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SuperDecisions Help

Copyright © 2012 Creative Decisions Foundation

Table of Contents

Cover Page ...................................................................................................................................................... 2Software Systems Supported and Specifications....................................................................................... 3

Overview

Overview AHP and ANP in SuperDecisions ......................................................................................... 4Example of an AHP Hierarchical Model.................................................................................................. 22

Example of an ANP Network Model ....................................................................................................... 27

The Fundamental Scale of the AHP........................................................................................................ 30

Building a Simple Network Model

Definition of a Network ............................................................................................................................ 31

Inner and Outer Dependence .................................................................................................................. 35

Starting a New Model................................................................................................................................ 36

Creating Clusters ....................................................................................................................................... 38

Creating Nodes .......................................................................................................................................... 43

Selecting Fonts/Colors/Icons .................................................................................................................. 48

Connecting Nodes ..................................................................................................................................... 54

Connecting Clusters .................................................................................................................................. 56

Node Pop-up Menu ................................................................................................................................... 57

Cluster Pop-up Menu ................................................................................................................................ 58

Making Judgments

Making Pairwise Comparisons................................................................................................................. 59

Entering Direct Data.................................................................................................................................. 64

Improving Inconsistency.......................................................................................................................... 65

Cycling through Comparisons ................................................................................................................. 67

Rating Alternatives .................................................................................................................................... 70

Synthesize to get Results .............................................................................................................................. 78

Performing Sensitivity ................................................................................................................................... 79

Saving Models and Subnetworks ................................................................................................................. 92

Sample Models................................................................................................................................................ 94

Documents Attached inside Models........................................................................................................ 95

Hierarchical Model to Choose a CAr ....................................................................................................... 96

Hamburger market share model ............................................................................................................. 97

Predicting Turnaround for US Economy........................................................................................... 98

Tutorials........................................................................................................................................................... 99

Important Additional Information

Math of the AHP ........................................................................................................................................ 100

Index ................................................................................................................................................................ 116

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Super Decisions Software Guide

Version 2.2

Decision Support Software

based on

The Analytic Hierarchy Process and The Analytic Network Process

developed by

Thomas L. Saaty

Software designed by William J. Adams

Creative Decisions Foundation 4922 Ellsworth Avenue Pittsburgh, PA 15213 www.creativedecisions.org

SuperDecisions Software

SuperDecisions Help

SuperDecisions Manual

©2012

Copyright © 2012 Creative Decisions Foundation

2

The SuperDecisions software implements the Analytic Network Process (ANP) for decision making. The ANP

is a framework for handling very complex oftentimes messy problems that are not reducible to quantitative

analysis because of the intangible considerations. It allows you to decompose a problem systematically and

incorporate judgments on intangible factors alongside tangible factors.

Software Versions

The Analytic Network Process for decision making was created by Thomas L. Saaty. The SuperDecisions

software was designed by William J. Adams.

We currently have distributions for:

1) Windows 2000 to 7.

2) Mac OS X (version 10.4 and above). PPC and Intel are both supported.

3) Ubuntu 12.04 (with a repository for apt automatic updating).

4) Generic Linux (x86 or amd64).

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This section was written to serve as a complete story of the fundamental ideas of the AHP and ANP and

how they are implemented in the SuperDecisions software. To jump directly to one of the sections below

mouse over the name and press <Ctrl><Left Click>. But we do suggest you scan the story quickly to get a

comprehensive overview.

A Hierarchical Model for Choosing the Best Car

..........................................................................................................................................................................

How a Hierarchy Appears in the SuperDecisions Software

.......................................................................................................................................................................... Showing Children of a Parent Node

.......................................................................................................................................................................... The

Fundamental Scale of the AHP and ANP

.......................................................................................................................................................................... The

Pairwise Comparison Matrix

.......................................................................................................................................................................... Deriving Priorities from the AHP Pairwise Comparison Matrix

.......................................................................................................................................................................... The

Mathematics behind the Pairwise Comparison Matrix

.......................................................................................................................................................................... The

SuperDecisions Pairwise Comparison Matrix View

The Supermatrix

.......................................................................................................................................................................... The

Unweighted Supermatrix

.......................................................................................................................................................................... The

Limit Supermatrix

.......................................................................................................................................................................... Getting the Answer from the Limit Supermatrix

Sensitivity Analysis

Changing from AHP Hierarchical Thinking to ANP Network Thinking

A Network Model for Choosing the Best Car

.......................................................................................................................................................................... Feedback

The Hamburger Network Model for Market Share

.......................................................................................................................................................................... The hierarchical models of the AHP have a goal at the top, criteria influencing the goal in the next level

down, possibly sub-criteria in levels below that and alternatives of choice at the bottom of the model.

Judgments are made on pairs of elements throughout the structure and synthesized to prioritize the

alternatives.

The network models of the ANP are made up of clusters of elements and connections among them. There

is no goal and any factors in the model can be linked to any other factors The ANP can offer a more

realistic way to model the relative world we live in than can the top-down approach of the AHP.

Priorities are established in both AHP and ANP for the factors in the model and they are then synthesized

to give the overall priorities for the alternatives of the decision. Often, though not always, the priorities

are established by pairwise comparing factors using judgments.

In this overview there is an example of an AHP model and an ANP network model. The purpose here is to

show how these decision making processes

A Hierarchical Model for Choosing the Best Car

In the Analytic Hierarchy Process a decision problem is structured as a hierarchy with a goal node at the

top, criteria influencing the goal in the level below (there may also be several additional levels of

sub-criteria), and the alternatives of the decision in the bottom level. Here is a graphic of a hierarchical

decision model to choose the best car. The criteria will be pairwise compared for importance to establish

their priorities with respect to the goal. The cars will be pairwise compared for preference to establish

their priorities with respect to each criterion. The results of all these comparisons will be combined to give

the best car overall; that is, the car with the highest priority.

This is an example of a decision model to choose the best car. The goal is connected to the criteria that

will be considered in choosing a car and each criterion is connected to the cars.

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The goal and criteria are one comparison group with the goal as the parent and the criteria as the

children. The criteria will be pairwise compared with respect to the Goal for importance.

Each criterion connected to the cars forms a comparison group with that criterion as the parent and the

cars as children. The cars will be pairwise compared with respect to the criterion for preference. There are

5 comparison groups for this model.

Below is a figure showing the hierarchy under construction; thus far the goal is connected to the criteria

and the first criterion, Prestige, is connected to the cars, so at this point two comparison groups have

been established.

Below the figures are shown as they are typically presented in the AHP theory. The second figure is a

"complete" hierarchy, that is, every node in a level is connected to every node in the level below. Not all

hierarchies have this same complete form.

A Hierarchy under Construction

The Finished Complete Hierarchy

A Hierarchy in the SuperDecisions Software

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In the SuperDecisions software a decision model is made up of clusters, nodes and links. Below is a

screenshot of the car choice hierarchy as it appears in the software.

Clusters are groupings of nodes which are logically related factors of the decision. Connections are made

among nodes to establish comparison groups and when nodes are connected links automatically appear

between their clusters. Though there are no levels the clusters may be arranged to look like a hierarchy by

dragging and dropping them to stack them.

In a hierarchy the links go only downward: from the goal node to the criterion nodes and from each

criterion node to the alternative nodes.

Note: Numbers are sometimes used to preface the cluster and node names because in the supermatrix (discussed a

few lines below) they are in alphabetical order and if you want to control the order, numbering is the best way to do it.

SuperDecisions Hierarchical Model Screenshot

This model can be found in Help>Sample Models>Tutorial_Models with the name:Tutorial_1_Acura_Relative_Model.sdmod

Showing Children of a Parent Node

The arrow from one cluster to another is merely an indicator that some parent node or nodes in the cluster

at the base of the arrow, the "from" cluster, are linked to some node or nodes in the cluster at the point

of the arrow, the "to" cluster, but it does not specifically indicate which nodes are connected. The parent

node or nodes are in the "from" cluster and their respective groups of children are in the "to" cluster.

6

Turn on the "show connections" mode as shown in the screenshot below. Holding the cursor over a node

will cause its children nodes to be outlined in red. If the entire cluster window of the children nodes is also

outlined in red it means the pairwise comparisons for that family have been finished and marked as

completed.

Screenclip Showing Children Nodes

The Fundamental Scale of the AHP and ANP

The pairwise comparison judgments used in the AHP pairwise comparison matrix are defined as shown in

the Fundamental Scale of the AHP below.

THE FUNDAMENTAL SCALE OF THE AHP

Intensity of importanceDefinition Explanation

1 Equal importance Two elements contribute equally to the objective

3 Moderate importance Experience and judgment slightly favor one element over another

5 Strong importance Experience and judgment strongly favor one element over another

7 Very strong importance An activity is favored very strongly over another

9 Absolute importance The evidence favoring one activity over another is of the highest possible order of affirmation

7

2, 4, 6, 8 Used to express intermediate values

Decimals 1.1, 1.2, 1.3, …1.9 For comparing elements that are very close

Rational numbers Ratios arising from the scale above that may be greater than 9

Use these ratios to complete the matrix if consistency were to be forced based on an initial set of n numerical values

Reciprocals If element i has one of the above nonzero numbers assigned to it when compared with element j, then j has the reciprocal value when compared with i

If the judgment is k in the (i, j) position in matrix A, then the judgment 1/k must be entered in the inverse position (j, i).

To compare n elements in pairs construct an n x n pairwise comparison matrix A of judgments expressing dominance. For each pair choose the smaller element serves as the unit and the judgment that expresses how many times more is the dominant element .Reciprocal positions in the matrix are inverses, that is, aij= 1/aji.

The Pairwise Comparison Matrix

The goal is the parent node of the criteria and they comprise one of the comparison groups in this model.

The criteria will be pairwise compared with respect to the goal. The pairwise comparison judgments are

made using the Fundamental Scale of the AHP and the judgments are arranged in a matrix, the pairwise

comparison matrix.

The numbers in the cells in an AHP matrix, by convention, indicate the dominance of the row element over

the column element; a cell is named by its position (Row, Column) with the row element first then the

column element. In the AHP pairwise comparison matrix below the (Price, MPG) cell has a judgment of 3 in

it, meaning Price is 3 times as important as Miles Per Gallon (MPG). So logically this means MPG has to be

1/3 as important as Price so 1/3 is automatically entered in the (MPG, Price) cell.

Only the judgments in the unshaded area need to be made and entered because the inverse of a

judgment, for example, (Price, MPG) is automatically entered in its transpose cell (MPG, Price). The

diagonal elements are always 1, because an element equals itself in importance. Matrices with this

property are called inverse matrices.

Only judgments in the unshaded area need to be made and entered. There will be 6 judgments required

for these 4 elements. If the number of elements is n the number of judgments is n(n-1)/2 to do the

complete set of judgments. It is possible to make less than this number of judgments and obtain a rough

estimate, but there must be a minimum of n -1 judgments.

AHP Pairwise Comparison Matrix

GOAL Prestige Price MPG Comfort

Prestige 1 1/4 1/3 1/2

Price 4 1 3 3/2

MPG 3 1/3 1 1/3

Comfort 2 2/3 3 1

Deriving Priorities from the AHP Pairwise Comparison Matrix

Priorities for the criteria are obtained by calculating the principal eigenvector of the above matrix. A short

computational way to obtain this vector is to raise the matrix to powers. Fast convergence is obtained by

successively squaring the matrix. The row sums are calculated and normalized. The computation is

stopped when the difference between these sums in two consecutive calculations of the power is smaller

than a prescribed value.

The eigenvector of the above matrix to four significant decimals is:

Criteria Priorities

Prestige 0.0986

8

Price 0.425

MPG 0.1686

Comfort 0.3077

Total 1.000

The Mathematics behind the Pairwise Comparison Matrix

The priorities of an AHP pairwise comparison matrix are obtained by solving for the principal eigenvector of

the matrix. The mathematical equation for the principal eigenvector w and principal eigenvalue λmax of a

matrix A is given below. It says that if a matrix A times a vector w equals a constant (λmax is a constant)

times the same vector, that vector is an eigenvector of the matrix. Matrices have have more than one

eigenvector; the principal eigenvector which is associated with the principal eigenvalue λmax (that is, the

largest eigenvalue) of A is the solution vector used for an AHP pairwise comparison matrix.

w = Aw λmax

Perron's Theorem shows that for a matrix of positive entries, there exists a largest real eigenvector and its

eigenvector has positive entries. This is an important theorem that supports the use of the eigenvector

solution in AHP theory to obtain priorities from a pairwise comparison matrix. The book, Fundamentals of

Decision Making and Priority Theory by Thomas L. Saaty, gives more details about the mathematics of the

pairwise comparison matrix.

For more on the mathematics of the AHP click here.

The Pairwise Comparison Matrix View

There are five modes for making assessments: judgments drawn for the Fundamental Scale of the AHP are

used in the graphical, verbal, matrix, and questionnaire modes and direct data is used in the Direct mode.

The matrix mode is shown in the screenclip below. Only the judgments in the unshaded cells of the AHP

Matrix above need to be entered so the cells shown in the software are limited to these.

The (Prestige, Price) cell has a red 4 in it with the arrow pointing up to Price, indicating Price is 4 times

more important than Prestige (the value 1/4 is used in the computations.) And 4 is used for the (Price,

Prestige) cell though it is not displayed. Prestige does not even show up as a column heading because the

entire column is greyed out in the AHP Matrix view. However, Prestige does appear as a row heading so it

is involved in as many comparisons as any other node.

The priorities for the criteria are the results shown in the panel at the right below and are entered in the

supermatrix in the column of the parent node of the comparison, the goal node in this case. Interpret

numbers in red as fractions, and numbers in blue as whole numbers; for example the red 4 in the (Prestige,

Price) cell represents the 1/4 in the AHP matrix above. The arrow points to the dominant factor for each

cell.

The Inconsistency of 0.077 is given above the derived priorities in the Results panel below. The

inconsistency should be less than 0.10, that is, 10%.

Matrix Comparison Mode as Shown in the SuperDecisio ns Software

9

Note that the priorities in the Results panel are the same as those given mathematically above for the AHP

comparison matrix. The inconsistency is also given. In this instance it is .07684 which is satisfactory. A

rule of thumb is that the inconsistency should be less than 0.10.

The Supermatrix

The AHP uses a data structure called a supermatrix that contains priorities from the comparison groups

with the priorities of the children from a comparison group of children and parent appearing in the column

of the parent. The name supermatrix is used for this matrix not because it is specially wonderful, but

because it is made up of column vectors of priorities, each of which was obtained from a matrix! So in a

way it is a matrix of matrices.

There are three supermatrices:

1) The unweighted supermatrix contains all the pairwise comparison results.

2) The weighted supermatrix, weighted by the importance of clusters, is important only in network models.

The weighted supermatrix is the same as the unweighted supermatrix for hierarchies. In the weighted

supermatrix all columns must sum to zero.

3) The limit supermatrix is the final version of the supermatrix obtained by raising the weighted supermatrix

to powers (with modifications depending on the model structure).

In this example, the priorities of the criteria are arranged beneath the goal in the first column and the

priorities of the cars are arranged beneath each criterion. The initial supermatrix of the derived priorities is

called the unweighted supermatrix. In the ANP component blocks of the supermatrix are multiplied by

constants so that the columns will sum to 1. This is the weighted supermatrix.

The Unweighted Supermatrix

The simple way to get the answer for a hierarchy is to multiply the priority of each element in the

hierarchy (derived through pairwise comparisons) by the weight of its parent element and sum the bottom

level priorities of the alternatives to get the answers. However, the solution may also be obtained using a

supermatrix.

The Unweighted Supermatrix for the Car Hierarchy

10

The Limit Supermatrix

The SuperDecisions software uses a special algorithm to remember and display additional priorities in the

Limit isupermatrix that appeared in successive powers of the matrix and give useful informatioin. The final

overall priorities for the alternatives, in raw unnormalized form, appear in the column beneath the goal. The

priorities for the criteria in the goal column, when normalized, are the original priorities derived by pairwise

comparison.

The weighted supermatrix is raised to powers until it converges to the limit supermatrix which contains the

final results, the priorities for the alternatives, as well as the overall priorities for all the other elements in

the model. It happens that the weighted supermatrix is the same as the unweighted supermatrix for an

AHP hierarchy, so raise the matrix above to powers.

A hierarchy is a special kind of network with a goal cluster from which an arrow only goes out, called a

source, and an alternatives cluster with arrows only coming in, called a sink. This type of model is actually

more difficult when it comes to finding the results than a more typical network model with connections

going every direction. To do the computations yourself, using Microsoft Excel for example, raise the

unweighted supermatrix to powers until all the cells go to zero, then back up to the previous power to find

the final non-zero priorities for the alternatives. All the rest of the cells will be zero.

Thomas Saaty uses a different method in his books that is more theoretically correct. With the method

above it is simply an observation that at the power when the matrix goes to zero, there must have been

priorities in it in the previous step, so back up and use those. But with the Saaty method each alternative

is connected to itself, resulting in an unweighted supermatrix containing an identity matrix of 1's in the

(alternative, alternative) block of cells. This matrix will not go to zero, instead it reaches a steady state

containing only the priorities of the alternatives with all the other cells zero as it is raised to powers.

SuperDecisions Limit Matrix with Final Priorities

11

Getting the Answer from the Limit Supermatrix

The result we are seeking is the priorities of the alternatives, the Raw numbers shown in the third column

are directly from the limit supermatrix: Acura TL 0.172133, Toyota Camry 0.100103, and Honda Civic

0.227764. To convert them to the priorities in the Normals column, they are normalized to 1, which means

sum the raw numbers and divide each by this sum. The Normals priorities add up to 1. The priorities in the

Ideals column are obtained by dividing each Raw number by the largest, 0.172133, resulting in the "Ideal"

alternative having a value of 1.

The synthesis command does the work for you, extracting the raw numbers from the limit supermatrix and

idealizing and normalizing them. From the perspective of the person who made the judgments in this model,

the Honda Civic is the best choice. This is, of course, a subjective outcome.

Synthesizing to show that the Honda Civic is the Be st Choice

12


To perform sensitivity analysis one asks what the decision would be if the priorities of the criteria were

different. Below is a screenshot of the dynamic sensitivity barchart showing that if the priority of the

Prestige criterion increases considerably from its original .04 value, the more prestigious car, the Acura TL,

represented by the red bar, becomes the best choice.

Note: To get to this screen select the Computations>New Sensitivity command, the Horz Barchart tab,

set the "Node for sensitivity" to Prestige and click and drag the Parameter button to the right.

Sensitivity Barchart increasing Priority of Prestig e

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Changing from AHP Hierarchical Thinking to ANP Network Thinking

In the Analytic Network Process (ANP), which is a generalization of the AHP, the decision elements are

organized in a network of clusters with links between the elements going in either direction or both

directions. The simplest type of network is a feedback model between criteria and alternatives. To

illustrate what this means we will convert the hierarchical car model to a network model.

A hierarchy is a network too, but a special kind of network with a goal cluster from which all the arrows

lead away, and a sink cluster (the alternatives) that all the arrows lead into. Links go only downward in a

hierarchy. In a typical network one has neither sinks nor sources; and the links can go in any direction. A

network can more faithfully represent the relative world we really live in. One does not buy a car by

determining in the abstract the importance of the criteria before going shopping and looking at a few cars.

The available cars determine how important the criteria are. And when new cars are added to those being

considered, the importance of the criteria may change.

A Network in the SuperDecisions Software for the Car Choice

In this network each car is linked to all the criteria and they are evaluated for their importance in each car

resulting in what one might call profiles of the cars with the priorities of the criteria being evaluated in

terms of the available cars - this is feedback.

14

Step 1. Compare the alternatives with respect to each criterion (as is done in AHP models).

Step 2. Compare the criteria with respect to each of the cars to get a profile of the importance of the

criteria in that car.

Step 3. Synthesize to get the overall priorities for the cars, and, incidentally, overall priorities for the

criteria. Note that the overall criteria priorities very much depend on the cars that have been included for

consideration. They are specific for this group of cars.

The main difference between this ANP model and the AHP model is that the importance of the criteria has

been derived from the available alternatives (using feedback) not established top down in an abstract way

from the goal.

Deriving the Priorities of the Criteria through Feedback

A typical pairwise comparison question to determine priorities of the criteria from the alternatives, known

as feedback, instead of from their importance to the goal, would be: “For the Acura TL, which do you like

better, its prestige or its price?”, “Its prestige or its MPG?” etc. The result of these questions for the

Acura TL is a profile of the importance of the criteria for the Acura. Such a profile is developed for each

car, then the profiles are combined in the limit supermatrix to give overall priorities for the criteria.

Profile of Criteria for the Acura TL

15

Priorities of the criteria for the Acura TL

Similar profiles can be obtained for the Toyota Camry and Honda Civic as shown below:

Toyota Camry

Honda Civic

16

The Unweighted Supermatrix for the Car Network

The Limit Supermatrix for the Car Network

The Final Synthesized Result for the ANP Model

Notice that the best car is no longer the Honda Civic as determined in the AHP model, but is now the more

prestigious Acura TL. This is because the criterion of Prestige is more important in the ANP model than we

estimated in the AHP model by evaluating the importance of the criteria directly with respect to the goal.

In the ANP where the priorities of the criteria are determined by feedback, getting information about the

criteria from the cars rather than from a goal, we find out it is more important than we knew.

Use the Synthesis command to get the final results, either the Syn shortcut or Computations>Synthesize

from the menu.

17

Showing Priorities of all the Nodes in the Model

In contrast to the solution using an AHP model where the Prestige node had the lowest priority at 0.09

when pairwise comparing with respect to the goal, here the Prestige node has the highest priority in the

criteria cluster at 0.317. The Limiting Priorities of all the nodes in the model sum to 1.0 and are shown

below in the right hand column. The limiting priorities are normalized for each cluster to sum to 1.0 shown

in the left hand column.

Limiting Priorities of all Nodes in Model

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An Example of a Larger Network Model

To build a network model, considers first what the decision is about. What is the goal? Then try to

determine factors that seem to play a part in the decision and what the alternatives will be. The factors

are arranged in some logical way into groups. In the software the term for the logical groupings is clusters

and the term for a factor is node.

The Hamburger network model shown below is for estimating relative market share of three hamburger

restaurant chains. It has factors customers might think about in choosing one of the restaurants

represented by nodes in the model and the nodes are logically grouped into 4 clusters.

Hint: To load one of the sample model files, click on the Help command on the main menu, then click on the name

of the model you want. This model is named Hamburger.sdmod.

1 Alternatives

1 McDonald's

2 Burger King

3 Wendy's

2 Advertising

1 Creativity

2 Promotion

3 Frequency

3 Quality of Food

1 Nutrition

2 Taste

3 Portion

4 Other

1 Price

2 Location

3 Service

4 Speed

5 Cleanliness

6 Menu Item

7 Take-out

8 Reputation

Below is the model as it appears in the SuperDecisions software. It is still referred to as a simple network

model as all the clusters and nodes are in the same window. This is one of the sample introductory models.

To load it go to: Help>Samples>1_Introductory_Models>Hamburger.sdmod

SuperDecisions Simple Network Model

19

This model can be loaded from Samples>Tutorial Models>hamburger.sdmod.

Links are made among the nodes to indicate influence. To show which nodes are connected from a given,

click on the "show connections" icon to turn on the "show connections" mode. Mouse over any node to

see the nodes to which it connects outlined in red.

Showing Creativity Node linked to Nodes in many Clu sters

20

For example, hold the cursor over the Creativity node to see it is linked to nodes in three clusters; its own

2 Advertising cluster (inner dependence), the 4 Other cluster and the 1 Alternatives cluster. Thus there

are three comparison groups involving the Creativity node. Similar views of the children of other nodes can

be seen by moving the cursor from node to node in the model. A parent node can have children in many

different clusters and be involved in many sets of pairwise comparisons as the parent.

Comparison Groups and PrioritiesA comparison group consists of a parent node and its children nodes. The children nodes of a comparison

group must all be in the same cluster, though a parent node can have children in several different clusters.

The children nodes in each cluster are pairwise compared with respect to the parent node. The results are

priorities for the children nodes. Priorities sum to 1.000 for each group of children nodes and are combined

throughout the model to give an overall answer for the alternatives of the decision.

Final Synthesized ResultsFor this model, the results are a set of priorities that reflected the relative market share of the restaurants

quite accurately at the time the model was done. The Raw values are obtained directly from the Limit

Supermatrix, normalized to give the Normals column, and divided by the largest of them to give the Ideals

column.

Synthesis Results

21

An Example of a Hierarchical Model

The SuperDecisions software can be used to create both hierarchical and network models. In the Analytic

Hierarchy Process a decision model is structured as a hierarchy with a goal node at the top, criteria

influencing the goal in the level below (there may also be several additional levels of sub-criteria), and the

alternatives of the decision in the bottom level. Here is a graphic of a hierarchical decision model to

choose the best car. The criteria will be pairwise compared for importance to establish their priorities with

respect to the goal. The cars will be pairwise compared for preference to establish their priorities with

respect to each criterion. The results of all these comparisons will be combined to give the best car

overall; that is, the car with the highest priority.

This is an example of a decision model to choose the best car. The goal is connected to the criteria that

will be employed in choosing a car and they form a comparison group with the goal being the parent and

the criteria being the children. The criteria will be pairwise compared with respect to the Goal for

importance.

Each of the criteria is connected to the three cars. Thus altogether there are five sets of pairwise

comparisons in this model: the criteria for importance with respect to the goal and the cars with respect

to each of the 4 criteria for preference. Below is a figure showing the hierarchy under construction; thus

far the goal is connected to the criteria and the first criterion, Prestige, is connected to the cars, so two

comparison groups have been defined.

The Finished Hierarchy

22

A Hierarchy in the SuperDecisions Software

In the SuperDecisions software a decision model is made up of clusters, nodes and links. Below is a

screenshot of the car choice hierarchy as it appears in the software.

Clusters are groupings of nodes which are logically related factors of the decision. Connections are made

among nodes to establish comparison groups and when nodes are connected links automatically appear

between their clusters. Though there are no levels the clusters may be arranged to look like a hierarchy by

dragging and dropping them to stack them.

A Hierarchy under Construction In a hierarchy the links go only down, from the goal node to the criterion nodes and from each criterion

node to the alternative nodes.

Note: Numbers are sometimes used to preface the cluster and node names because in the supermatrix they are in

alphabetical order and if you want to control the order, numbering clusters and nodes is the best way to do it.

SuperDecisions Hierarchical Model Screenshot

23

Go to Help>Sample Models>Tutorial_Models to load this car choice model:Tutorial_1_Acura_Relative_Model.sdmod

The arrow from one cluster to another is merely an indicator that some parent node or nodes in the cluster

at the base of the arrow, the "from" cluster, are linked to some node or nodes in the cluster at the point

of the arrow, the "to" cluster, but it does not specifically indicate which nodes are connected. The parent

node or nodes are in the "from" cluster and their respective groups of children are in the "to" cluster.

Turn on the "show connections" mode as shown in the screenshot below. Holding the cursor over a node

will cause its children nodes to be outlined in red. If the entire cluster window of the children nodes is also

outlined in red it means the pairwise comparisons for that family have been finished and marked as

completed.

Pop-up Descriptions Click on the ? icon to select it and depress it to turn on the "show descriptions"

mode. Then holding the cursor over a cluster or node for a few seconds will cause its description to pop

up, if one was entered when they were created.

How to show which children nodes are connected from a parent node

24

In this hierarchical model the Goal node is the parent of the criteria nodes, and each criterion node is the

parent of the alternatives. Criteria are compared using for importance to the goal; the cars are compared

for preference with respect to each of the criterion nodes.

Thus there are 4 sets of comparisons to be made for this model.

Deriving Priorities from the AHP Pairwise Comparison Matrix



Prestige 1 1/4 1/3 1/2

Price 4 1 3 3/2

MPG 3 1/3 1 1/3

Comfort 2 2/3 3 1

Priorities for the criteria are obtained by calculating the principal eigenvector of the above matrix. A short

computational way to obtain this vector is to raise the matrix to powers. Faster convergence can be

obtained by successively squaring the matrix. The row sums are calculated and normalized at each

iteration. The computation is stopped when the difference between these sums in two consecutive

calculations of the power is smaller than a prescribed value.

The principal eigenvector of the above matrix to four significant decimals is:

Criteria Priorities

Prestige 0.0986

25

Price 0.425

MPG 0.1686

Comfort 0.3077

Total 1.000

What does this result mean, in everyday language? It says the most important thing in buying a car is the

price (presumably one would prefer a less expensive car - though if a Russian billionaire were making the

decision he might prefer the most expensive car, still price is a big factor) at 42.5% and the second most

important thing is Comfort at 30.77%. The least important thing about a car is its prestige.

Remember, this is top-down thinking, the kind of thinking where your judgments reflect what you believe

to be the politically correct proper answer. It is not necessarily the way you think when you go to the

dealer showroom and see a few cars. Somehow one's opinion about the importance of prestige may take a

dramatic swing upwards. People continue to buy those fancy cars, and the ANP is a more realistic decision

process that captures this reasoning because it allows feedback; that is, you must see and evaluate a

few alternatives before you can actually determine the importance of the criteria. This is called feedback.

The ANP can integrate the top-down view with a bottom-up view.


To perform sensitivity analysis one asks what the decision would be if the priorities of the criteria were

different. Below is a screenshot of the dynamic sensitivity barchart showing that if the priority of the

Prestige criterion increases considerably from its original .04 value, the more prestigious car, the Acura TL,

represented by the red bar, becomes the best choice.

Note: To get to this screen select the Computations>New Sensitivity command, the Horz Barchart tab,

set the "Node for sensitivity" to Prestige and click and drag the Parameter button to the right.

Sensitivity Barchart showing Results of increasing Priority of Prestige

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To create an ANP decision model first consider the decision problem to come up with a collection of

factors that seem to represent the issues in the decision, define the purpose of the decision carefully, and

determine some possible alternatives of choice. In the software the term for a factor is node and the

nodes are logically arranged into groups called clusters.

What is a Model?

The term model is sometimes used as a verb meaning to create a framework in some way that represents

something in the real world. Models range from things like statues created by artists to miniatures of

buildings that architects use to a mental understanding of a situation. It is the latter meaning that we

have in mind when talking about a decision model; the model starts with a conception of what the decision

is about, what the alternatives are and what factors should be taken into account in someone's mind. The

mental model is then built using the software resulting in a computer file. Information about the mental

model the decision maker has in mind is transferred into the software, including the factors and

alternatives of the problem, and their structure, how they are grouped and linked together. SuperDecisions

models are files with the extension .sdmod.

After the software implementation of the mental model of the decision we usually refer to it as "the

model"; it is a representation of how we see the decision: what the decision is about, the factors that

come into play, and the alternatives of the decision. The decision maker(s) then make assessments using

judgments or data about the elements of the problem and the software combines these judgments to

prioritize the alternatives, or rank them. This is the end result, what we were looking for when we

conceptualized and created the decision model.

Market Share Models as Validation Exercises

Models to estimate the market share of products or companies involve both tangibles and intangibles and

have been found to be a good means of validating the ANP decision process. One may ask, "Why try to

estimate market share when the incomes and other market measures of most public companies are widely

available on the internet?" Early on when the ANP was first being applied, it was important to find real-life

situations that had measurable outcomes against which the model results could be checked to validate the

ANP as a decision making process. The success of a market share model does depend on the familiarity of

the person who is designing the model and making the judgments with the product.

The Hamburger Market Share Model

The Hamburger model was created for the purpose of estimating the relative market share of three fast

food hamburger chains. The factors become the nodes in the model and they are grouped into clusters of

nodes that are similar in some way. They become the nodes in the model and include tangible things such

as price and soft intangible things such as creativity of advertising, location of restaurants, and other

things customers might think about in deciding where to eat, and they are logically grouped into the 4

clusters listed below.

Hint: Sample models are under the Help command on the main menu. This sample model can be loaded by double-clicking on it from Help>Sample Models>Introductory Models>Hamburger.sdmod

1 Alternatives

1 McDonald's

2 Burger King

3 Wendy's

2 Advertising

1 Creativity

2 Promotion

3 Frequency

3 Quality of Food

1 Nutrition

2 Taste

3 Portion

4 Other

1 Price

2 Location

3 Service

4 Speed

5 Cleanliness

6 Menu Item

7 Take-out

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8 Reputation

Below is a screenshot of the hamburger model. This is referred to as a simple network because all the

clusters are in the same window.

SuperDecisions Network Model for to Estimate Market Share of Hamburger Places

Links are made among the nodes to indicate influence. It is not possible from this view of the model to

determine precisely which nodes are connected. An arrow from one cluster to another is a general

indicator that some node(s) in the cluster at its base are connected to some node(s) in the cluster at its

point.

To show which nodes are connected from a given, click on the "show connections" icon, the fan-shaped

icon next to the ? help icon, to turn on the "show connections" mode. With this mode turned on mousing

over any node in the model will show the nodes it is connected to outlined in red.

For example, the Creativity node is linked to nodes in three clusters; its own 2 Advertising cluster, the 4

Other cluster and the 1 Alternatives cluster.

Showing Which Nodes are Linked from the Creativity Node

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Comparison Groups

A comparison group consists of a parent node and its children nodes that are in the same cluster. The

Creativity node in the screenshot above is the parent in three different comparison groups as it is

connected to children nodes in three clusters: 1 Alternatives, 4 Other and 2 Advertising clusters.

The children nodes are pairwise compared with respect to the parent node for dominance. Dominance may

be expressed in terms of importance, preference or likelihood. The results are priorities for the children

nodes. Priorities sum to 1.000 for each group of children nodes. Priorities are combined and synthesized for

all the nodes throughout the model to give the answer, the priorities of the alternatives of the decision.

For this model, these priorities reflect the relative market share of the restaurants.

For example, 1 McDonald's, 2 Burger King and 3 Wendy's must be pairwise compared with respect to 1

Creativity to establish priorities for them and thus rank them for creativity in advertising. See the section

on pairwise comparing for more details.

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The pairwise comparison judgments used in the AHP are defined in the Fundamental Scale of the AHP

shown below. Elements may be pairwise compared with respect to importance, preference or likelihood.

Most comparisons can be broadly categorized as one of these three types.

THE FUNDAMENTAL SCALE OF THE AHP

Intensity of importanceDefinition Explanation

1 Equal importance Two elements contribute equally to the objective

3 Moderate importance Experience and judgment slightly favor one element over another

5 Strong importance Experience and judgment strongly favor one element over another

7 Very strong importance An activity is favored very strongly over another

9 Absolute importance The evidence favoring one activity over another is of the highest possible order of affirmation

2, 4, 6, 8 Used to express intermediate values

Decimals 1.1, 1.2, 1.3, …1.9 For comparing elements that are very close

Rational numbers Ratios arising from the scale above that may be greater than 9

Use these ratios to complete the matrix if consistency were to be forced based on an initial set of n numerical values

Reciprocals If element i has one of the above nonzero numbers assigned to it when compared with element j, then j has the reciprocal value when compared with i

If the judgment is k in the (i, j) position in matrix A, then the judgment 1/k must be entered in the inverse position (j, i).

To compare n elements in pairs construct an n x n pairwise comparison matrix A of judgments expressing dominance. For each pair choose the smaller element serves as the unit and the judgment that expresses how many times more is the dominant element .Reciprocal positions in the matrix are inverses, that is, aij= 1/aji.

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A network is composed of clusters, nodes and links among the nodes in a window. Hierarchies are special cases of

networks in which the links point from the goal to the criteria to the alternatives. In general networks the links may

go in any direction, and there is no goal.

Node Comparison Groups and Network Links

A comparison group is comprised of a parent node and its children nodes in a given cluster that will be compared

with respect to it. The children nodes must all be in the same cluster, though a parent node may be the parent of

several groups and have sets of children in different clusters. A parent node may even have children in its own

cluster, and thus be involved in several comparison sets.

A node may serve as a parent in one comparison group and a child in another.

An arrow will appear from one cluster to another when there is at least one link from a node in the first cluster to a

node in the second cluster. Most commonly a node in the first cluster will be connected to several nodes in the

second cluster. When a node is linked to a node or nodes in its own cluster, the arrow becomes a loop on that

cluster.

The arrows that indicate links automatically appear whenever nodes are linked. Clusters cannot be linked in any other

way.

Cluster Comparisons

Cluster comparisons arise in networks when 2 or more clusters are connected by arrows from a given cluster. They

must be pairwise compared for impact on the given cluster. Select the Cluster tab in the comparisons mode to

perform the pairwise comparisons. To see the results of all the cluster comparisons select Computations>Cluster

matrix. Each column gives the relative impact of the clusters connected from the column heading cluster.

Mathematical Note: Each value in the cluster matrix is multiplied times all the entries in the corresponding component

of the unweighted supermatrix to yield the weighted supermatrix in which all the columns sum to 1. The software

performs this computation automatically.

Complex Models

A model in which all the clusters and nodes are in a single window is a simple network model. Nodes in a simple

network can have subnetworks attached to them, and the resulting model is called a complex model. Complex

models may have any number of cascading levels, but usually are limited to three for a BOCR (Benefits,

Opportunities, Costs and Risks) model. When a subnetwork is created for a node a blank window appears and a

simple network of clusters, nodes and connections is built there.

2-Level Complex Model

An example of a 2-level BOCR (benefits, opportunities, costs, and risks) model is shown below for the sample

introductory model Car_BOCR.sdmod. This is a model to choose whether to buy an American, Japanese or European

car. There are 4 subnets, one for each of the BOCR nodes (known as merit nodes). The priority vectors for the cars

obtained in the subnets are combined in the top level window using a formula. The additive(negative) formula is

being used here: bB+oO-cC-rR. The b,o,c and r stand for the priorities of the Merit Nodes in the top level window,

while the B,O,C, and R stand for the synthesized vectors of priorities of the cars in the respective subnets.

Top Level of Two-level BOCR Network

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The judgments in a subnet are made with respect to the controlling node, the 1Benefits node in the top level window, so all the judgments are with respect to which of the nodes in a pairwise comparison has the more positive benefit.

Subnet under 1Benefits node Judgments in this subnetwork are made with respect to benefits

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Subnet under 2Opportunities node Judgments made with respect to opportunities. Opportunities are possibilities that may be available in the future as opposed to benefits which are things that are known and for which the benefits can be estimated with confidence.

Judgments in the costs and risks subnets are made posing the question as to which is the more costly or risky. The highest priority alternative should be the most costly and most risky in order for the formula to work correctly.

Subnet under 3Costs node Judgments made with respect to costs (in the present). Judgments are made so the most costly is given the higher priority or preference. Synthesizing in a costs subnet should result in the most costly alternative having the highest priority.

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Subnet under 4Risks node Judgments made with respect to risks (possible things that might happen in the future). Judgments are made so the most risky has the highest priority when the priorities for the alternatives are synthesized.

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Outer DependenceOuter Dependence is when a cluster is connected to a cluster other than itself by an arrow. Such a connection means at least one node in the first cluster is the parent for a comparison group of children nodes in the other cluster.

Inner Dependence

Inner Dependence is when a cluster is connected to itself, which occurs when the parent node and the children nodes forming a comparison group are in the same cluster. A loop will appear on the cluster in this case.

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The Design CommandTo build a model you must create clusters, nodes within the clusters and connections among the nodes.

When the software starts the blank window shown below will appear.

d

There are two ways to begin building a model:

• Use the Design menu command

• Use the File>New command

Using the Design menu command directly begin creating clusters, nodes and connections in the blank

window.

• Design>Cluster to create, edit and modify cluster appearance

Shortcut: Press N (Capital N. Hold down the Shift key and press letter n)

• Design>Node to create, edit and modify node appearance

Shortcut: Press n (Press the letter n)

• Design>Node connexions from to make node connections.

The Design Command

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The File>New command opens the template window which has a number of possibilities:

• Select the template you want to use

• Full Template - creates a complex BOCR model with three levels: the top Level contains

Benefits, Opportunities, Costs and Risks nodes, (the BOCR nodes); a second level of control

hierarchies; and a third level of decision subnets containing the alternatives.

• Small Template - creates a two level complex model with BOCR nodes in the first level

attached directly to decision subnets (containing the alternatives) in the second level.

• Open File opens a browser for loading a pre-existing model

• Simple Network does nothing except display the error message below. In the opening

screen you must create a network of clusters, nodes and connections using the Design

menu commands.

• File>Recent Files shows a list of recently used models.

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To build a model you must create clusters, nodes within the clusters and make connections among the

nodes.

Creating a Cluster in a New Model

From the main menu:

• Design>Cluster brings up the new cluster dialog box where you can to create, edit and modify

cluster appearance

• Keyboard Shortcut Press <Shift> <n> (i.e. capital N) with the cursor located anywhere on the

background of the main screen to bring up the New Cluster Dialog Box. Upon saving, the new

cluster window will be located where the cursor was when you pressed the keys.

• Mouse Shortcut Press <Shift> and <Left-Mouse-Button> to bring up the New Cluster Dialog Box.

Upon saving the new cluster window will be located where the cursor was when you implemented

the shortcut.

In the New Cluster Dialog box shown below type the name of the cluster in the Name field. Press <Tab>

or click in the next field with the mouse to move to the next field. Type a description of the cluster in the

Description field, or leave it blank.

For a complete description of setting Fonts, Icons, Pictures and Colors click here.

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Click the Create Another button at the bottom to save and begin creating the next cluster immediately.

Click Save to save the current cluster and return to the main screen.

The name of the previous cluster remains selected in the Name field when you create the next cluster

immediately so you can begin typing the new name. Pressing <Enter> before typing anything will result in

the software trying to save a new cluster with the old name and you will get the warning shown below.

Click the Save button to stop creating clusters and return to the main window.

Pop-ups Menu Icon (the ? menu icon)

The description of a cluster will appear whenever the cursor is held anywhere over the cluster background

for a few seconds if the Popups mode for showing descriptions has been activated. Left-click with the

mouse to depress it and activate the pop-ups.

Left-click the Pop-ups icon on the menu bar to toggle the "show descriptions" mode on and off.

See Selecting Fonts/Colors/Icons.

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Cluster MenuThe basic Cluster commands may be accessed from the Main Menu.

Select Design/Cluster from the main menu for the cluster menu.

Design/Cluster/New or Edit to bring up the Cluster Dialog box shown below where you can set the

properties of the cluster.

Design/Cluster/Remove brings up a list of the clusters in the current network. Click the one you want to

remove. Watch out! It happens right away with no confirming query. The cluster, the nodes in it and all

links will be removed.

Design/Cluster/Set all Clusters title or icon font Sets the font on the title bar of the expanded cluster

window, or on the cluster icon ; choose this network, this network and subnets, everywhere in model.

Double-click a cluster to turn it into an icon or to expand it full size again.

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Set all clusters color Click to bring up a dialog box to change the color of the cluster background for all

the clusters in the current network. Click OK when satisfied with the color in the Color/Solid box.

Shortcuts to bring up the Cluster Dialog Box

Locate the mouse somewhere on the main screen, make sure no cluster or node is selected, and press the

Shift key and n to bring up the New Cluster Dialog box. The new cluster will be located on the screen in

the same location as your cursor was when you performed the command.

To bring up the Cluster Dialog box for editing, press the Shift key and e. Your cursor may be located

anywhere on the screen when you perform the command. A list of clusters will appear from which you

must select the one you wish to edit. If a cluster is selected (left-click on a cluster to select it) when you

key in the command, the Edit Cluster Dialog box will appear with the selected cluster characteristics,

ready to be edited. If no cluster was selected the New Cluster Dialog box will appear.

To remove a cluster press the Shift key and r from the Main screen to bring up a list of clusters. Select

the one to be removed. Be CAREFUL. It will be removed with no further confirming queries. If a cluster is

already selected when the command is issued a confirming query will appear and when you answer yes the

cluster will be removed.

To set the characteristics for the cluster see the section on Selecting fonts/colors/icons for Nodes and

Clusters.

Cluster pop-up menus with commands that apply to th is cluster

Right-click the mouse anywhere on the background of a cluster or Left-click the button at top left to pop

up a cluster menu with commands that relate to that cluster as shown below. .

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Iconify/Expand cluster Select Iconify/Expand to shrink the cluster to an icon as shown below, or double

left-click anywhere on a cluster. All connections will still exist and be shown when you double left-click on

the icon to expand it again. Double-clicking the cluster named Criteria above will produce the icon below.

If a picture has been selected for it, that will appear.

Cluster comparisons Select Cluster comparisons to bring up the cluster comparison screen with the first

alphabetical node in this cluster as the parent of the comparison.

Create node in cluster n Select Create node in cluster or press the letter n anywhere on the background

of the main window to bring up a list of clusters. The Node Dialog box will then appear and you can create

a new node.

Remove self loop Select Remove self loop to remove all loops on the cluster (these are links from a node

in the cluster to other node(s) in the same cluster).

Organize Nodes Select the Organize Nodes command to choose to have the nodes arranged horizontally,

vertically or packed in a rectangular pattern. In all cases the arrangement is in alphabetical order

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There are several ways to bring up the Node Dialog box that is used to create a new node.

Main MenuFrom the main menu choose Design>Node>new. When the list of possible clusters appears select the cluster within

which the new node is to be located.

Cluster Selector Menu

Keyboard Shortcuts to Create a NodePress Alt and n with your mouse located over the background of the cluster window in which you wish to create the

node to bring up the Node Dialog Box.

or

Press <Shift> and left-click with your mouse located over the background of the cluster window in which you want

to create the node to bring up the Node Dialog Box.

or

Press n with your mouse located on the background of the main window and with no cluster or node selected to

bring up the Node Dialog box.

Note: Click on a cluster or node to select it. The title bar of a selected cluster will be darker than the others, a selected

node will be outlined in black.

Mouse Shortcut to Create a Node

Press Shift and left click with the mouse located in a cluster window in the exact location where you want the node

to be created. The Node Dialog box will come up and upon saving the new node will be placed where the mouse

pointer was.

Node Dialog boxThe Node dialog box is used to type a name and description for the node and to set display characteristics such as

font size and type, the icon, and the color for the node name text box.

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Type the name of the node in the Name area and a description for it in the Description area. There is no theoretical

limit to the number of characters, but long names will be truncated in some displays. The name may include letters,

numbers, spaces and characters.

Click the "Save" button to save the new node and exit to main screen; select the "Create another" button at the

bottom of the dialog box to save the current node and continue creating new nodes; and select the "Cancel" button

to cancel the current selection and return to the main screen.

Hint: Make the beginning of node names different, not the end, if you are using similar names. Otherwise you may

not be able to identify the nodes when in the comparison node as names may be abbreviated.

Pop-ups to show Node DescriptionBelow is an example of a cluster named Criteria containing one node named Price. Holding the cursor over the node

for a few seconds will bring up its description, The cost of the car", if the "show popups" mode is on. To turn it on

left-click the Popups icon and the icon will be slightly depressed on the main window menu. To turn it off

left-click again and it will spring back to normal display.

The Design>Node Menu Commands Node Menu Commands

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New brings up a list of existing clusters. Select the cluster where the new node or node to be edited can be found

and the Node Dialog box yo create a node in the selected cluster will appear. Proceed to set the properties of the

node.

Edit brings up a list of all the nodes in the current network. Select the node to be edited to bring up the Edit Node

Dialog box.

Remove brings up a list of all the nodes in the current network. Click on the one to be removed. Watch out! It

happens right away with no confirming query. The node and all the links to and from it will be removed. It might be

a good idea to back up your model beforehand with the File>Backup command just in case you change your mind.

Shortcuts to remove node

Keyboard

Your mouse must be over the node you wish to remove. With your mouse over the node, press the Alt

key simultaneously with the r key and confirm that you wish to remove the node.

Menu Command

Choose the Design menu. From that menu choose the Node submenu. Finally choose the Remove

menu item from the Node submenu. It will bring up a dialog box asking you to select the cluster and the node

in that cluster to be removed.

Popup Menu Command

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With your mouse over the node you wish to edit, click the right mouse button. This will bring up a

popup menu for nodes. Choose the Remove node menu item.

Set all nodes font to bring up a dialog box to change the fonts of the nodes so they are all the same in 1) this

network, 2) this network and subnets 3) everywhere in model.

Set all nodes color to bring up a dialog box to change the color of the node background for all nodes in the current

network. Click OK when satisfied with the color in the Color/Solid box.

Node Pop-up Menu

Right-click any node in the model to bring up a pop-up menu for it.

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Selecting Fonts/Colors/Icons for Nodes and Clusters

Fonts, background colors and icons can all be selected or changed for nodes and clusters.

The same dialog box appears for clusters and nodes to set the fonts, background color of cluster window,

the icon to represent the cluster and so on. In this section we describe the settings that control the

cluster or node appearance.

• Select Design>Cluster (Node)>New (Edit) to bring up the Cluster/Node Dialog Box

New Cluster/Node Dialog Box

Name of Cluster Enter the name of the cluster in the first field. You may enter an unlimited number of characters.

Description of Cluster Enter a description for the cluster in the second field. You may enter an unlimited number of characters.

They will appear as the pop up description.

Selecting font for title Select the style of font, size of font and style of font(italics, bold, etc.) for the cluster window title using

the three buttons here.

Selecting font for icon Select the type of font, size of font and style of font(italics, bold, etc.) for the cluster icon title using the

three buttons here.

Changing the Icon Click the Change Icon button with your mouse and select a new icon by double clicking on the one you

want from among those that appear in the Icon Chooser dialog box. You can also browse for and select

fonts from a different directory than the default one which is the c:/Program Files/Super Decisions/Icons

directory.. The selected new icon will appear in the Editing dialog box. Save to exit the dialog. To switch

back and forth from the icon to the window double click on the icon or the background of the cluster

window.

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Double click the icon you want to select it. Browse to a different directory to select an icon of your own

(.jpg files work best)

Changing Window Color Click the Change Color button with your mouse and select a new color by double clicking on the one you

want from the Color dialog box that appears. Select the OK button to effect the color change.

Saving Changes Click the Save button to permanently change to the creating/editing selections you have made for this

cluster window. If the dialog box disappears before you have had a chance to save the changes, it may

have become a minimized icon on the Windows menu bar at the bottom of the screen. Look on the

Windows menu bar to find it, then double-click on it to open it again and then click the Save button.

Details of Commands

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Edit Dialog boxA dialog box like the one shown below is used to create and edit node and cluster names and

appearances. To cause it to appear for editing the Design>Edit menu may be used or right-click on the

node or cluster and select Edit Node or Edit Cluster from the menu that appears.

Note: the Make Connections icon must not be turned on (i.e., depressed) in order for the Editing

dialog box to appear.

Name of Node/Cluster Type the name in the first field. You may enter as many characters as you like, including spaces and

special characters like $, though it is recommended that you stick to 12 or fewer for display purposes. Use

the description field to elaborate. Names will be shown truncated as required for space in the comparison

screens. Tab to move to the description field or click with the mouse.

Description of Node/Cluster Type a description in the area labeled Description. There may be an unlimited number of characters. A

description copied to the Windows clipboard in another application may be pasted here by clicking with the

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mouse in the area then simultaneously pressing the keys <Ctrl><v>.

Main Font selection To change the type of font used for the name of the cluster window or node, use the three buttons here:

type of font, size of font, style of font.

Font selection for Cluster/Node IconsA cluster window can be minimized to an icon by double-clicking on it (and restored by double-clicking

again).

Icons can be selected and displayed for nodes as well.

Select the type of font, size of font and style of font(italics, bold, etc.) for the icon title using the three

buttons here.

Changing Icon Click the Change Icon button with your mouse and select a new icon by double clicking on the

one you want from among those that appear in the Icon Chooser dialog box. You can also browse

for and select fonts from a different directory than the default one. The selected new icon will

appear in the Editing dialog box. Save to exit the dialog. To switch back and forth from the icon to

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the window double click on the icon or the node text box window.

Changing Node Text Box Color Click the Change Color button with your mouse and select a new color by double clicking on

the one you want from the Color dialog box that appears. Select the OK button to effect the color

change.

Saving Changes Click the Save button to permanently effect the creating/editing selections you have made for

this node. If the dialog box disappears before you have a chance to save your changes, it has

probably been minimized and is now located on the bottom Window Menu bar. Double-click on it to

open it up again and click on the Save button.

Selecting Fonts/Colors/Icons for Clusters

Main Font

This is the font used for the title of the expanded window view of the cluster. Left click the buttons under

Main Font to select font characteristics. See the cluster named Voters below to see the currently

selected font: times roman, 12 pt., Normal.

You can double-click anywhere on a cluster to shrink it to an iconized view shown below.

Icon Font This is the font used for the title of the iconized view of the cluster shown below. Left click the

buttons under Icon Font to select font characteristics for the icon. Double-click on the icon to expand it

again to the window view.

Change Icon Double-click the Change Icon button to select an icon image. You can also select a blank icon

for no image. The current images are in the directory c:/Program Files/Super Decisions/icons and are .gif

files. Other picture formats such as .jpg can also be used. To use your own image files click the Change

Directory button, then type the path to the directory where your image files are located. Click the Update

List button, then click the Done button to see the images. You can now select one and double-click to

make it the new icon image.

Change Color Double-click the Change Color button to select a new background color for the cluster

window.

Cluster window with above selected color

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After you have created clusters and nodes in your network, they must be connected. One way to go about

deciding what nodes to connect is to start by holding a node in a cluster in mind, then examine the other

clusters in turn, asking if the nodes in them influence that node (or are influenced by that node). If they do,

make links from that node to them. A comparison group is formed this way that will have that node as the

parent node and the nodes in the other cluster to which it is linked as children. The children nodes of such a

comparison group may all be in the same cluster as the parent (inner dependence) or a different cluster

(outer dependence). A node may serve as a parent in more than one comparison group. And a node may serve

a double role as a child in another comparison group.

Connections may only be created between nodes. Clusters become connected as a consequence of node

connections.

Node Connections

To form a comparison group select a node, the node that you wish to become the parent node, right click on it

to drop down a menu, select the "Node connexions from" command and follow the instructions to connect to

the desired children nodes in another cluster (or in the same cluster). The children nodes must be together in

the same cluster. It is not allowed for some of them to be in one cluster and some in another cluster, and still

belong to the same comparison set. You may make two (or more) different comparison sets for the same

parent. Shortcut: Left-click on the parent node and right-click on each of the children nodes in turn. You may

make more than one comparison set at once by left-clicking once on the parent node, then right-clicking on

nodes in one cluster after another til you finish making all the connections for that parent.

Cluster connections appear automatically when nodes in them are linked

When a parent node in one cluster has children nodes in another cluster, they are automatically connected

and a line appears going from the parent node’s cluster to the children nodes’ cluster. This is called outer

dependence.

To get the link going the opposite direction, you must connect a parent node in the second cluster to some

children nodes in the first cluster.

When the children nodes are in the same cluster as the parent node, the cluster is connected to itself and the

link appears as a loop on the cluster. This is called inner dependence.

Nodes may be connected in several ways

• Use the Design/Node connections from command on the main menu and follow the instructions to

select the parent node and its children nodes.

• Place the cursor over the node that is to be the parent node, right-click on it to get a drop-down menu

of node commands and select the "Node connexions from" command. Follow the instructions to

connect the current node to its children nodes.

• Turn on the Do connexions mode by left clicking on the menu icon with the three arrows. When this

icon is depressed you can connect nodes by left-mouse click on the node that will become the parent

and right-mouse clicking on the children nodes. Similarly, you can disconnect by left-clicking on the

parent node and right-clicking on some previously connected children nodes.

Tip: Be Careful! You will lose any judgments you have entered for nodes when you disconnect them.

To select several nodes at once:

• To connect all nodes in the "from" cluster to all nodes in the "to" cluster shift left-click on any node in

the "from" cluster; shift right-click on any node in the "to" cluster.

• To select only some nodes in a cluster, hold down the Ctrl key and left-click on each node to be

selected.

Tip: Do not make spurious links where you do not plan to make judgments; the priorities will be set

to default equal values - and there will be a very different result from not making any judgments at

all.

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Make Cluster Connections

There is no way to directly create a cluster connection. A cluster connection appears when a node (or

nodes) in the cluster at the origin of the arrow is connected to a node or nodes in the cluster at the

terminal end of the arrow. The cluster connection automatically appears when the nodes are linked.

A cluster connection is merely an indicator that some node or nodes in the originating cluster are

connected to some node or nodes in the terminal cluster.

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Node Popup Menu

Right-click a node to bring up the Node Editing menu for that particular node.

Edit node brings up the Edit Node dialog box

Remove node will remove the node and all its connections and subnetworks (instantly!)

Node connections from will allow you to select nodes you wish to connect from this node. Left click to

select nodes from list of all nodes in model (left click again to deselect).

Hilight nodes connected to this one

Hilight nodes connected from this one

Unhighlight Nodes (tip: same node selected node when hilighted must be selected to turn off.)

Node compare interface launches the pairwise comparison mode

Make/Show subnetwork will create a blank subnet attached to current selected node or open an existing

one. Subnetworks are always linked from a node in the network above.

Remove Subnetwork will remove any existing subnetwork (and all its subnetworks) attached to selected

node.

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Popup Menu

With your mouse hovering over the background of the cluster you wish to edit, click the right mouse

button. This will bring up the cluster popup menu of commands.

Or bring it up by left-clicking the button at the top left of the cluster window.

Edit cluster brings up the Edit Cluster dialog box for setting cluster properties.

Remove cluster will remove the cluster and all nodes and node connections.

Iconify/Expand cluster will toggle between the icon for a cluster and the window view of it.

Create node in cluster will bring up the new node dialog box and add the new node to the cluster.

Remove self loop will remove a self loop from the cluster; i.e. it will remove all links among nodes in that

cluster.

Organize Nodes will re-arrange the nodes:

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When all of the cluster and node connections have been created, you will want to compare/assess nodes and clusters in each network. Nodes are compared with respect to another node and clusters are compared with respect to another cluster.

Judgments/assessments are done for the nodes in each network. Nodes that form a comparison group must be in the same cluster. These nodes, the children nodes are connected from the same parent node, and are assessed with respect to how they influence that node, or how that node influences them. The parent node may be in a different cluster from the children, or the same cluster.

Influence must be treated consistently, how the parent influences the children, or vice-versa, but the flow direction should be kept the same throughout the network and throughout the model.

The Fundamental Scale of the AHP gives the numerical judgments one can use when making

pairwise comparisons and their verbal equivalents. Briefly speaking the numbers from the fundamental scale range from 1 to 9. These are absolute numbers; that is, when assessing the dominance of A over B, and a judgment of 7 is given, it means A is 7 times B, or Very Stronglybetter than B. Absolute numbers cannot be changed to a different number with the same meaning in the way that ratio scale numbers can; e.g. a measurement in yards or meters.

To start the comparison process:

• Select the command Assess/compare from the main menu of the network window. • Select the sub-menu command Cluster comparisons or Node comparisons to start the

comparison process. If a cluster or node has been previously selected, by left-clicking on it, the comparison process will start there. If nothing is selected, the process begins with the first cluster or node (alphabetically organized).

• Left-click on the Do Comparison button and select the tab in the first pane to do either Node or Cluster comparisons.

Troubleshooting Tips Question: Why am I not able to do cluster comparisons when there are only two clusters? Answer: There are

no comparisons possible with only two clusters. If one is the parent, the other is the only child and there is

nothing to compare it to.

Solution: Do not try to do cluster comparisons when the network has only two clusters. The cluster weights

are automatically set at 1/2 for each.

Shortcuts to the Comparison Window

Click the menu icon shortcut to start the comparisons with the currently selected node as the parent of the

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Click the menu icon shortcut to start the comparisons with the currently selected node as the parent of the

comparison group:

or

Pop-up the cluster/node menu by right mouse-clicking the cluster background or the node. This will launch th

Comparison mode with the cluster on the background in a cluster. Select the command Cluster comparisons to

enter the Comparison Window in the Questionnaire mode

In every comparison mode, a verbal comparison phrase for the selected pair appears at the top of the

window. It shows the dominance order with the more dominant of the pair first, and the less dominant

second. The node at the beginning is the more dominant: for example, more important, more preferred, more

likely, more beautiful. Enter whatever comparison word you wish to have appear in the verbal comparison

phrase.

The current judgment is shown by the depressed square in the tracking boxes at the right-hand side of the

screen. If there are n comparison elements, then there are n(n-1)/2 judgments. It is not necessary to enter all

the judgments, though it is better to do so in order to extract the most information from the user about the

real dominance relationships. You can shorten the judgments by entering only those in the first row, or last

column, or the diagonal. At a minimum there must be one judgment in every row and column.

The first four modes are for making pairwise comparisons. The fifth mode is for entering data directly. To

change modes click the tabs at the top in Panel 2.

The first line of the Comparative phrase shows the parent node and parent's cluster. The second line shows

the nodes being compared and the verbal expression for the judgment.

The comparative word can be changed from importance to something else by left-clicking on the highlighted

portion. Select Other to type in a word of your own more appropriate for this set of comparisons, e.g. beautiful.

The Five Modes for entering Judgments

Questionnaire Mode

Select the numerical judgment that best expresses your judgment by right-mouse-clicking on the number

closest to the dominant element. Since Price is moderately more important that Miles per Gallon, click the

blue 3 on the Price side.

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Graphic Mode

The ratio of the bar lengths is used for the judgment for the current pair of elements. To change the bar

lengths, left click with the mouse on a radius line in the pie chart display and drag to change the relative

sizes of the areas. The ratio of these areas is the same as the ratio of the bars. Click with the mouse on the

judgment tracking boxes at the right of the screen to advance to the next pair. To reset to equal, click the "No

Comparison" button. The judgments are continuous (unlike in the questionnaire mode where they are limited

to integers) and thus can assume decimal values as well as the whole numbers of the AHP Fundamental

Scale.

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Verbal Mode

In the verbal mode words are used on the scale. The lines on the scale represent the judgment

numbers of the AHP Fundamental Scale. Left-mouse-click to bring a red bar up to your selected

judgmen. As in the Graphic Mode, the values are continuous and are not limited to whole numbers from

1 to 9, but can be decimals if the red bar does not rest exactly on one of the lines representing a whole

number.

Matrix Mode

Each cell in the matrix mode corresponds to the pair of elements being compared. The judgments

are shown as decimal numbers, rounded off to the nearest tenth.When a judgment is displayed in

red,and the arrow points up, the element at the top is dominant, in blue, with the arrow pointing left,

the element at the side is dominant. To invert the dominance order, double-click with your mouse on

the arrow in the cell you wish to invert. Improving Inconsistency is always done from the matrix mode.

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Direct Mode

The Direct Mode can be used to enter data. It is not necessary to have the numbers sum to 1 as the

software will normalize the numbers entered. If higher numbers are worse than lower numbers, the

priorities must be inverted, so click the Invert checkbox to reverse the priorities. In the example below

an example of inverting priorities is shown. The Avalon is the cheapest car, and has the highest

priority at 0.40678.

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Direct Mode

The Direct Mode can be used to enter data. Select the Direct tab and type in the data. It is not necessary to

have the numbers sum to 1 as the software will normalize the numbers entered. If higher numbers are worse

than lower numbers, the priorities must be inverted, so click the Invert checkbox. In the example below shows

priorities that are in the reverse order of the data. The Avalon is the cheapest car, and has the highest priority

at 0.40678.

When should you invert priorities? If a larger number means less desirable such as for price, or heat, or miles

left to go, invert the priorities.

Technical note: To invert priorities so that the one that is largest becomes the smallest and vice versa, take the

inverse of each priority, sum these inverses and divide each by the sum.

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Improving inconsistency must be done in the Matrix Mode of the comparison/assessment process. The

inconsistency should be about 0.10 or less. The current value of 0.1975 is too high. Click on the

Inconsistency button at the top left of the matrix for help in finding the inconsistent judgement(s).

• Inconsistency of Current gives a suggested judgment for the current cell that would produce the

best overall consistency for this set of comparisons. The currently selected cell is the one with the

cursor in it, in this case Price versus Miles per Gallon. Entering 7.6566 (type it in and press the

enter key or mouse to another cell) is the value that would improve the inconsistency the most.

• However, it is risky to pick one judgment and change it because it may not be the really logically

inconsistent judgment. You might have another judgment that is logically wrong if you re-examine

it, or that was accidentally inverted, that is merely causing the current judgment to be

inconsistent. In general it is better to use the method below and let the computer find the most

inconsistent judgment.

• Basic Inconsistency Report This report shows how changing the judgment of any pair of nodes in

the model will change the Inconsistency. The pairs are listed in the order of most improvement.

• In this report the most effect would be gained by changing the Price versus Comfort judgment from

7 to 1. You may change the judgment as you like; it is not necessary to accept the computed Best

Value. the Current and Best Value cells are hyperlinked to the judgment, so clicking on a value

takes you directly to that judgment in the matrix and you can enter a new judgment. Or leave it as

it is and consider changing the second one listed.

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You do not have to use the suggested judgment. Here entering a judgment of 3 for Price versus Comfort

(a compromise between the original judgment and what the computer suggests) results in a much

improved inconsistency value of 0.06560 which is well under the tolerable limit of 0.10, and 3 might reflect

your understanding better than the suggested value of 1.0.

Note the "Copy to clipboard" button which allows you to paste the pairwise comparison matrix values into

Excel. They would appear as shown in the following table.

1Price 2Miles per

Gallon

3Prestige 4Comfort

1Price 1 3 4 3

2Miles per Gallon 0.333333 1 2 0.333333

3Prestige 0.25 0.5 1 0.25

4Comfort 0.333333 3 4 1

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Select the Assess/Compare command from the main menu of a network to get into the pairwise

comparison screen. Select either cluster comparisons or node comparisons.

If there is no currently selected cluster or node, the comparison cycle begins with the first cluster in alphabetical order and the first node in that cluster in alphabetical order that is linked to children nodes. If a node is selected (left-click on a node to select it) when the comparison mode is entered, the comparisons will start with that node. By default, the node comparison tab is selected as shown below. There are three panels that appear side-by-side in the software:

1) Choose the Node or Cluster tab for either node comparisons or cluster comparisons. The Node comparison tab is selected by default when the comparison mode is first entered. The Choose Node arrow moves to the next node in alphabetical order, or click on the button to the right under it to see a list of all nodes in the model and select the one you want. The cluster chooser works in a similar manner. The Restore button at the bottom will restore the original judgments that were there as this set of pairwise comparisons came up.

2) The second panel is where judgments are made using one of the five modes: Graphical, Verbal, Matrix, Questionnaire or Direct. Click here for more detailed information about these modes.d

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3) The third panel gives the priorities derived from the pairwise comparisons and the inconsistency (too high here!). The blue arrows move to the next (or last) node in alphabetical order with children to be compared. The blue arrows with ? on them move to the next (or previous) node that has not yet been compared.

Selecting the + at the top of the results panel will expand the view of the priorities. Click the - sign to collapse the view.

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Expanded View of Priorities

If you are doing node comparisons, marking the comparison as completed by selecting yes will cause the cluster to be outlined in red when "show completed comparisons" is turned on and the parent node is moused over.

The judgments are saved when the move is made to the next set of comparisons or when all have been completed and the mode is closed.

You do not have to deliberately "save" the judgments.

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The Design>Ratings command on the main screen menu is used to create a Ratings Spreadsheet attached to

the current network. Ratings spreadsheets can be removed with the Design>Remove Ratings command.

Ratings models have the word Ratings on the title bar and the alternatives are not in the main screen view,

but in the attached spreadsheet where they are rated against standards for the criteria. This is referred to as

absolute measurement. Having the alternatives in the main screen, linked from the bottom level covering

criteria, and pairwise comparing them against each other with respect to the criteria is referred to as relative

measurement.

An Example of Rating Alternatives

Example of a Rating Spreadsheet

Multiply and add across a row in the Ratings Window to get the total for that row. For example, for Jim

Kendall multiply 1.0 (for Outstanding) times 0.077272 (priority of Dependability) and so on across the row to

get the total. His total happens to be 1.0000 which means he scored in the top category for every criterion. An

alternative that gets a total score of 1.0000 is sometimes referred to as the Ideal candidate.

Tip: Selecting View>Category Display>Names and Priorities will turn on the display of both names and

priorities for possible ratings when a cell is clicked on, then select the category from the list.

Tip: You can drag and drop the columns if you wish to re-arrange the order of the criteria.

Completed Ratings Spreadsheet

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Starting a New Ratings Spreadsheet

Design, Ratings from the main screen menu brings up this initial view of the Ratings Spreadsheet. You must

fill it out by selecting the criteria, entering the alternatives, constructing the categories for each criterion and

rating each alternative under every criterion.

Startup View of Spreadsheet

Edit>Criteria>New to add criteria from the list of all nodes in the main screen that appears. Right click on the

names of the criteria to become the columns in the spreadsheet. The criteria in the lowest level above the

alternatives in the main screen view should be selected. These are called the "covering criteria". Note that

the Goal is not a criterion to be used in rating and is not included. Neither is Work. Its two subcriteria,

Quantity and Quality, are used instead.

Spreadsheet with the Criteria in Place

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Edit>Alternatives>New and enter the names of the alternatives.

Spreadsheet with Alternatives in Place

Create Ratings Categories and Pairwise Compare Them for Each Criterion

Step 1. Use the menu command Edit, Criteria Categories and select the criterion from the drop down list for

which you want to create categories and end up in the Category Editor

or

Right-Click on a column heading and select Edit Categories to get into the Category Editor.

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Step 2. Click on New and type the category names as shown below:

Step 3. Click on the Comparisons button to bring up the pairwise comparison screen for Ratings. Enter the

judgments as to how much you prefer one category over another. For example, Outstanding is preferred by 2

to Very Good.

Step 4. Select Computations>Ideal Priorities to see the resulting category priorities. Priorities for categories

are idealized by dividing by the largest so that 1.00 is always the top priority, and the others are a proportion

of it. This prevents undue priority dilution for a criterion for which there just happen to be many categories.

Using the usual normalized priorities would give very small priorities for all of them.

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An Inconsistency index of 0.00 is shown for this example in the Priorities window above, so no improvement is

necessary. If the inconsistency index is greater than 0.10, go to the Matrix Mode and select the command

Computations>Basic Inconsistency Report to improve the consistency. <Ctrl> <Left Click> how to improve

inconsistency for more information.

Step 5. Close the Inconsistency Report window with the X in the upper right corner to return to the comparison

screen. Close the comparison screen with the X to return to the Category Editor. If you wish to save and

re-use these categories and their priorities for another criterion, select the File>Save Template command and

give the template a name. Templates are .rcp type files and are saved in the c:/Program Files>Super

Decisions/Samples directory. You can retrieve it With the File>Load Template command. Close the Category

Editor to return to the spreadsheet. Ratings can now be made for the cells in the first column.

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The Results of the Ratings in the Spreadsheet

The results in the Ratings spreadsheet are given in the Totals column and the Priorities column (which are the

normalized Totals). You can also see them with the Calculations>Synthesize command.

View Menu Commands

Totals shows the scores in a column next to the Alternatives without normalizing.

Priorities shows the scores normalized by dividing by the total of all the alternatives’ scores in a column next

to the Alternatives.

The priorities are the results that are equivalent to the Normals column of the Synthesized results in a relative

model with alternatives in the main screen.

To turn off the columns displaying Totals and Priorities, left click and uncheck on the drop-down menu.

Matrix Priorities displays the numbers in each cell and the column priorities in a table that can be copied to

the Windows clipboard and then pasted into Excel (see below).

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Column Priorities shows the priorities of the criteria and these can also be copied and pasted into Excel.

Alternative Priorities and Totals Shows Priorities and Totals as bar graphs.

Cell Display gives the option to display names of selected categories or numeric priorities in the spreadsheet.

Category Display gives the option to display only category names or both names and priorities in the drop

down menu of categories for a column. To use the Numeric Value command enter a number between 0 and 1.

For example, .5 means you are giving a rating of 50% on a category.

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Other Spreadsheet Menu Commands

File Menu

File>Save Model will save the whole model including the main screen and any subnets.

File>Close will delete an existing spreadsheet and replace it with the startup spreadsheet instead.

Edit Menu

Edit>Alternatives and Criteria - use these commands to add, rename and delete alternatives and criteria.

Edit>Copy Alternatives and Paste Alternatives - these are very useful commands in multilevel models where

the alternatives are rated and show up in many subnets. Names need to be entered exactly the same to

synthesize correctly in multi-network models, so the copy and paste commands are essential.

Calculations>Synthesize and Synthesize Whole Model To show the synthesized results for the associated main

screen only select the Calculations>Synthesize command; to show the overall results for a multi-level model

select the Calculations>Synthesize Whole Model command.

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The main results of an AHP or ANP model are the overall priorities of the alternatives obtained by

synthesizing the priorities of the alternatives from all the subnetworks. To get the overall results use the

Computations menu in the top-level model. To get intermediate results for a particular subnet you may use

the Computations menu in that subnet.

Synthesis Select the Computations Synthesize command in the main screen top level to

synthesize the results throughout the model or the alternatives. There must be a cluster named

alternatives in the model to obtain results with the synthesis command. In a simple model they are

in the main screen; in a complex model they are in the bottom level subnets. If the results are zero,

it may be because you don't have any cluster(s) named "alternatives".

Synthesis Shortcut Icon

In models in languages other than English, please name the alternatives cluster with the English

word alternatives, because the software needs that as its cue to the nodes to be synthesized.

The synthesis command will give the priorities of the alternatives for the network where the

command is invoked, all of its subnetworks, sub-subnetworks, etc.

Priorities Select the Computations Priorities command to determine the priorities of all the

nodes in a network, normalized by cluster (organized by cluster with the sum of the priorities of the

nodes in the cluster adding up to 1.0), and limiting with respect to the network (the sum of the

priorities of all the nodes in the network adding up to 1.0)

The commands for obtaining final results are:

Computations Priorities This command gives a list of the priorities of all nodes in the current

network shown in two ways: normalized by cluster so that the weights of the nodes in each cluster

add up to one, and limiting priorities with the weights of all the nodes in the network adding up to

one .

Limit Matrix Option If the network has a hierarchical structure with all influence and links

feeding down to the alternative nodes in a bottom cluster, you must use a special command to see

the priorities of the nodes that are not in the bottom cluster. Select the Computations, Limit Matrix

Options command and click on New Hierarchy without limit. This will then show the priorities

of the intermediate nodes when the Computations, Priorities command is used. The normal limit

matrix option is Calculus type. This is the default option for each network and subnetwork and will

automatically be the one used each time you enter that network. You must reset the option each

time the network is entered if you want a different one.

Computations Synthesize Using the Synthesize command in a bottom level subnetwork, gives

the synthesis of alternative priorities for that network only; from an intermediate network it

synthesizes for that network and all the subnets of that network. Synthesizing from the top-level

network gives the synthesis of priorities of the alternatives throughout the entire model. You might

think of the priorities of the alternatives rolling up from the bottom level decision networks to the

networks they are attached to, and from there rolled on up to the next network and so on to the

top level.

Note: In multilevel complex models the alternatives are in the bottom level subnets only.

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Performing Sensitivity in a Hierarchysave2

Below is a screenshot of a model to choose the best car. You can load this hierarchical model Tutorial_1_Acura_Relative_Model.sdmod from

Help>SampleModels>Tutorial Models.

There are two commands under the Computations menu command that can be used for hierarchical sensitivity analysis: Node Sensitivity and Sensitivity. We will go through the Node Sensitivity commands first.

Node SensitivitySelect Computations>Node Sensitivity from the menu of the main screen model. Sensitivity can be done on both relative models, in which alternatives are pairwise compared, and on ratings models, in which they are rated against standardized scales.

Hierarchical Model to Choose the Best Car

The results of the model are shown in the synthesis screenshot below:

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Sensitivity studies in a hierarchy are carried out by selecting a criterion then varying its priority (redistributing the changes proportionately among the other criteria) and observing how the priorities of the alternatives change.

Step 1. Make sure the sensitivity parameter is set to Smart P0 to perform hierarchical sensitivity correctly using Node Sensitivity. Use this command Computations>Influence/Sensitivity>Options> and click on Smart P0 to select it.

Step 2. Select Computations>Node Sensitivity

Step 3. Click the "Node for Sensitivity" button and select the criterion for which you wish to perform sensitivity. Prestige is selected below with the Plot display tab. The priority of Prestige is read from the x-axis in the Plot display. As Prestige increases the Acura becomes the best choice. The colored lines represent the priorities of the alternatives at the value on the x-axis of the priority of Prestige. The red line is for the prestigious car, the Acura. It becomes more and more the favored car as the priority of Prestige increases from 0 to 1.

The black dots represent the synthesized priorities of the cars for a priority of 0.1 for Prestige. The Copy button at the bottom will copy the data used for the plot to the Windows clipboard from which it can be pasted into Excel.

Node Sensitivity - Plot

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There are three other kinds of sensitivity: Barchart, Piechart and Horizontal Barchart. To perform "what-if" sensitivity analysis click on the Parameter button and slide it to the right.

Barchart Sensitivity

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Note that the priority of the Acura TL has increased from .305 to .344 as the priority of Prestige has been increased by sliding the parameter button right to 0.097.

Effect of Increasing the Priority of Prestige

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Piechart Sensitivity

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Horizontal Barchart Sensitivity

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Sensitivity (for hierarchies)This is a different type of what-if sensitivity that allows you to select combinations of dependent variables. The initial sensitivity screen that appears has the first node (in alphabetical order) selected as the independent variable. In the tutorial model, which is a 3-level hierarchy, the goal should be the independent variable. In a 4-level hierarchy one of the criteria might be selected as the independent variable.

Step 1. Select Computations > Sensitivity to get into the sensitivity analysis screen and click on

Independent Variable.

Sensitivity Analysis Screen

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.

Step 2. The Sensitivity Input Selector dialog box will appear. Click on the name of the node (Acura TL) and then click Edit.

Sensitivity Input Selector box

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Step 3. This leads to the Edit Parameter dialog box.

Click on the Wrt node ("with respect to" node) selector button on the right and select the goal node of the model (it happens to be named Goal Node in this model!).

Edit Parameter Box

Step 4. Select the Parameter Type of Supermatrix and click the Done button at the bottom. This takes you back to the Sensitivity Input Selector box where you click the Update button at the bottom to get the final sensitivity screen.

Edit Parameter Box

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Edit Parameter box for the SuperMatrix parameter and 1st other node set to Prestige

Step 4. The sensitivity graph for Prestige will appear. To perform sensitivity for a different criterion return to the Edit Parameter box and select a different criterion for the 1st other node.

Sensitivity Graph for Prestige

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Step 5. Interpretation of the graph: The priority of Prestige is read from the intersection of the dotted line with the x axis and is initially set to .5. The priorities of the alternatives are read from where the dotted line intercepts the slanted lines. They are also given as bar graphs below. To change the priority of Prestige to the value it has in the model drag the dotted line left to 0.1. At a priority of 0.1 for prestige, the priorities of the alternatives will be the original synthesized values.

Exporting the Graph Data

Select the File>Save command and give the file a name, for example, prestige data. It will be

saved as a .txt file and can be opened using Word's Notepad, or by importing it into Excel.

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Sensitivity (for networks)

Select the Independent Variable from the Edit menu in Sensitivity Analysis to see the Sensitivity input dependent variable list. It always starts with a default initially selected dependent variable that is first in the list of nodes for that network.

To add a new dependent variable select New or to edit one click on it and select Edit to bring up the New parameter dialog box shown below in Figure 45. You can select a mix of dependent variable types, networks, number of steps, etc., but you must set the parameters for each.

1) Type of parameter:

Priorities

Supermatrix

Comparison

2) Network where the sensitivity is being carried out.

The zero network is the top level one, or you could choose:

Benefits subnetwork

Costs subnetwork

Risks subnetwork

3) Enter the starting and ending value of the parameter selected for the range where sensitivity is to be plotted.

4) Enter the number of steps - (don’t get carried away - about 10 to 20 is adequate). The calculations can get huge if you have selected too many dependent variables and have asked for lots of steps.

At any particular experiment, the values for the dependent variables and the alternatives are shown below the graph. You may also elect to have the corresponding values of other parameters shown by selecting the Extra Params command on the Sensitivity Analysis Edit menu.

Exporting Sensitivity Data To export the data points of the sensitivity plot to Excel, select

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Exporting Sensitivity Data To export the data points of the sensitivity plot to Excel, select File Save from the Sensitivity Analysis menu. Use a name ending in .txt when asked for a file name. Then load Excel, select the File, Open command and open the file. Be sure to change the type of file to *.* to display names of all files so the .txt file will show up. The Excel import wizard will then come up. Step through it selecting OK or Next to Finish at the end. The Excel spreadsheet containing the sensitivity data points will then appear. You can use it then for displaying the data in more ways and for data manipulation - such as arranging one of the alternative columns in decreasing order, so you can see the maximum value obtained.

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The goal is the parent node of the criteria and they comprise one of the comparison groups in this model.

The criteria will be pairwise compared with respect to the goal. The pairwise comparison judgments are

made using the Fundamental Scale of the AHP and the judgments are arranged in a matrix, the pairwise

comparison matrix.

The numbers in the cells in an AHP matrix, by convention, indicate the dominance of the row element over

the column element; a cell is named by its position (Row, Column) with the row element first then the

column element. In the AHP pairwise comparison matrix below the (Price, MPG) cell has a judgment of 3 in

it, meaning Price is 3 times as important as Miles Per Gallon (MPG). So logically this means MPG has to be

1/3 as important as Price so 1/3 is automatically entered in the (MPG, Price) cell.

Only the judgments in the unshaded area need to be made and entered because the inverse of a

judgment, for example, (Price, MPG) is automatically entered in its transpose cell (MPG, Price). The

diagonal elements are always 1, because an element equals itself in importance.

Only judgments in the unshaded area need to be made and entered. There will be 6 judgments required

for these 4 elements. If the number of elements is n the number of judgments is n(n-1)/2 to do the

complete set of judgments. It is possible to make less than this number of judgments and obtain a rough

estimate, but there must be a minimum of n -1 judgments.



Prestige 1 1/4 1/3 1/2

Price 4 1 3 3/2

MPG 3 1/3 1 1/3

Comfort 2 2/3 3 1

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The AHP Matrix as it appears in the SuperDecisions Software

There are five modes for making assessments: judgments drawn for the Fundamental Scale of the AHP are

used in the graphical, verbal, matrix, and questionnaire modes and direct data is used in the Direct mode.

The matrix mode is shown in the screenclip below. Only the judgments in the unshaded cells of the AHP

Matrix above need to be entered so the cells shown in the software are limited to these.

The (Prestige, Price) cell has a red 4 in it with the arrow pointing up to Price, indicating Price is 4 times

more important than Prestige (the value 1/4 is used in the computations.) And 4 is used for the (Price,

Prestige) cell though it is not displayed. Prestige does not even show up as a column heading because the

entire column is greyed out in the AHP Matrix view. However, Prestige does appear as a row heading so it

is involved in as many comparisons as any other node.

The priorities for the criteria are the results shown in the panel at the right below and are entered in the

supermatrix in the column of the parent node of the comparison, the goal node in this case. Interpret

numbers in red as fractions, and numbers in blue as whole numbers; for example the red 4 in the (Prestige,

Price) cell represents the 1/4 in the AHP matrix above. The arrow points to the dominant factor for each

cell.

The Inconsistency of 0.077 is given above the derived priorities in the Results panel below. The

inconsistency should be less than 0.10, that is, 10%.

Matrix Comparison Mode

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This program comes with several sample models. Super Decisions models fall into two general categories: 1) simple network models in which all the elements of the model appear in a single network of clusters containing nodes (nodes are the most basic elements of any Super Decisions model); and 2) complex models consisting of a control model, containing control criteria nodes to each of which may be attached a simple subnetwork. The sample models fall into these two broad categories and were chosen to illustrate the most important basic concepts of Super Decisions models. The models currently included are located in the samples subdirectory. To quickly load a sample model select the Help command on the menu, then select Sample Models and the model you want.

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Attaching Files inside Models

You can attach other files inside any network by using the paperclip icon. This Resource Allocation sample

model has two attached files as you can see from the 2 on the paperclip icon. Click Attachments.

Double-click to directly launch the attachment or click the attachment, then File>Export to save it.

Many of the sample models have a rich collection of associated files. If there are attached files, the

paperclip icon will have a small number beside it indicating how many.

We are grateful to the many MBA students at the University of Pittsburgh Graduate School of Business

and students in other courses taught by Professor Saaty who have given us permission to share their

final projects for educational purposes. These projects represent hours of work and the tremendous

real-world experience and expertise of students most of whom had full-time jobs in industry.

Note that attached files belong to the network or subnet they are attached to, though normally you would

probably attach files in the main top-level network. Files you might include:

Word files(with .doc extensions) containing background information and reports

Powerpoint files (with .ppt extensions) containing powerpoint presentations

Excel files (with .txt or .xls extensions)containing other data, exported from the model, for import to the

model, or with external calculations

Other SuperDecisions model files (with .sdmod extensions) containing related Super Decisions models;

there might be several models for a complex project.

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9.5 Car Choice (Hierarchy)The car choice model is a hierarchical model that is re-done here as a simple network model. With a model that is a hierarchy, the alternatives at the bottom level are a "sink". That is, all links go into that cluster and none come out. One must link every node in this cluster to itself. This has the effect of introducing an identity matrix for the alternatives cluster into the supermatrix. It appears in the lower right hand corner of the supermatrix. It "catches" the priorities flowing into it. The supermatrix has the wonderful property that the synthesis of the alternatives with respect to the elements of every cluster appear as the bottom level components in the Limiting Supermatrix. This is very apparent in the car model which is also one of the sample models in the Expert Choice hierarchical decision making software. The results here are the same as in the Expert Choice model: the priorities of the cars Volvo, Mercedes, and so on, and they appear at the bottom of the leftmost column in the supermatrix labelled Goal at the top. All components are zero, except those across the bottom level for the alternatives which give all the possible syntheses with respect to what are intermediate levels in the Expert Choice model, but are clusters in the Super Decisions model.

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9.1 Hamburger market share modelThe goal of this model is to estimate market share among fast food hamburger places for the three competitors,

McDonald’s, Burger King and Wendy’s. It is a simple network model containing feedback and self-loops among the

clusters. Because of its structure, there need to be cluster comparisons. So the weighted supermatrix differs from

the unweighted supermatrix.

There is an implicit control criterion with respect to which all judgments are made in this model: Market Share. For

example, when comparing the clusters indirect competitors to marketing mix, one asks which is more important for

gaining market share. Similar kinds of questions are asked when node comparisons are being made.

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US economy model for predicting turnaround time

The US Economy example is a simple network model with all its elements in a single network. It is a holarchy, meaning it resembles a hierarchy from the top down, but the cluster containing the bottom level of alternatives links back to the cluster containing the main criteria. The purpose of this model was to estimate how long, in months, it would be from the time the model was done until the US Economy turned around. The priorities for 3 months, 6 months, 18 months and 24 months represent likelihoods. For example, a value of .159 for 3 months means that there is a 15.9% likelihood that the economy would turn around in 3 months. To get the exact time in months use the formula: .159x3 +.129x6+.237x18+.475x24.

The 1991 model that is include in the sample models is named: Predicting_US_Economy2001.sdmod

It has attached files inside - click the paperclip icon - including a paper that was published and the Excel file of expected value computations to get the number of months until the turnaround from the priorities of the model.

This model was done in April, 2001, and predicted that the economy would turn around in October-November, 2001. This proved to be a correct prediction as determined by the NBER a couple of years later.

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Links to online powerpoint tutorials and their associated sample models.

Tutorial_1_AHP_Hierarchy

Sample_Model

Step-by-step instructions for how to create a simple AHP

decision model as a hierarchy with three levels to decide which

car to purchase. This is a relative model in which alternatives are

pairwise compared against each criterion.

Tutorial_2_AHP_Rating_Model

Sample_Model

Step-by-step instructions for how to build an AHP Ratings model

in which the alternatives are rated against standards developed

for each criterion rather than pairwise compared.

Tutorial_3_AHP_Sensitivity Shows how to use sensitivity analysis on completed model.

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THE SCALE, CONSISTENCY, AND THE EIGENVECTOR

(This writing on the math of the AHP is extracted from: The Fundamentals of Decision Making and Priority

Theory with the Analytic Hierarchy Process

byThomas L. Saaty, 478 pages, RWS Publications, Pittsburgh, PA, 2011 revision, ISBN 0-9620317-6-3, or 978-0-9620317-6-2, paperback, $30.)

1. Introduction

In chapters 1 and 2 we introduced some basic concepts of the AHP, including the comparison matrix and the principal eigenvector, which contains the derived scale. Here we will justify our choice of the scale when there are obviously many alternatives. We will also examine inconsistency and give a table of inconsistency measures for random judgments. There is an overall measure of inconsistency in hierarchies and another in networks. We then illustrate the impact of changes in judgment on inconsistency. We use the notion of consistency to show why people cannot deal with a large number of alternatives simultaneously, an observation already known to psychologists on empirical grounds. In case of incomplete judgments, we describe a method due to Harker for completing some or all of them. Finally, we give two methods to determine the most inconsistent judgments and how to improve them if one should desire to do so.

2. The Scale

In the paired comparison approach of the AHP one estimates ratios by using a fundamental scale of absolute numbers. In comparing two alternatives with respect to an attribute, one uses the smaller or lesser one as the unit for that attribute. To estimate the larger one as a multiple of that unit, assign to it an absolute number from the fundamental scale. This process is done for every pair. Thus, instead of assigning two numbers wi and wj and forming the ratio wi/wj we assign a single number drawn from the fundamental 1-9 scale to represent the ratio (wi/wj)/1. The absolute number from the scale is an approximation to the ratio wi/wj. The derived scale tells us what the wi and wj are. This is a central observation about the relative measurement approach of the AHP and the need for a fundamental scale. The names of Ernest Heinrich Weber (17951878) and Gustav Theodor Fechner (180187) stand out as one considers the subject of stimulus, response, and ratio scales. In 1846 Weber formulated his law regarding a stimulus of measurable magnitudes. He found, for example, that people while holding in their hand different weights, could distinguish between a weight of 20 g and a weight of 21 g, but not if the second weight is only 20.5 g. On the other hand, while they could not distinguish between 40 g and 41 g, they could between the former weight and 42 g, and so on at higher levels. We need to increase a stimulus s by a minimum amount ∆s to reach a point where our senses can first discriminate between sand s + ∆s. ∆s is called the just noticeable difference (jnd). The ratio r = ∆s/s does not depend on s. Weber's law states that change in sensation is noticed when the stimulus is increased by a constant percentage of the stimulus itself. This law holds in ranges where ∆s is small when compared with s, and hence in practice it fails to hold when s is either too small or too large. Aggregating or decomposing stimuli as needed into clusters or hierarchy levels is an effective way for extending the uses of this law. In 1860 Fechner considered a sequence of just noticeable increasing stimuli. He denotes the first one by s0. The next just noticeable stimulus [1] is given by

having used Weber's law. Similarly

In general

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

α )s -s (

= n 0n

log

loglog

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

0a b,+ s a = M ≠log

We assume that the stimuli arise in making pairwise comparisons of relatively comparable activities. We are interested in responses whose numerical values are in the form of ratios. Thus b = 0, from which we must have log s0 = 0 or s0= 1, which is possible by calibrating a unit stimulus. This is done by comparing one activity with itself. The next noticeable response is due to the stimulus αα = s = s 01

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

α 202 s = s

which yields a response of 2. In this manner we obtain the sequence 1, 2, 3,... . Our ability to make qualitative distinctions is well represented by five intensities: equal, moderate, strong, very strong, and extreme. We can make compromises between adjacent intensities when greater precision is needed. Thus we require nine values which, according to the previous discussion, should be consecutive. The following refinement of the above is also possible: equal, tad, weak, moderate, moderate plus, strong, strong plus, very strong, very very strong, and extreme. The resulting scale would then be validated in practice. We use the scale of absolute values shown in Table 3.1 to make the comparisons. We deal with widely varying measurements of alternatives by grouping those of similar magnitude in clusters and then uniformizing the measurement with a pivotal alternative that appears in two adjacent clusters. We do this because in using judgments, people are usually unable to accurately compare the very small with the very large. But they can make the transition gradually from clusters of smaller elements to clusters of larger ones. This approach is the valid way to extend the 1-9 scale as far out as one wants. In any case one does not need to go too far out on the scale to set priorities against one's personal goals. Thus the problem is to find the right numbers to represent comparisons of objects that are close on some property. The 1-9 scale is a simple scale that serves well. If the need arises for a judgment larger than 9, the larger element could be placed in another homogeneous set of comparisons and the 1-9 scale is also applied to that set. Clustering is used to combine different homogeneous groups. When we have a situation like ai

preferred to aj by 3 and aj preferred to ak by 5, implying that ai is preferred to ak by 15, we need not conclude that there is a need for a wider scale but that there is inconsistency in the judgments already given, because we know from the homogeneity requirement that there is no need for a number outside the scale. No matter what finite scale one chooses to represent the outer limit of perception, 1 to 9 or 1 to αn, inconsistency might seemingly require an even larger number. Instead of extending the scale one should look for better understanding of the inconsistency of judgments.

Table 3.1 The Fundamental Scale

Intensity of Importance

Definition Explanation

1 Equal Importance Two activities contribute equally to the objective

2 Weak

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3 Moderate importance

Experience and judgment slightly favor one activity over another

4 Moderate plus

5 Strong importance Experience and judgment strongly favor one activity over another

6 Strong plus

7 Very strong or demonstrated importance

An activity is favored very strongly over another; its dominance demonstrated in practice

8 Very, very strong

9 Extreme importance The evidence favoring one activity over another is of the highest possible order of affirmation

Reciprocals of above

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

A reasonable assumption

Rationals Ratios arising from the scale

If consistency were to be forced by obtaining n numerical values to span the matrix

There are many situations where elements are close or tied in measurement and the comparison must be made not to determine how many times one is larger than the other, but what fraction it is larger than the other. In other words there are comparisons to be made between 1 and 2, and what we want is to estimate verbally the values such as 1.1, 1.2, ..., 1.9. There is no problem in making the comparisons by directly estimating the numbers. Our proposal is to continue the verbal scale to make these distinctions so that 1.1 is a "tad", 1.3 indicates moderately more, 1.5 strongly more, 1.7 very strongly more and 1.9 extremely more. This type of refinement can be used in any of the intervals from 1 to 9 and for further refinements if one needs them, for example, between 1.1 and 1.2 and so on. Let us note that we are unable to distinguish between object sizes as they become very small or very large. Sometimes we can with the aid of an instrument like the microscope or the telescope which bring things to our range of abilities, but we must then relate the new magnitudes to those we know best in our daily experience. The idea of a logarithmic scale arises from saturation of the ability to make distinctions which happens at both ends of the scale. What we have done with the 1-9 scale, which we are compelled to use, as we can use no other scale with clear understanding of magnitudes, is to piecewise linearize the logarithmic idea in a limited operational range of magnitudes. Naturally if one uses actual measurements to form the ratios, one gets them back by solving for the derived scale because in the consistent case one gets back exactly what one puts in. In fact Expert Choice, the implementation software package of AHP, uses fractional values. They allow one to put very close approximations to whatever number one may think of between 1 and 9. But what arbitrary values would one assign to feelings. It is better to have a well tested protocol to go from comparisons to words to numbers validated to work in known situations, than to guess at widely disparate numbers assigned to firm judgments associated with perception. There are undoubtedly a few situations with known scales whose numbers can be used by an experienced person to form ratios and one need not use the scale 1-9. But AHP was developed to set

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priorities involving subjective understanding on all sorts of alternatives and even on ranges of measurements of alternatives. In that case the decision maker must exercise care in assessing distinctions in his judgments and feelings and must put them in smaller homogeneous ranges for which he is well equipped to make unambiguous distinctions.

Figure 3-1 Pairwise compare five areas for size

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

instance is the power scale, . ..., , , n10 ααα There are several problems in using such a scale. Clearly one can choose the numbers 1, 2, 22, 23, or 1, 3, 32, or similar subsets that are already in the 1-9 scale. A rule is needed to identify the value of α, the base in the geometric scale, to be associated with a verbal expression. Note that if we find one counterexample that produces a poor result with such a scale, for whatever α one may choose, then we would no longer be tempted to try a power scale. It is difficult to see how a power scale would be as natural a representation of the semantic scale when comparing homogeneous alternatives. If we extend the 1-9 scale in an extreme case to 100, the scale 1, α, ... , αn with n = 100 can give astronomically large values even for small values of α near one. We not only have difficulty estimating such values, but lower down on the scale we could have a problem distinguishing between some of the small values.

Considern = 1 2 3 4 5 6 7 8 9 10 ... 100

then2n/2 = 21/2 21 23/2 24/2 25/2 26/2 27/2 28/2 29/2 210/2 ... 2100/2

or 2(n-1)/2= 1 21/2 21 23/2 24/2 25/2 26/2 27/2 28/2 29/2 ... 299/2

The last value is beyond our ability to compare it semantically, say with 210/2. If instead of α = 2 we use α= 1.009 for example, it is difficult to distinguish between the scale values for very small n. When the use of one of the two scales gives rise to a consistent matrix, the use of the other would make it inconsistent. There are situations where both matrices are inconsistent and there is reversal in the ranks of the first and second alternatives as represented by the two eigenvectors as in the following example

Scale Matrix Eigenvector

1 2 3 1 .3477 1-9 1/2 1 6 1 .2939 1/3 1/6 1 1/3 .0808 1 1 3 1 .2775

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C.R. = .062

Scale Matrix Eigenvector 1 21 / 2 21 1 .2966 .707 1 22 . 5 1 .3334 2( n - 1 ) / 2 1/2 .176 1 1/2 .1064 1 1 2 1 .2775 C.R. = .065

3. The Eigenvector

From linear algebra we know that for a given matrix A and a given vector b, the equation Ax = b has a solution if the inverse A-1 exists which is the case if and only if A 0. The fact that a homogeneous system of linear equations, Ax = 0, has a non-zero solution if A = 0 plays an important role in the judgment matrix of the AHP.

The matrix of paired comparisons in the AHP leads to the condition w = Aw λmax or 0, = I)w-(A λmax a homogeneous system in the matrix I.-A λmax A nonzero solution implies that the

determinant |I-A| λmax is equal to zero. But this determinant is an nth degree polynomial in ,λmax where n is the order of A. This polynomial is equal to zero when λmax is a root of the equation obtained by setting the determinant equal to zero and known as the characteristic equation of A. Such a root is known as a characteristic root or eigenvalue of the matrix A. Thus when λmax is an eigenvalue of A, the solution vector w is not identically equal to zero. Consider the homogeneous system Aw = λw. The characteristic polynomial of an n by n matrix A has n zeros, λ1, λ2, ..., λn. In the following we use the vector e = (1, 1, ..., 1)T. All other vectors are column vectors.

Theorem 3.1 If A > 0, w1 is its principal eigenvector corresponding to the maximum eigenvalue λ1, λi λj for all i and j, and wi is the right eigenvector corresponding to λi then

cw =

eAe

eA 1kT

k

klim

∞→

where c is some constant.

Proof Because w1, ..., wn are linearly independent, we have: e = a1w1 + ... + anwn where ai, i = 1,...,n are constants. On multiplying both sides on the left by Ak we have:

w a + + w a + wa = wa + + wa = eA n

1

n

k

n21

2

k

211k1n

knn1

k11

k

λλ

λλλλλ KK

and again

multiplying on the left by ekT we have:

Since w1 > 0, b1 0, the theorem follows on putting . The proof of this theorem can be generalized to a nonnegative matrix, some power of which is positive. Because of its central relevance we need the following:

Definition - A matrix is irreducible if it cannot be decomposed in the form where A and C are square matrices and 0 is the zero matrix.

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Definition - A nonnegative, irreducible matrix A is primitive if and only if there is an integer m 1 such that Am > 0. Otherwise it is called imprimitive.

The graph of a primitive matrix has a path of length m between any two vertices. From the work of Frobenius (1912), Perron (1907), and Wielandt (1950), we know that a nonnegative, irreducible matrix A is primitive, if and only if, A has a unique characteristic root of maximum modulus, and this root has multiplicity 1.

Theorem 3.2 For a primitive matrix A

Aee A cw, =

A

eA Tkk

k

infinityk

≡→

____lim

where c is a constant and w is the eigenvector corresponding to λmax λ1.

The actual computation of the principal eigenvector in Expert Choice is based on Theorem 3.1. It says that the normalized row sums of the limiting power of a primitive matrix (and hence also of a positive matrix) gives the desired eigenvector. Thus a short computational way to obtain this vector is to raise the matrix to powers. Fast convergence is obtained by successively squaring the matrix. The row sums are calculated and normalized. The computation is stopped when the difference between these sums in two consecutive calculations of the power is smaller than a prescribed value. There is literally an infinite number of ways to estimate the ratio wi/wj from the matrix (aij). But we have already shown that our formulation with particular emphasis on consistency leads to an eigenvalue problem. What is an easy way to get a good approximation to the priorities? Multiply the elements in each row together and take the nth root where n is the number of elements. Then normalize the column of numbers thus obtained by dividing each entry by the sum of all entries. Alternatively normalize the elements in each column of the judgment matrix and then average over each row. We would like to caution the reader that for real applications one should only use the eigenvector derivation procedure because it can be shown that the approximations described above can lead to rank reversal in spite of its closeness to the eigenvector.

4. Consistency

Classically transitivity in ordered sets means that j, i ≥ kj ≥ implies k i ≥ (> preferred).

Because we require that 1 aij ≥ for i,j ≥ consistency implies transitivity for 1,a=>j i ij ≥≥

1 a=> kj jk ≥≥ and because of consistency we have aijajk = aik 1 and hence i k. Thus transitivity is a necessary condition for consistency. It is obviously not sufficient as, for example, when aij = 2, ajk = 3, and aik = 4 shows. The AHP does not require that judgments be consistent or even transitive. When A is consistent the entire set of entries can be constructed from a set of n judgments that form a chain across the rows and columns. Such a chain of interconnecting entries is given by:

One might ask, if the judgments are totally random in nature, what kind of consistency would the AHP interpret them to have? The consistency of a matrix of such random judgments should be much worse than the consistency of a matrix of informed judgments. The measure can be used to compare and evaluate the goodness of the consistency of informed judgments. The user of the AHP can often be perfectly consistent if he/she wishes to preserve the relation aijajk = aik by using redundant judgments, from relations between previously given judgments. This method is something we do not necessarily recommend. Second, in theory, by deriving the scale of relative measurement from the fundamental scale through the principal eigenvector, one is able to capture second, third and higher order effects whose entries contain magnitudes for relations between the alternatives that can differ considerably from the entries of the original matrix. Reciprocal, nonnegative matrices may have complex eigenvalues, but their sum is a real number. We note that since the maximum eigenvalue lies between the largest and the smallest row sums, a

105

matrix whose columns are identical has an eigenvalue which is equal to the sum of any of its columns. A small perturbation of a consistent matrix leaves its maximum eigenvalue λmax close to its value n. The remaining eigenvalues are perturbed closely to their zero values. The choice of perturbation most appropriate to describe the effect of inconsistency on the eigenvector depends on what is thought to be the psychological process involved in pairwise comparisons of a set of data. We assume that all perturbations of interest can be reduced to the general form aij = (wi/wj)εij. For example:

εαα ij

j

iij

i

j

j

iij

j

i

w

w w

w + 1

w

w = + w

w

≡

Consistency occurs when εij = 1. Let us now develop a few elementary but essential results about consistent matrices. Writing out the system Aw = λmaxw in detail, we have for the ith equation:

w

wa =

i

j

ij

n

j=1∑λmax

We define:

λµ i

n

2=i1-n

1 - = ∑

and note that

n =

1=i

n

iλ∑

because n is the sum of the diagonal elements known as the trace of A. This

fact follows by expanding )-)...(-( = |A-I| n1 λλλλλ and equating coefficients. Now n = i

n

=1i

λ∑ implies

that λλλµ 1 ;

1-n

n- = ≡max

max

and since

w

wa = 1-

i

j

ij

ij∑

≠λmax

we have

∑

≤≤ w

wa+

w

wa = n- n

j

iji

i

jij

nj<i1

λmax

and therefore

Substituting, aij = (wi/wj)εij,εij >0 we arrive at the equation

We observe that as εij 1, i.e., as consistency is approached, µ0. Also, µ is convex in the εij. This follows from the observation that εij +1/εij is convex (and has its minimum at εij = 1), and from the fact that the sum of convex functions is convex. Thus, µ is small or large depending on εij being near to or far from unity, respectively (i.e., near to or far from consistency). Finally, if we write εij = 1 + δij, with δij > -1 we have

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∑

≤≤ δδδµ

ij

3ij2

ijnj<i1 +1

-1)-n(n

1 =

We want µ to be near zero, or λmax to be near to its lower bound n, and thus to approach consistency. It is interesting to note that (λmax - n)/(n - 1) is related to the statistical root mean square error. Indeed, let us assume that δij < 1 (and hence that δ3

ij/(1 + δij) is small compared with δ2ij). This is a

reasonable assumption for an unbiased judge, who is limited by the "natural" greatest lower bound -1 on δij (since aij must be greater than zero), and who would tend to estimate symmetrically about zero in the interval (-1,1). Now, µ 0 as δij 0. Multiplication by 2 gives the variance of the δij. Thus, 2µ is this variance.

How to Estimate λmax There is a simple way to obtain the exact value (or estimate) of λmax when the exact value (or estimate) of w is available in normalized form. Add the columns of A and take the scalar product of the resulting vector with the vector w. Thus from:

w = wa ijij

n

j=1

λmax∑

we obtain

λλ maxmax =w = wa = wa i

n

=1ijij

n

=1i

n

j=1jij

n

j=1

n

=1i∑∑∑∑∑

A Measure of Random Inconsistency The consistency index of a matrix of comparisons is given by C.I. =( λmax-n)/(n-1). Theconsistency ratio (C.R.) is obtained by forming the ratio of C.I. and the appropriate one of the following set of numbers, each of which is an average random consistency index computed by E. Forman [2] for n 7 for very large samples (and by a number of other people for n > 7 for smaller samples). They create randomly generated reciprocal matrices using the scale 1/9, 1/8, ..., 1, ...8, 9 and calculate the average of their eigenvalues. This average is used to form the Random Consistency Index R.I.

n 1 2 3 4 5 6 7 8 9 10

R.I. 0 0 .52 .89 1.11 1.25 1.35 1.40 1.45 1.49

n 11 12 13 14 15

R.I. 1.51 1.54 1.56 1.57 1.58

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

R.I. 0 0 .52 .89 1.11 1.25 1.35 1.40 1.45 1.49 1.51 1.54 1.56 1.57 1.58

DeSchutter's Conjecture John DeSchutter has conjectured the following relationship between the index R.I. and n, the size of the matrix:

where 1.98 is the average value of the ratio of each value computed so far from n = 3 to n = 15 divided by (n-2)/n for the corresponding value of n. Here (n-1) is the minimum number of judgments needed to measure consistency and n(n-1)/2 is the number elicited for redundancy. Alternatively, a plot of values of

R.I. against n shows that the resulting curve approaches 1.98 as an asymptote, i.e., One may suspect that this value is actually equal to 2.

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Why Tolerate 10% Inconsistency Inconsistency may be thought of as an adjustment needed to improve the consistency of the comparisons. But the adjustment should not be as large as the judgment itself, nor so small that using it is of no consequence. Thus inconsistency should be just one order of magnitude smaller. On a scale from zero to one, the overall inconsistency should be around 10%. The requirement of 10% cannot be made smaller such as 1% or .1% without trivializing the impact of inconsistency. But inconsistency itself is important because without it, new knowledge that changes preference cannot be admitted. Assuming that all knowledge should be consistent contradicts experience which requires continued adjustment in understanding. Thus the object of developing a wide ranging consistent framework depends on admitting some inconsistency. By noting the Random Inconsistency Indices for different values of n, we distribute 10% proportionally by multiplying it by each of the R.I. values. These percentages are suggested by the R.I. values above. In other words, if the R.I. is equal to 1 for all n, we would allow 10% in all cases. However, for n = 3 and 4, the values are 0.52 and 0.89 and for n 5 it is greater than one. By convention we require the consistency ratio to be 5% and 8% for n = 3 and 4 respectively and 10% for all values of n 5. A plausible approach to the measurement of inconsistency has also been given by Golden and Wang [3].

5. Why Compare About Seven Elements

There are two explanations which one can give to justify the use of not many more than seven elements in a comparison scheme.

Consistency Explanation In making pairwise comparisons, errors arising out of inconsistency in judgments affect the final answer. We distribute inconsistency proportionately among the alternatives because in computing the eigenvector, transitivities of all order are considered and the average is taken over all these transitivities. If the number of elements is small their relative priorities would be large. These priorities would be less affected by inconsistency adjustment. Thus for example if there are 10 homogeneous elements, each would have a relative priority of .10 and is unaffected by a 1% inconsistency distributed among the 10 so that its value would now be .10 .01. If the number of elements is large, the relative priority of each would be small and would be more affected by inconsistency. Still the number should be large enough to enable one to make redundant judgments to improve the validity of the outcome. For this reason seven elements is found to be a reasonable choice for high average priority and high validity.

Neural Explanation This explanation has to do with the brain limit on the identification of simultaneous events. The perception or simultaneity span is the ratio of the buffer-delay time to the attentional integration time. Some psychologists have found that the more intense the stimulus the greater the perception or simultaneity span. The reason is that with increased intensity the time of rapid integration is reduced. The most common duration time estimate for the short term memory (buffer-delay) is 750 milliseconds and that for item-integration time is 100 milliseconds. Their ratio is about seven. Both forgoing explanations have the following mathematical justification.

Stability of the Eigenvector Requires a Small Numbe r of Homogeneous Elements The question often arises, how sensitive the priorities give by the eigenvector components are to slight changes in the judgment values. Clearly, it is desirable that the priorities do not fluctuate widely with small changes in judgment. There are essentially three ways to test this sensitivity: (1) by finding a mathematical estimate of the fluctuation; (2) by deriving answers based on a large number of computer runs appropriately designed to test the sensitivity; (3) by a combination of the two, particularly when it is not possible to carry out the full demonstration analytically. We have already pointed out, in the case of consistency, that λmax is equal to the trace of the matrix which consists of unit entries. In this case one would expect the eigenvector corresponding to the perturbed matrix to undergo an overall change by an amount inversely proportional to the size of the matrix. In general, the eigenvalues of a matrix lie between its largest and smallest row sums. Changing the value of an entry in the matrix changes the correspondence row sum and has a tendency to change

108

λmax by an equal amount. However, since a change in the eigenvector should also be influenced by the size of the matrix, we expect that the larger the matrix, the smaller the change in each component. We begin the analytical treatment of the question by considering a matrix A with the characteristic equation

0 = a++a+ = I)-(A n1-n

1n

Kλλλdet

Following standard procedures, let B+A ε be the matrix obtained by introducing a small perturbation in A. The corresponding characteristic equation is

0 = )(a++)(a+ = I)-B+(A n1-n

1n ελελλε Kdet

where )(ak ε is a polynomial in ε of degree (n-k), such that a)(a kk →ε as 0.→ε Let λ1 be the maximum simple eigenvalue corresponding to the characteristic equation of A. It is known in matrix theory that for small ε , there exists an eigenvalue of B+A ε which can be expressed as the sum of a convergent power series, i.e.,

K+k+k+ = )( 22111 εελελ

Let w1 denote the eigenvector of A corresponding to λ1 and let )(w1 ε be the eigenvector of B+A ε corresponding to ).(1 ελ The elements of )(w1 ε are polynomials in )(ελ and ε , and, since the

power series for )(1 ελ is convergent for small ε , each element of )(w1 ε can be represented as a convergent power series in ε . We may write

K+z+z+w = )(w 22

111 εεε If the matrix A has linear elementary divisors, then there exist complete sets of right and left eigenvectors w1, w2, .., wn and v1, v2, ..., vn, respectively, such that

ji 0 = wv jTi ≠

Note that wj and vj are the jth eigenvectors (right and left), and not the jth components of the vectors. The vectors zi can be expressed in terms of the wj as

ws = z jij

n

j=1

i ∑

which, when substituted in the formula for )(wi ε , gives

wt+w = )(w i

jij

n

j=1

n

2=i11 εε ∑∑

where the tij are obtained by dividing the sij by the coefficient of w1.

The first order perturbations of the eigenvalues are given by the coefficient k1 of ).(1 ελ We now derive the expression for the first order perturbations of the corresponding eigenvectors. Normalizing the vectors wj and vj by using the euclidean metric we have

1 = |w| |v| jTj

We know that

If we substitute the expressions for and we obtained above and use Aw1 = λ1w1, we have

Multiplying across by and simplifying, we obtain

and

where, as noted above, k sub 1 is the first order perturbation of λ1 and

109

where [B] is the sum of the elements of B.

Thus for sufficiently small ε the sensitivity of λ1 depends primarily on wv .wv 1T11

T1 might be

arbitrarily small. The first order perturbation of w1 is given by

BA wherew)wv)-/(wA)(v( =

w)wv)-/(wBv(

wt = w

jjTjj11

Tj

n

2j=

jjTjj11

Tj

n

2j=

jj

n

2j=1

=

1

ελλ

λλ

ε

ε

≡∆∆

∆

∑

∑

∑

The eigenvector w1 will be very sensitive to perturbations in A if λ1 is close to any of the other

eigenvalues. When λ1 is well separated from the other eigenvalues and none of the wv iTi is small, the

eigenvector wi corresponding to the eigenvalue λ1 will be comparatively insensitive to perturbations in A. This is the case, for example, with skew-symmetric matrices (aji = -aij).

The wv iTi are interdependent in a way which precludes the possibility that just one

n1,2,...,=iwv1/ iTi is large. Thus if one of them is arbitrarily large, they are all arbitrarily large.

However, we want them to be small, i.e., near unity. To see this let

wd = v and vc = w jij

jjjij

ji ∑∑

where n. ..., 2, 1, = i 1, = |b| = |w| ii It is easy to verify by substitution that

wv/ww = c jTji

Tjij

and

wv/vv = d jTji

Tjij

Then

wv)/vv)(ww( =

vcwd = wv

jTji

Tji

Tj

j

jij

j

Tjij

ji

Ti

∑

∑∑

for i = j

1 = vv = w iTi

Ti

and

)wv)(vv)(ww(+)wv( = wv

-1j

Tji

Tji

Tj

ij

-1i

Tii

Ti ∑

≠

Since

we have

which must be true for all i = 1, 2, .., n. This proves that all the must be of the same order.

We now show that for consistent matrices cannot be arbitrarily large. We have in the case of consistency

110

Therefore

n> ]w1/[n/ =

]w1//)1,...,w)(w,...,1/w[(1/ = )wv(

1-1i

n

=1i

1-1i

n

=1i

T1n111n11

1-1

T1

∑

∑

since /n. w< w1/n/ 1i

n

=1i1i

n

=1i∑∑

Now )wv( -11

T1 is minimized when all w1i are equal since

1. = w1i

n

=1i∑

In practice, to keep )wv( -11

T1 near its minimum we must deal with relatively comparable activities

so that no single w1i is too small. To improve consistency the number n must not be too large. On the other hand, if we are to make full use of the available information and produce results which are valid in practice, n should also not be too small.

If, for example, we reject the values 0.1, wv 1T1 ≤ then we must have n 9.

Under the assumption that the number of activities being compared is small and that they are relatively comparable, i.e., their weights differ by a multiple of their number, we can show that none of the components of w1 is arbitrarily small and none of those of v1 is arbitrarily small, and hence the scalar product of the two normalized vectors cannot be arbitrarily small. With large inconsistency one cannot guarantee that none of the w1i is arbitrarily small. Thus, near-consistency is a sufficient condition for stability. Note also that we need to keep the number of elements relatively small, so that the values of all the w1i are of the same order.

Illustrative Matrices Here are examples of inconsistent matrices with their inconsistency indices: 1) a12 = 2, a13 = 9, a23 should be 9/2 but is 9 (twice what it should be)

.051

.367

.582

11/91/9

91.5

921

λmax = 3.054, C.I. = .027, C.R. = .046 (acceptable).

2) If we change just a12 from 2 to 3, we have λmax = 3.136, C.I. = .068, C.R. = 0.117.

3) Interchange an element (a12) with its reciprocal in

λmax = 4 C.R. = 0 λ max = 4.250 C.R. = .092

6. A Method for Incomplete Comparisons

When not all judgments are available, either because of the limitation or because the judge is unwilling or unsure to make a comparisons of two elements, Harker [4] provides the following suggestions:

Have the decision maker provide judgments, such that at least one judgment is answered in each column, yields a matrix with some unknown ratio elements.

111

Enter zero for any missing judgment in that matrix, and add the number of missing judgments in each row to the diagonal element in the row, producing a new matrix A.

Calculate the weight w:

cw =

eAe

eAk kT

k

∞→lim

Use the resulting wi/wj as a suggested value for the missing judgments to make it consistent with the judgments already provided.

If needed, the decision maker can be guided to make additional judgments, that have the greatest impact on the weight w. One computes the larger absolute value of the gradient of w with respect to the (i,j) calculated using the following formula:

x:right principal eigenvector = w, in AHP notation Ax = λmaxx y:left principal eigenvector ytA = λmaxy

[ ]j>i a)/xy(-)xy( =

,ji,| = D

2ijijji

ij

A

∂∂ λ

λmax

max

where y is normalized so that y'x = 1.

Then

∂∂

j>i|x

= Dij

Ax

is the matrix of gradients for the weights x and is given by:

o

z-xD

e

I-AA-1 ~~~~

maxmax λλ

where I = n x n identity matrix e = n dimensional row vector of ones z = (zk)=n dimensional column vector defined by:

otherwise 0

j = k if a/x-

i = k if x

= z 2iji

j

k

~ denotes the matrix or vector with its last row deleted.

The choice of the next (i,j) value is made according to:

where Q is the set of unanswered comparisons and denotes or the Chebyshev norm.

Harker's Example

If:

112

then λmax= 3.0092027 x= (0.5396145, 0.2969613, 0.1634241) y= (0.1634241, 0.2969613, 0.5396145)

After normalizing y so that yTx = 1, we have:

y= (0.6177249, 101224806, 2.0396826)

The derivatives of the principal eigenvector are:

000

0.0320137 00

0.0213425-0.03201370

= DAλmax

111

22.0092027-1/2

322.0092027-

=

e

I - A -1~~maxλ

0.16342410.2157468 0.1350274

0.29696130.2695597-0.0807193

0.53961460.0538129 0.2157468-

=

and

0

0.0052318

0.0060190

=

0

xDA~

maxλ

For each question, z and x/ij are:

(1,2)

∂∂

0.0090518-

0.0612519-

0.0703116

= x

0

0.1349036-

0.2969613

=z 12

(1,3)

(2,3)

Therefore, (1,2) should be asked next. The decision maker may decide to stop the questioning, or continue according to:

a.)If the maximum absolute difference in the attribute weights from one question to the next (wk

113

and wk+1) is less than or equal to a given constant α% and |w-w|

n i 1

max arg = k

i1+k

i≤≤l

then the

procedure would stop at the (k+1) comparison if α≤

w

|w-w|k

k1+k

l

ll

b.)The next question derived from the gradient procedure would only be asked if it appears that the ordinal ranking could be reversed.

Due to the computational complexity of this task, a simplification can be made by using a sample of random spanning trees to calculate: aij = the current value of the (i,j)th question which has just been chosen.

aij =the largest path intensity in the set of all elementary paths connecting i and j aij = the smallest path intensity

Let uij = max [1, aij - aij] Lij = min [1, aij - aij] and define P(w) to be a function which returns the ordinal ranking inherent in the cardinal ranking

w; that is P: Rn Zn where Zn is the n dimensional space of natural numbers. For example, if w = (.15, .3, .2, .35)T then P(w) = (4, 2, 3, 1)T. Three rankings can be defined with this function:

P1 = P(w) P2 = P(w + w/ij (aij + uij)) P3 = P(w + w/ij (aij - Lij))

P1 = current ordinal ranking P2, P3 =approximation to the ordinal ranking if the (i,j)th comparison achieved its max and

min deviation, respectively.

If P1 = P2 = P3, the next comparison is unlikely to alter the ordinal ranking.

Nonlinear responses In the standard theory it is assumed that aij is an approximation to the ratio wi/wj. However, one could have situations in which aij is an approximation to some function of this ratio f(wi/wj)α with α > 0.

The eigenvector problem can be written as w = wA αα λmax where ).w ..., ,w ,w( = w n21αααα

Defining v to be equal to wα leads to Av = λmaxv, a standard eigenvalue problem in the AHP.

7. Improving the Consistency of Judgments

There are two ways to identify the most inconsistent judgment and improve its value. One is to compare each aij with the corresponding ratio wi/wj from the eigenvector. That aij for which aijwj/wi is largest (smallest) is the most inconsistent. The proposal is to change that aij to wi/wj. This adjustment would improve the overall consistency. In fact any change in aij in the direction of wi/wj improves inconsistency. A second method that uses the gradient is due to Harker [5]. Although using the gradient is a more sophisticated way of computing inconsistency, it is less efficient because generally the judgment

matrices are of small order. As in the previous section, it involves computing the matrix . The most inconsistent judgment has the largest absolute value and is the one to improve by changing it in a direction closer to the ratio of the corresponding components of the eigenvector.

References

1.Batschelet, E., 1973, "Mathematical Recreations and Essays", MacMillan, New York.

2.Forman, E.H., 1990, "Random Indices For Incomplete Pairwise Comparison Matrices", European

114

Journal of Operational Research 48, 153-155.

3.Golden, B.L., E.A. Wasil, and P.T. Harker, editors, The Analytic Hierarchy Process, Springer-Verlag, New York, 1989.

4.Harker, P.T., 1987, "Incomplete Pairwise Comparisons in the Analytic Hierarchy Process," Mathematical Modelling 9/11, 837848.

5.Harker, P.T., 1987, "Alternatives Models of Questioning in the Analytic Hierarchy Process", Mathematical Modelling 9, 353-360.

6.Saaty, Thomas L., The Analytic Hierarchy Process, McGraw Hill, 1980. Reprinted by the author, 1988 at Pittsburgh.

115

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Index

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A B C D E F G H I J K L M N O P

Q R S T U V W

Index Entry Section

$$ 3. Formulas

Aadditive(negative) formula Small Network Template

additive(probabilistic) formula Small Network Template

additive(reciprocal) formula Small Network Template

advanced save 2.9 Saving Models

AHP fundamental scale 4.0.1 Making Comparisons

assessments in ratings Ratings Assessments

assessments, performing 4. Making Judgments/Assessments

attaching files 8. Other Important Features

automatic backup 2.9 Saving Models

BBOCR merit nodes 2.6 Basic Control Networks

BOCR nodes 2.5 Subnetworks

BOCR, formula for combining 5. Supermatrix Computations

bridge sample model 9.4 Bridge model to choose between

two bridges

building models 2. Building Models

Ccalculating totals in ratings 2.4 Rating networks

calculating totals in ratings Ratings Assessments

calculations command in ratings 2.4 Rating networks

calculations command in ratings Ratings Assessments

calculations, supermatrix 5. Supermatrix Computations

calculus type 5. Supermatrix Computations

calculus type, limit matrix power

option

6. Synthesizing for Results and

Sensitivity

car choice hierarchy 9.5 Car Choice (Hierarchy)

carmaker 9.6 Carmaker

categories in Ratings 2.4 Rating networks

Category Editor 2.4 Rating networks

category templates 2.4 Rating networks

cluster color, changing Selecting Fonts/Colors/Icons for

Clusters

cluster comparisons 2.1.1 Creating Clusters

cluster description, entering Selecting Fonts/Colors/Icons for

116

Index

$ $

2 2-level complex model, 31

A a hierarchy in SuperDecisions, 92Adams, William J., 3additive(negative) formulaadditive(probabilistic) formulaadditive(reciprocal) formulaadvanced saveAHP fundamental scaleassessments in ratingsassessments, performingattaching filesautomatic backup

B best car, network model for, 4BOCR complex model (2-level or 3-level), 31BOCR merit nodesBOCR nodesBOCR, formula for combining subnet prioritiesbridge sample modelbuilding models

C calculating totals in ratingscalculations command in ratingscalculations, supermatrixcalculus typecalculus type, limit matrix power optioncar choice hierarchycar decision, network model for, carmakercategories in RatingsCategory Editorcategory templateschanging from AHP thinking to ANP thinking, 4Children nodes, 4Cluster, 4cluster color, changingcluster comparisonscluster description, enteringcluster font, selectingcluster icon font, selectingcluster icon, changingcluster menu, 38cluster menuscluster name, entering

117

cluster, comparisonscluster, connectionsclusters, changing all title fonts,, 38clusters, creatingclusters, deletingclusters, editing, 38clusters, general, 38clusters, organizing nodes inclusters, removing, 38color of cluster, changing, 38color of node, changingcolors, for nodescommand to create nodecommand to edit cluster, 38command to edit nodecommand to remove nodecommand, to create cluster, 38comparing clusterscomparisons, completedcomparisons, cycling throughcomparisons, enteringcomparisons, show completedcomplete hierarchy, 4complex models, 31compressed mode, savingcomputations, full reportcomputations, generalcomputations, prioritiescomputations, supermatrixcomputing prioritiesconnecting nodesconnections, clusterconnections, generalconnections, nodeconnections, showingconsistency improvementconsistency of judgmentscontrol criteria nodescontrol networkcontrol nodescreate cluster, shortcut, 38create cluster, using mouse, 38create node in cluster, 43create node, mouse command, 43create node, mouse shortcut, 43creating modelscreating rating networkscreating subnetworksCreative Decisions Foundation, 2criteria, 4cycling thru comparisons

D data, entering directlydata, for comparisons

118

Decision Lens Software, Decision Model, decision networksdesign menu, create nodedesign menu, remove clusterDesign>Cluster menu, 38dynamic sensitivity, 79

E edit cluster, popup menuedit node, menu commandedit node, popup menuentering comparisonsentering judgmentsexporting sensitivity dataextra params

F feedback, file print commandfiles, attachingfonts, for nodesformula, additive(negative)formula, additive(probabilistic)formula, additive(reciprocal)formula, for combining BOCRformula, inverted values informula, multiplicativeformula, multiplicative(power weighted)formula, templateformulaic control networksformulaic networkformulaic networksformulasformulas, conventions infull modelsfull report, printingfull templatefundamental scale, 30Fundamental Scale of the AHP, 4fundamental scale of the AHP and ANP, 4, 30

G graphic comparison modegraphical view, supermatrix

H hamburger modelhamburger network model, market share, 4hardtech, model with sinkshelp command, loading sample models fromhierarchy, 4hierarchy, as displayed in software, 22hierarchy, constructing, 4Hierarchy, traditional way to display, 22holarchy, US economy model

119

I Iconify/Expand clusterIdealIdealAltIdealsimporting files into Excelimproving consistencyinconsistency, 92influence, 4inner dependenceinverse matrix, 4inverted values, in formulainverting alternative results

J judgments, making

K key to remove nodekey used to create clusterkey used to create nodekey used to edit clusterkey used to edit nodeskey used to remove cluster

L limit matrixlimit matrix optionslimit matrix power option, default selectionlimit power typeslinking nodeslinks, 4

M main windowmaking assessments in ratingsmaking connectionsmaking judgments/assessmentsMathematics of pairwise comparisons, 4matrix comparison modemenu, design commandmenu, remove clustermenus for cluster operationsmerit nodesmodel, definition of, 27models, buildingmodels, creatingmodels, opening simultaneouslymodels, savingmouse shortcut, create nodemouse, create clustermouse, use to create nodemultiplicative formulamultiplicative(power weighted) formulamultiply selecting nodes

120

N National Missile Defense modelNetnetwork model for best car, 4networks with formulasNetworks with Ratingsnetworks, simplenew hierarchy, limit matrix power optionnew model, templates forNode, 4node color, changing, 43node description, entering, 43node icon font, selecting, 43node icon, changing, 43node name, entering, 43node sensitivity, 79node title font, selecting, 43node, command to create, 43node, comparisonsnode, connectionsnode, create from design menu, 43node, values dialog box, 43nodes, creating, 43Nodes, creating in cluster, 43nodes, deleting, 43nodes, editing, 43nodes, general, 43Nodes, organizing in a cluster, 43nodes, removing, 43nodes, selecting multiplenodes, setting fonts and colors, 48NormalNormalAltNormalsnumber of judgments in pairwise comparison matrix, 4

O open file templateopening models simultaneouslyOrganize nodes in a clusterouter dependence

P pairwise comparison matrix, 4pairwise comparison matrix in SuperDecisions, 92pairwise comparison modes, 4parent node, 4Perron's theorem, 4pop-up descriptions of nodes and clusters, 22popup menu to edit nodepopup menu, edit clusterpopup menu, remove clusterpopup menu, remove nodeprincipal eigenvalue, 4

121

principal eigenvector, 4printingpriorities from pairwise comparisons, 4priorities of nodes, computationspriorities, computingpriorities, of all nodes in network,

Q questionnaire, in comparisonsquick access, sample models

R RatRating module, creatingRating module, removingrating networksrating networks, creatingratings assessmentsratings calculations commandRatings Category EditorRatings networksRawremove clusterremove cluster, design menuremove cluster, popup menuremove node, menu commandremove node, popup menuresults, synthesizing, 78rules for constructing formulas

S Saaty, Thomas L., 3sample models, accessing from Helpsample models, Help menusample models, quick accesssample models, where they aresamplessave interval, settingsave, advancedsaving modelssaving subnetssaving, compressed modescale, fundamentalSelected clusterselecting multiple nodessensitivity, 4sensitivity analysis, 79sensitivity barchart, 22shortcut key, edit clustershortcut key, edit nodeshortcut key, remove nodeshow completed comparisonsshow connections, 22showing children of a parent node, 4, 22simple network, 57

122

simple network templatesimple networkssmall templateSmartAltSoftware Systems Supported, 3stochastic matrixstochastic supermatrix columnssubcriteria, 4subnet, importing structuresubnet, importing templatesubnets, savingsubnetworkssubnetworks, accessingsubnetworks, creatingSuper Decisions Main windowsupermatrixsupermatrix calculationssupermatrix computationssupermatrix, graphical viewsupermatrix, text viewsynthesissynthesize for a control networksynthesize for a sub-network

T template formula, defaulttemplate, for simple networktemplate, fulltemplate, Open filetemplate, smalltemplate, use to make subnettemplatestemplates for categoriestext view, supermatrixthe supermatrices, 4TotalTotalAlttotals, calculating in ratingsTutorials, 99

U unweighted supermatrixUS economy, sample modelusing templates for categories

V Vargas, Luisverbal comparison mode

W weighted supermatrix

123

helpsmith - superdecisions help

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anp network model

ahp hierarchical model

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