assessment criterion. The intensities, Very High,
High, Medium, Low, and Very Low were themselves prioritized in the usual pairwise
comparison matrix to determine their priorities. We then assigned the appropriate
intensity for each merit on all assessment criteria. The outcome is as found in the bottom
row of table.




                                                                   29
Table Priority Ratings for the Merits: Benefits, Costs, Opportunities, and Risks
Intensities: Very High (0.42), High (0.26), Medium (0.16), Low (0.1), Very Low (0.06)
                                     Benefits Opportunities      Costs        Risks
 Economic Growth (0.19)               High       Medium       Very Low      Very Low
   (0.56)     Equity (0.37)          Medium        Low           High         Low
              Regional (0.03)          Low       Medium        Medium         High
  Security Non-Proliferation         Medium       High         Medium         High
   (0.32)     (0.08)
              Threat to US (0.21)     High        High        Very High Very High
  Political Constituencies (0.1)      High       Medium       Very High       High
   (0.12)     American Values         Very         Low           Low         Medium
              (0.02)                   Low
  Priorities                           0.25        0.20          0.31         0.24
We are now able to obtain the overall priorities of the three major decision alternatives
listed earlier, given as columns in the table below which gives three ways of synthesize
for the ideal mode, we see in bold that PNTR is the dominant alternative any way we
synthesize as in the last two columns.
Table Two Methods of Synthesizing BOCR Using the Ideal Mode




                                                                                                   bB + oO - cC - rR
                                          Opportunities
   Alternatives




                           Benefits




                                                                                           BO/CR
                                                               Costs




                                                                          Risks




                             (0.25)          (0.20)             (0.31)            (0.24)
                  PNTR                1                   1        0.31           0.54     5.97             0.22
                  Amend
                                0.51               0.43            0.50           0.53     0.83         -0.07
                   NTR
                  Annual
                                0.21               0.13            0.87           0.58     0.05         -0.33
                  Exten.



NETWORK VALIDATION EXAMPLES

Turn Around of the U.S Economy
Let us consider the problem of the turn around of the US economy and introduce 3, 6, 12,
24 month time periods at the bottom (see Blair et. al). Decomposing the problem
hierarchically, the top level consists of the primary factors that represent the forces or
major influences driving the economy: “Aggregate Demand” factors, “Aggregate
Supply” factors, and “Geopolitical Context.” Each of these primary categories was then
decomposed into subfactors represented in the second level. Under Aggregate Demand,
we identified consumer spending, exports, business capital investment, shifts in consumer
and business investment confidence, fiscal policy, monetary policy, and expectations


                                                              30
with regard to such questions as the future course of inflation, monetary policy and fiscal
policy. (We make a distinction between consumer and business investment confidence
shifts and the formation of expectations regarding future economic developments.)

Under Aggregate Supply, we identified labor costs (driven by changes in such underlying
factors as labor productivity and real wages), natural resource costs (e.g., energy costs), and
expectations regarding such costs in the future. With regard to Geopolitical Context, we
identified the likelihood of changes in major international political relationships and major
international economic relationships as the principal subfactors. With regard to the
subfactors under Aggregate Demand and Aggregate Supply, we recognized that they are, in
some instances, interdependent. For example, a lowering of interest rates as the result of a
monetary policy decision by the Federal Reserve should induce portfolio rebalancing
throughout the economy. In turn, this should reduce the cost of capital to firms and
stimulate investment, and simultaneously reduce financial costs to households and increase
their disposable incomes. Any resulting increase in disposable income stimulates
consumption and, at the margin, has a positive impact on employment and GNP. This
assumes that the linkages of the economy are in place and are well understood. This is what
the conventional macroeconomic conceptual models are designed to convey.

The third level of the hierarchy consists of the alternate time periods in which the
resurgence might occur as of April 7, 2001: within three months, within six months,
within twelve months, and within twenty-four months. Because the primary factors and
associated subfactors are time-dependent, their relative importance had to be established
in terms of each of the four alternative time periods. Thus, instead of establishing a
single goal as one does for a conventional hierarchy, we used the bottom level time
periods to compare the two factors at the top. This entailed creation of a feedback
hierarchy known as a "holarchy" in which the priorities of the elements at the top level
are determined in terms of the elements at the bottom level, thus creating an interactive
loop. The figure below provides a schematic representation of the hierarchy we used to
forecast the timing of the economic resurgence.




                                              31
                         Figure Overall View of the “2001” Model
To obtain our forecast, we subsequently multiplied each priority by the midpoint of its
corresponding time interval and added the results (as one does when evaluating expected
values):

     Time Period       Midpoint of Time           Priority of Time    Midpoint x
                           Period                      Period          Priority

                      (Expressed in
                      months from
                      present, with the
                      current month as 0.)
  Three months        0 + (3 – 0)/2 = 1.5            0.30581            0.45871
  Six months          3 + (6 – 3)/2 = 4.5            0.20583            0.92623
  Twelve months       6 + (12 – 6)/2 = 9.0           0.18181            1.63629
  Twenty-four         12 + (24 – 12)/2 =             0.30656            5.51808
  months              18.0
  TOTAL                                                                 8.53932

We interpret this to mean that the recovery would occur 8.54 months from the time of the
forecasting exercise, or in the fall. The Wall Street Journal of July 18, 2003, had the
following to say about the turnaround date:




                                             32
             The Wall Street Journal
             Friday, July 18, 2003
             Despite Job Losses, the Recession
             Is Finally Declared Officially Over
             JON E. HILSENRATH


             The National Bureau of Economic Research           When calling the end to a recession,
           said the U.S. economic recession                   the NBER focuses heavily on two economic
           that began in March 2001 ended eight               indicators: the level of employment
           months later, not long after the Sept. 11          and gross domestic product, or the
           terrorist attacks.                                 total value of the nation's goods and services.
             Most economists concluded more than              Since the fourth quarter of 2001, GDP has
           a year ago that the recession ended in late        expanded slowly but consistently-rising 4 %
           2001. But yesterday's declaration by the           through March of 2003.
           NBER-a private, nonprofit economic research          Employers, however, have eliminated
           group that is considered the official              938,000 payroll jobs since November 2001.
           arbiter of recession timing-came after a           In addition, 150,000 people have dropped
           lengthy internal debate over whether               out of the labor force because they are
           there can be an economic recovery if the           discouraged about their job prospects,
           labor market continues to contract.                according to the government.
           The bureau's answer: a decisive yes.




Market Shares for the Cereal Industry (2002)


The following is one of numerous validation examples done by my graduate students in
business most of whom work at some company. Many of the examples are done in class
in about one hour and without access to data. The answer is only found later on the
Internet. The example below was developed by Stephanie Gier and Florian John in
March 2002. They write: To become familiar with the Super Decision software we have
chosen to estimate the market shares for the Ready-to Eat breakfast cereal industry. This
idea was born after and delicious breakfast with Post’s OREO O’s. To see how good our
assumptions were, we compare our calculated results with the market shares of 2001.
First we created the model. We identified 6 major competitors in the ready to eat cereal
market, Kellogg, General Mills, Post, Quaker, Nabisco and Ralston as our alternatives.
There were more companies in this market having an actual cumulative market share of
roughly about 6% that it turned out later that we had left out. Since we were only
concerned with deriving relative values, the relative shares of other residual companies
do not matter.

Major impacts on the companies’ market shares are:
    Price of the products offered (named cost for the consumer)
    Advertising / Sales Ratio (how much money is spend for advertising)
    Shelf Space (places where the products are located in the stores)
    Tools (Selling Tools used to increase sales and market shares)
    Distribution / Availability (major distribution channels used to sell the product)

These five major impacts (clusters) are further divided in the following nodes:



                                                         33
Tools: (Coupons, trade dealing, in-pack premiums, vitamin fortifications)
Distribution: (Supermarket Chains, Food Stores, Mass Merchandiser)
Shelf Space: (Premium Space, Normal Space, Bad Space)
Cost: (Expensive, Normal, Cheap)
Advertising: (<15%,<14%,<13%,<12%,<11%,<5%)
Their interactions are depicted in the figure below. Second we carried out comparisons
and performed calculations to obtain the final result (see later). Third we compared our
calculated market shares with the real market shares for 2001. The table that follows lists
estimated market share values and the actual ones taken from the website of the
International Data Corporation.

                                                           SHELF SPACE



                                                       Premium space
                                                       Normal space
                                                       Bad space
                               TOOLS                                                      COST


                   Coupons
                   Trade dealing                                                  Expensive
                   In-pack premium                                                Normal
                   Vitamin-fortification                                          Cheap




                         DISTRIBUTION/
                                                                                        ADVERTISING
                         AVAILABILITY

                                                                                    < 15 %
                    Supermarket chains                                              < 14 %
                    Food stores                                                     < 13 %
                    Mass merchandiser                                               <12 %
                                                                                    < 11 %
                                                                                    <5%
                                                           ALTERNATIVES

                                                       Kellogg
                                                       General Miles
                                                       Post
                                                       Quker
                                                       Nabisko
                                                       Ralston




                                           Figure Cereal Industry Market Share


                                       Table Overall-Results, Estimated and Actual

 Alternatives   Kellogg               General Mills        Post           Quaker              Nabisco    Ralston
 Estimated       0.324                     0.255         0.147            0.116                  0.071    0.087
 Actual          0.342                     0.253          0.154           0.121                  0.057    0.073
Compatibility index value: 1.01403 (very good). It is obtained by multiplying element-
wise the matrix of ratios of ones set of data, by the transpose of the matrix of ratios of the
other set, adding all the resulting entries and dividing by n2 and requiring that this ratio
not be more than 1.1.
Let us describe the calculations needed to derive the result in the “Estimated” column of
the table. From the pairwise comparison judgments we constructed a super matrix, done
automatically by the software Super Decisions. Then we weighed blocks of the


                                                             34
supermatrix by the corresponding entries from the matrix of priority vectors of paired
comparisons of the influence of all the clusters on each cluster with respect to market
share shown in the following table. This yielded the weighted supermatrix that is now
stochastic as its columns add to one. We raised this matrix to limiting powers to obtain
the overall priorities of all the elements in the figure.
                                          Table Cluster Priority Matrix
                                                                      Distrib./
                            Advertising      Alternatives    Cost                   Shelf Space   Tools
                                                                     Availability
         Advertising          0.000             0.184        0.451     0.459          0.000       0.000
         Alternatives         0.000             0.000        0.052     0.241          0.192       0.302
         Cost                 0.000             0.575        0.000     0.064          0.044       0.445
         Distribution /
                              0.000             0.107        0.089     0.000          0.364       0.159
         Availability
         Shelf Space          0.000             0.071        0.107     0.084          0.297       0.000
         Tools                0.000             0.062        0.302     0.152          0.103       0.095



Market Shares for the Airline Industry (2001)

James Nagy did the following study of the market share of eight US airlines. Nowhere
did he use numerical data, but only his knowledge of the airlines and how good each is
relative to the others on the factors mentioned below. Note that in three of the clusters
there is an inner dependence loop that indicates that the elements in that cluster depend
on each other with respect to market share. The table gives the final estimated and the
actual relative values that are again very close. The figure below shows the network
model of clusters and their inner and outer dependence connections that produced these
results. Nagy writes: “I initially chose the airline industry for the assignment because I
was a frequent traveler. My study group at Katz helped me make the comparisons
between airlines that I did not have first hand experience as a passenger. Otherwise, I
used my personal experience and perception of consumer sentiment towards the airlines
to make the comparison. I was equally surprised at the results. In fact, I initially
questioned how they could be so close. I would like to see the results of a study using
today’s consumer perception. A lot has changed in the industry since the 9/11 tragedy in
the year 2001. You could divide the class up into 4 to 5 small groups and let them do the
comparisons as individual groups and compare the results.”

                          Table Market Share of Airlines, Actual and Predicted
                                                  Actual (yr 2000)    Model Estimate
                           American                      23.9             24.0
                           United                        18.7             19.7
                           Delta                         18.0             18.0
                           Northwest                     11.4             12.4
                           Continental                    9.3             10.0
                           US Airways                     7.5              7.1
                           Southwest                      5.9              6.4
                           American West                  4.4              2.9




                                                        35
                  Figure Airline Model from the ANP Super Decisions Software



The Adjacency and Path Matrices

The question often arises as to why raising the matrix of judgments or sometimes the
supermatrix to a power determines for its entries the number of paths between the
corresponding vertices, whose length is equal to that power. The vertex matrix is a first
step for understanding this concept. We define a vertex (or adjacency) matrix for both
directed and undirected graphs. The element in the (i,j) position of the matrix is equal to
the number of edges incident with both vertex i and vertex j (or directed from vertex i to
vertex j in the directed case). Thus for the directed graph of a network we might have:

                                         v1 v2 v3 v4 v5 v6
                                    v1 0    0 0 0 0 0
                                    v 2 0   0 1 0 0 0
                                                     
                                   v 3 0   1 0 0 0 0
                                 V                  
                                    v 4 0   0 0 0 0 0
                                    v5 1    1 0 1 1 1
                                                     
                                    v6 1    0 0 0 0 0

In general, we have the following theorem regarding the vertex matrix V of a graph:

Theorem: The matrix V n gives the number of arc progressions of length n between any
two vertices of a directed graph.

Proof: If aik is the number of arcs joining vi to vk and akj is the number of arcs joining vk
to vj, then a ik a kj is the number of different paths each consisting of two arcs joining vi to


                                              36
vj and passing through vk. If this is summed over all values of k, that is, over all the
intermediate vertices, one obtains the number of paths of length 2 between vi and vj. If
we now use aij to form a ija jm, we have the number of different paths of length 3
between vi and vm passing through vj, and so on. Thus if we assume the theorem is true
for V n-1 , then the coefficients of V n = V n-1 V give the number of paths of length n
between corresponding vertices. This completes the proof. A similar theorem holds for
undirected graphs.


Relationship Between the Supermatrix and the ANP and Input-Output Econometric
Analysis
The following proof of the relationship between the supermatrix and Leontieff’s Input-
Output Model is due to Luis G. Vargas. The figure below depicts interdependence in
economic input-output models.


                                         Sectors

                              Value                   External
                              Added                   Demand


                                          External
                                          Markets


                                Figure Input-Output Network


Let A be the matrix of relative input-output coefficients. Let v and d be the relative value
added and the relative final demand vectors. Let xij the amount of resource that sector j
receives from sector i. Let v j be the value added corresponding to sector j and let d i be
the final demand of sector i. The value added of a sector includes wages and salaries,
profit-type income, interest, dividends, rents, royalties, capital consumption allowances
and taxes. The final demand of a sector includes imports, exports, government
purchases, changes in inventory, private investment and sometimes, household purchases.
Thus, the input-output matrix is given by:

                            Sectors      S1    S2      Sn
                               S1      x11   x12      x1n       d1 
                                                                   
                              S2       x21   x22      x2 n      d2 
                                                                   
                                                                   
                              Sn       xn1   xn 2     xnn       dn 
                                      v                         0
                                       1     v2       vn           


                                              37
Let I1 , I 2 ,    , I n be the total input to the sectors and let O1 , O2 ,                       , On be the total output of
                       n                                   n
the sectors, i.e.,    x
                       j 1
                              ij    di  Oi and          x
                                                          i 1
                                                                 ij    vj  I j .

                                                                                        xij
Let the relative input-output coefficients be given by aij     .
                                                             Ij
The relative final demand of a sector with respect to the other sectors is given by
            d                                                                     vj
ai ,n 1  n i . On the other hand, the relative value added is given by an 1,1  . Thus,
           dh
                                                                                  Ij
           h 1
the matrix of interactions represented by the network in Figure 1 is give by:

                                                         a11 a12                      a1n          a1,n+1 
                                                                                                            
                                                             a21 a22                   a2 n         a2,n 1 
                           A                 a,n 1  
                       W                                                                                
                           an 1,             0                                                          
                                                         an1 an 2                     ann          an ,n 1 
                                                        a                                            0 
                                                         n 1,1 an 1,2                an 1,n              

Because W is a stochastic irreducible matrix, lim W k is given by weT where w is a
                                                                          k 

(n  1) 1 vector that is the principal right eigenvector of W, and eT  (1,1,...,1) is the
1 (n  1) unit vector. Thus, we have Ww  w or

                                        n

                                       a w
                                       j 1
                                              ij   j     wn 1ai ,n 1  wi , i  1, 2,..., n

                                        n

                                       a
                                       j 1
                                              n 1, j   w j  wn 1

In matrix notation we have:
                                                        ( I  A) wn  wn 1a,n 1
                                                        an 1, wn  wn 1
where wn  (w1 ,..., wn )T . Thus, we have wn  wn 1 ( I  A)1 a,n1 and hence, we can write

                                                       w ( I  A) 1 a,n 1  T
                                            lim W k   n 1                  e .
                                            k 
                                                             wn 1           

Note that wn  wn 1 ( I  A)1 a,n1 is the relative output of the economy as given by
Leontieff’s model.




                                                                        38
Example of an Input-Output Matrix
Consider the following input-output matrix shown in the table below.
                                             Table Input-Output Matrix
                Agriculture        Manufacturing Services         Other         Final Demand         Total Output
Agriculture                   10              65            10              5              10                  100
Manufacturing                 40              25            35             75              25                  200
Services                      15               5             5              5              90                  120
Other                         15              10            50             50             100                  225
Value Added                   20              95            20             90               0

Total Input               100                200            120           225            225                  645




The supermatrix corresponding to this input-output model is given in the table below:

                               Table The Supermatrix for the Input-Output Matrix
                                    0.1            0.325    0.083333333         0.022222222     0.044444444
                                    0.4            0.125    0.291666667         0.333333333     0.111111111
W=                                 0.15            0.025    0.041666667         0.022222222             0.4
                                   0.15             0.05    0.416666667         0.222222222     0.444444444
                                    0.2            0.475    0.166666667                 0.4               0


and the limiting matrix lim W k is given in the table below:
                                   k 

                              Table Limiting Supermatrix of the Input-Output Table
                0.114942529          0.114942529 0.114942529 0.114942529                  0.114942529
                0.229885057          0.229885057 0.229885057 0.229885057                  0.229885057
                0.137931034          0.137931034 0.137931034 0.137931034                  0.137931034
                 0.25862069           0.25862069 0.25862069 0.25862069                     0.25862069
                 0.25862069           0.25862069 0.25862069 0.25862069                     0.25862069


      0.114942529 
                     
wn   0.229885057  and by normalizing to unity we can see that
      0.137931034 
                     
      0.25862069 
                   0.15503876        100 
                                         
               1
wn (1  wn 1 )   0.310077519   1  200 
                   0.186046512  645  120 
                                         
                   0.348837209       225 

which coincides with the normalized values of the total output of the economy.

The input-output table (supermatrix) of the Sudan economy (1976) with eigenvector
values as the estimates and the actual values in parentheses was computed by the Nobel
Laureate Lawrence Klein who participated in the study with his Wharton Forecasting


                                                            39
Associates. The results of this fairly complex exercise using paired comparison
judgments are generally close to those of the econometric forecasting model.




      Table Input-Output Table of Sudan Economy (1976) by Wharton Forecasting Associates
                 Agriculture   Public      Manufacturing   Transportation   Construction   Services
                               Utilities   and Mining      and
                                                           Distribution
Agriculture         .0079            0         .2331            .0008           .0699            0
                  (.00737)          (0)      (.21953)         (.00042)        (.06721)          (0)
Public              .0009            0         .0130            .0075              0          .0033
Utilities         (.00024)          (0)      (.01159)         (.00618)            (0)       (.00283)
Manufacturing       .0041            0            0             .0089           .0379         .0037
and Mining        (.00393)          (0)          (0)          (.00857)        (.04216)      (.00322)
Transportation      .0691         .1694        .1281               0            .1115         .0153
and               (.06993)      (.145360     (.12574)             (0)         (.09879)      (.00641)
Distribution
Construction          0              0            0               0                0         .0546
                     (0)            (0)          (0)             (0)              (0)      (.05402)
Services              0           .0117        .0224           .0224            .0039        .0004
                     (0)        (.01030)     (.02549)        (.02422)         (.00520)     (.000210


Desirability of Drilling for Oil in Alaska – the ANWR Model
In late 2002 a study was done [Emanuel, Cefalu, 2002] to find out whether drilling for oil
should be allowed in the Artic National Wildlife Reserve (ANWR) in northern Alaska.
Environmentalists, mostly living in the lower 48 US states have been blocking drilling in
the region. In the ANWR study the alternatives were: Do Drill, Do Not Drill. And the
study results were compared against a poll of Alaskan residents asking the question: “Do
you think we should drill, or not drill in ANWR?”

ANWR-Arctic National Wildlife Refuge covers 19 million acres on the Northern coast of
Alaska north of the Arctic Circle and 1,300 miles south of the North Pole. The consensus
of the geologic community is that the Coastal Plain of ANWR represents the highest
petroleum potential onshore area yet to be explored in North America. If explored, it is
estimated that it will take 15 years or more before oil and gas will reach the market.
Legislation was passed in the 1980’s that created a majority of the National Parks in
Alaska and expanded ANWR to its current size. The Reagan Administration was ready
to drill but was derailed by the Exxon Valdez catastrophe. The first Bush Administration
likewise wanted to drill, but was unsuccessful. The Clinton Administration designated it
as a protected area and it has been that way ever since then. The second Bush
Administration, in response to ongoing Middle East violence and the 9/11 terrorist


                                                40
attacks, sees drilling in ANWR as vital not only for economic but national security
reasons.     Several environmental groups consider ANWR a great American natural
treasure and one of the last places on the earth where an intact expanse of arctic and sub-
arctic lands remain protected. They feel the habitat, the wildlife, and the culture need to
be protected from the exploration of gas and oil. The top- level main network is shown in
Figure below.




                  Figure The Main Network of the ANWR Decision Model
The strategic criteria here are General Public Opinion, International Politics, and Amount
of Oil. They are first pairwise compared for importance, then used to rate the importance
of the top rated alternative for each of the Benefits, Opportunities, Costs and Risks, called
the merits of the decision, as shown in the table. The ratings categories themselves are
pairwise compared to establish priorities for High, Medium and Low. To select the
appropriate rating, keep in mind the highest priority alternative under the merit being
evaluated: Risks for example as shown in the figure. Do Not Drill is the highest priority
alternative under risk, meaning it is the most risky, so keeping that in mind the Risk merit
under General Public Opinion is evaluated as being low. The results of rating the merits
are also shown in the figure and the driving merit in this decision is Benefits at .407,
closely followed by Opportunities at .364. In this decision, Costs are found to be
unimportant, and Risks are about half as important as Benefits and Opportunities. These
values for the merits nodes are used to weight the values for the alternatives as
determined in the subnets they control to give the overall results for the alternatives.

Connected to each of the BOCR merit nodes is the subnet for Benefits that contains a
hierarchy of control criteria: Economic, Political, Social. The control criteria are
pairwise compared for importance.




                                             41
                    Figure Rating the BOCR under Strategic Criteria




                 Figure Subnet Containing Control Criteria under Benefits
The subnet for Benefits containing a hierarchy of control criteria is shown in the figure.
And connected to each control criterion node under benefits is a decision network
containing the alternatives of the problem. The decision subnet for the economic control
criterion under benefits is shown below in the figure. The final results are shown in the
figure where the results from the subnets are combined in the main network using the
formula Bb + Oo - Cc – Rr where b, o, c, and r are the values for the decision alternatives
from the control subnets, and B, O, C, and R are the priorities of the BOCR as
determined by rating them under the strategic criteria.




                                             42
                  Figure The Decision Subnet under Economic Benefits




                         Figure Final Results of ANWR Study


. In a recent poll conducted among native Alaskans they support opening ANWR to oil
and gas exploration 75% to 19% with 6% undecided. The question asked was “Do you
believe oil and gas exploration should or should not be allowed within the ANWR
Coastal Plain?” Assigning the 6% equally yields 78% to 22%, the results of the poll are
very close to the results of the model.




                                           43

				
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