Course Slides 1 Summer School 2007

8/14/2019 Course Slides 1 Summer School 2007

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SUMMER SCHOOL19-21 September 2007Gaeta (LT)

The Analytic Hierarchy Process (AHP)for Decision Making

PROF. THOMAS L. SAATY(University Of Pittsburgh Pennsylvania USA)


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The Analytic HierarchyProcess (AHP)

for

Decision Making

Decision Making involves

setting priorities and the AHP

is the methodology for doing

that.


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Decision Making1) Precedes Creative Thinking to

determine which area to brainstorm first

2) It also follows Creative Thinking todetermine which avenues to specialize

in first for implementation needs

Problem SolvingCreative Thinking and Decision

Making are the corner stones ofProblem Solving

Creative ThinkingPrecedes Decision Making to

brainstorm, connect and organizethe criteria needed tomake a decision


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Creative ThinkingWhat to brainstorm?

How to structure?

Opportunity FindingHelp to add value

Problem SolvingRemove obstacles

Decision MakingPresent and likely future?

Problem to solve?

Opportunity to seize?

Best action?

Resource allocation?

Solution Strategy

Implementation

Design, Planning and Action Physical (engineering) and Behavioral

(management)


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A Simple Example of Ranking

You have five people ranked on three

criteria: age (in years), wealth (in dollars)and health (in relative priorities), how do

you determine their overall ranks. Assume,in addition for this exercise, that the

younger a person is the more favorable it is

for his/her rank with respect to age.


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Part 1: Decision Making Challenges

Current State of Corporate Decision Making

Inefficient Meetings

Consequences

Benefits From a Structured Process for Decision Making Benefits of SuperDecisions (free!) software


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At least 50% of all decisions end in failure.

33% of all decisions made are never implemented.

50% of decisions implemented are discontinued after 2

years.

66% of decisions are based on failure prone methods. Decisions using high participation succeed 80% of the

time but occurs only 20% of the time.

Practically every decision failure is preventable

Source: Why Decisions Fail - Author Paul Nut - Publisher; Berret & Koehler 200220 year study of over 400 business decisions from Public, Private and Not-For Profit organizations in the United States, Canada & Europe.


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Only 37% of decision makers said they have a

clear understanding of what their organization

is trying to achieve.

Only 1 in 5 was enthusiastic about their teams

and organizations goals. Only 1 in 5 said they have a clear line of site

between their tasks and their organizations

goals.

SOURCE: Harris Poll of 23,000 respondents



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Inefficient Meetings

11 Million meetings in the U.S. per day

Most professionals attend a total of 61.8 meetings permonth

Research indicates that over 50 percent of thismeeting time is wasted

Professionals lose 31 hours per month inunproductive meetings, or approximately four workdays

A network MCI Conferencing White Paper.Meetings in America: A study of trends, costs and attitudes toward businesstravel, teleconferencing, and their impact on productivity (Greenwich, CT: INFOCOMM, 1998), 3.

Robert B. Nelson and Peter Economy, Better Business Meetings (Burr Ridge, IL: Irwin Inc, 1995), 5.


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Benefits from Structured Decision Making Increase the success of decisions made by at least 50%

Increase benefits by blending diverse perspectives

Achieve and sustain more excellent operational performance

Maximize return on investments in business change

Increase business leadership and stakeholder confidence and

commitment Achieve more consistent business performance

Sustain increasing business prosperity

Source: Why Decisions Fail - Author Paul Nut - Publisher; Berret & Koehler 200220 year study of over 400 business decisions from Public, Private and Not-For Profitorganizations in the United States, Canada & Europe.


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Benefits of Systematic Decision Making

Rapidly build consensus on goals, objectives and

priorities Align work activities to what is important for

organizational success

Objectives based budgeting versus resource basedbudgeting (save the salami and peanut butter for

lunch)

Get past corporate stovepipes (tear down the walls)


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Benefits of Systematic Decision Making (contd)

Improve speed and effectiveness of decisionmaking

Synthesize existing corporate data with

management priorities Achieve buy-in on major corporate decisions

Improve accountability and outcomes over time

Achieve cost savings through efficient allocation

of resources


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Real Life Problems Exhibit:

Strong Pressures

and Weakened Resources

Complex Issues - Sometimes

There are No Right Answers

Vested Interests

Conflicting Values


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Most Decision Problems are Multicriteria

Maximize profits

Satisfy customer demands

Maximize employee satisfaction

Satisfy shareholders

Minimize costs of production Satisfy government regulations

Minimize taxes

Maximize bonuses


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Knowledge is Not in the Numbers

Isabel Garuti is an environmental researcher whose father-in-law is a master chef

in Santiago, Chile. He owns a well known Italian restaurant called Valerio. He

is recognized as the best cook in Santiago. Isabel had eaten a favorite dish

risotto ai funghi, rice with mushrooms, many times and loved it so much that shewanted to learn to cook it herself for her husband, Valerios son, Claudio. So

she armed herself with a pencil and paper, went to the restaurant and begged

Valerio to spell out the details of the recipe in an easy way for her. He said it

was very easy. When he revealed how much was needed for each ingredient, he

said you use a little of this and a handful of that. When it is O.K. it is O.K. and itsmells good. No matter how hard she tried to translate his comments to

numbers, neither she nor he could do it. She could not replicate his dish.

Valerio knew what he knew well. It was registered in his mind, this could not

be written down and communicated to someone else. An unintelligentobserver would claim that he did not know how to cook, because if he did, he

should be able to communicate it to others. But he could and is one of the best.


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Knowing Less, Understanding More

You dont need to know everything to get tothe answer.

Expert after expert missed the revolutionarysignificance of what Darwin had collected.Darwin, who knew less, somehow understood

more.


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An elderly couple looking for a town to which theymight retire found Summerland, in Santa Barbara

County, California, where a sign post read:

SummerlandPopulation 3001Feet Above Sea Level 208

Year Established 1870

Total 5079

Lets settle here where there is a sense of humor, saidthe wife; and they did.

Arent Numbers Numbers?

We have the habit to crunch numberswhatever they are


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Do Numbers Have an Objective Meaning?

Butter: 1, 2,, 10 lbs.; 1,2,, 100 tons

Sheep: 2 sheep (1 big, 1 little)

Temperature: 30 degrees Fahrenheit to New Yorker, Kenyan, Eskimo

Since we deal with Non-Unique Scales such as [lbs., kgs], [yds,

meters], [Fahr., Celsius] and such scales cannot be combined, we needthe idea ofPRIORITY.

PRIORITY becomes an abstract unit valid across all scales.

A priority scale based on preference is the AHP way to standardize

non-unique scales in order to combine multiple criteria.


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Stability, Objectivity, and Measurement

We deal with many things whose stimuli are qualitative and changeaccording to our state of mind and our judgment about them is variable

and unstable and subjective. For some things we have created scales of

measurement that enable us to get the same readings at any time no

matter who we are. But the readings have to be interpreted as to their

significance, importance or priority. We call such things tangibles.

How can we deal with intangibles as we experience them directly likethe color red from sense data, beauty as we see a work of art and

respond to it using experience, or goodness and love that are abstract

modes of behavior? We can compare one shade of red with another,

one object of beauty with another, and one act of goodness or love withanother according to our preference or sense of importance or likelihood

of occurrence. The comparisons yield relative magnitudes of dominance

that are stable at the time we do them and we can also compare them

with what other people get in their evaluation.


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Nonmonotonic Relative Nature of Absolute Scales

Good for

preserving food

Bad for

preserving food

Good for

preserving food

Bad for

comfort

Good for

comfort

Bad for

comfort

100

0

Temperature


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OBJECTIVITY!?

Bias in upbring: objectivity is agreed upon subjectivity.

We interpret and shape the world in our own image. Wepass it along as fact. In the end it is all obsoleted by the

next generation.

Logic breaks down: Russell-Whitehead Principia; Gdels

Undecidability Proof.

Intuition breaks down: circle around earth; milk and coffee.

How do we manage?


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Organizing the Ways to Access,Connect, and Prioritize Information

We cannot use randomly gathered information very well without

thinking in organized ways. The best way to use the plethora ofsporadic, randomly acquired information in our memories is to createwell-organized complex structures and represent the basic conceptswith criteria and alternatives with which such information deals andthen do prioritization to determine what is important and what is not.

We quickly learn that we know much to use in predicting whathappens in the real world than we can do without such systematic waysto represent understanding. We have no better ways to do it. Themodern computer is a gift that we can use to deal with complexity toenhance the power of our minds. Without them it is extremely difficultto mange complexity in the integrated ways that now we can. It is a

new revolution for better decision making.


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Making a Decision

Widget B is cheaper than Widget A

Widget A is better than Widget B

Which Widget would you choose?


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Basic Decision Problem

Criteria: Low Cost > Operating Cost > Style

Car: A B B

V V V

Alternatives: B A A

Suppose the criteria are preferred in the order shown and the

cars are preferred as shown for each criterion. Which carshould be chosen? It is desirable to know the strengths of

preferences for tradeoffs.


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To understand the world we assume that:

We can describe it

We can define relations between

its parts and

We can apply judgment to relate the

parts according to

a goal or purpose that we

have in mind.


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GOAL

CRITERIA

ALTERNATIVES

Hierarchic

Thinking


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Relative MeasurementThe Process of Prioritization

In relative measurement a preference, judgmentis expressed on each pair of elements with respect

to a common property they share.

In practice this means that a pair of elements

in a level of the hierarchy are compared with

respect to parent elements to which they relatein the level above.


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If, for example, we are comparing two apples

according to size we ask:

Which apple is bigger?

How much bigger is the larger than the smaller apple?

Use the smaller as the unit and estimate how

many more times bigger is the larger one.

The apples must be relatively close (homogeneous)

if we hope to make an accurate estimate.

Relative Measurement (cont.)


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The Smaller apple then has the reciprocal value when

compared with the larger one. There is no way to escape this sort

of reciprocal comparison when developing judgments

If the elements being compared are not all homogeneous, they are

placed into homogeneous groups of gradually increasing relativesizes (homogeneous clusters of homogeneous elements).

Judgments are made on the elements in one group of small

elements, and a pivot element is borrowed and placed in the next

larger group and its elements are compared. This use of pivotelements enables one to successively merge the measurements of

all the elements. Thus homogeneity serves to enhance the accuracy

of measurement.

Relative Measurement (cont.)

P i i C i


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Pairwise ComparisonsSize

Apple A Apple B Apple C

SizeComparison


Apple A S1/S1 S1/S2 S1/S3

Apple B S2/S1 S2/S2 S2/S3

Apple C S3/S1 S3/S2 S3/S3

We Assess The Relative Sizes of theApples By Forming Ratios

P i i C i


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Pairwise ComparisonsSize


SizeComparison


Apple A 1 2 6 6/10 A

Apple B 1/2 1 3 3/10 B

Apple C 1/6 1/3 1 1/10 C

When the judgments are consistent, as they are here, any

normalized column gives the priorities.

ResultingPriorityEigenvector

Relative Sizeof Apple

C i


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Consistency itself is a necessary condition for a betterunderstanding of relations in the world but it is not

sufficient. For example we could judge all three of

the apples to be the same size and we would be perfectly

consistent, but very wrong.

We also need to improve our validity by using redundant

information.

It is fortunate that the mind is not programmed to be always

consistent. Otherwise, it could not integrate new information

by changing old relations.

Consistency (cont.)


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Consistency

In this example Apple B is 3 times larger than Apple C.

We can obtain this value directly from the comparisons

of Apple A with Apples B & C as 6/2 = 3. But if wewere to use judgment we may have guessed it as 4. In

that case we would have been inconsistent.

Now guessing it as 4 is not as bad as guessing it as 5 or

more. The farther we are from the true value the more

inconsistent we are. The AHP provides a theory forchecking the inconsistency throughout the matrix and

allowing a certain level of overall inconsistency but not

more.


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Consistency (cont.)

Because the world of experience is vast and we deal with it in pieces according to

whatever goals concern us at the time, our judgments can never be perfectly

precise.

It may be impossible to make a consistent set of judgments on some pieces that

make them fit exactly with another consistent set of judgments on other related

pieces. So we may neither be able to be perfectly consistent nor want to be.

We must allow for a modicum of inconsistency.

C i t


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How Much Inconsistency to Tolerate? Inconsistency arises from the need for redundancy.

Redundancy improves the validity of the information about the real world.

Inconsistency is important for modifying our consistent understanding, but it must not be too large

to make information seem chaotic. Yet inconsistency cannot be negligible; otherwise, we would be like robots unable to change our

minds.

Mathematically the measurement of consistency should allow for inconsistency of no more than

one order of magnitude smaller than consistency. Thus, an inconsistency of no more than 10%

can be tolerated.

This would allow variations in the measurement of the elements being compared without

destroying their identity.

As a result the number of elements compared must be small, i.e. seven plus or minus two. Being

homogeneous they would then each receive about ten to 15 percent of the total relative value in the

vector of priorities.

A small inconsistency would change that value by a small amount and their true relative value

would still be sufficiently large to preserve that value.

Note that if the number of elements in a comparison is large, for example 100, each would receive

a 1% relative value and the small inconsistency of 1% in its measurement would change its value

to 2% which is far from its true value of 1%.

Consistency (cont.)

Pairwise Comparisons using Inconsistent Judgments


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Pairwise Comparisons using Inconsistent Judgments

Nicer ambiencecomparisons

Paris London New York

Normalized Ideal

0.5815

0.3090

0.1095

1

0.5328

0.1888

Paris

London

New York

1

1

1

2 5

31/2

1/5 1/3

Pairwise Comparisons using Judgments and the Derived Priorities


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Pairwise Comparisons using Judgments and the Derived Priorities

1

0.4297

0.1780

0.6220

0.2673

0.1107

1 3 7

1/3 1 5

1/7 1/5 1

TotalNormalized

B. Clinton M. Tatcher G. Bush

Politician

comparisons

B. Clinton

M. Tatcher

G. Bush


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When the judgments are consistent, we

have two ways to get the answer:1. By adding any column and dividing each entry by the

total, that is by normalizing the column, any column

gives the same result. A quick test of consistency if allthe columns give the same answer.

2. By adding the rows and normalizing the result.

When the judgments are inconsistent we

have two ways to get the answer:

1. An approximate way: By normalizing each column,forming the row sums and then normalizing the result.

2. The exact way: By raising the matrix to powers and

normalizing its row sums


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Comparison of Intangibles

The same procedure as we use for size can be used to

compare things with intangible properties. For example,we could also compare the apples for:

TASTE

AROMA

RIPENESS

Th A l ti Hi h P (AHP)


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The Analytic Hierarchy Process (AHP)

is the Method of Prioritization AHP captures priorities from paired comparison judgments of the

elements of the decision with respect to each of their parent criteria.

Paired comparison judgments can be arranged in a matrix.

Priorities are derived from the matrix as its principal eigenvector,

which defines an absolute scale. Thus, the eigenvector is an intrinsic

concept of a correct prioritization process. It also allows for the

measurement of inconsistency in judgment.

Priorities derived this way satisfy the property of an absolute scale.

D i i M ki


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Decision Making

We need to prioritize both tangible and intangible criteria:

In most decisions, intangibles such as

political factors and social factorstake precedence over tangibles such as

economic factors and

technical factors It is not the precision of measurement on a particular factor

that determines the validity of a decision, but the importance

we attach to the factors involved.

How do we assign importance to all the factors and synthesize

this diverse information to make the best decision?

There are Primitive People Living in


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There are Primitive People Living in

our Time Who cant Count

I have to tell you something. I did a small research in Papua with the local tribe. Thesepeople still live literally in a stone age environment. They do not wear clothes, and

many live in the trees and practice cannibalism. They don't speak the Indonesianlanguage. It was so funny that people from the central bank who accompanied me tothis region noticed and took a picture where I am holding my laptop interviewing thesepeople using the AHP approach....a combination of stone age respondents with moderntechnology!!!!. I will send you the photos when I return to Indonesia next month. Forthis research the AHP questions had to be designed is such that they are very simple yet

straight to the most critical points that we want to know about the livelihood of thissociety. It was a fascinating experience. Those who accompanied me were so impressedwith the power of the AHP in conducting research on such a unique society. Accordingto the central bank people who paid attention to my interviews: "....the outcome is clearand quantified, and yet the respondents do not need to have any knowledge about anynumbers and their corresponding meaning. What it needs are just their honest

perceptions and expressions. They hardly know how to count. In fact they prefer to have 100rupiah than 1000 rupiah simply because the color of the money (red) is what they are used to hold.The extra zero on the bill does not mean much to them.

Love,

Iwan (Iwan Y. Azis is Indonesias greatest economist and is Professor at Cornell University)


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Verbal Expressions for MakingPairwise Comparison Judgments

Equal importance

Moderate importance of one over another

Strong or essential importance

Very strong or demonstrated importance

Extreme importance

Fundamental Scale of Absolute Numbers


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1 Equal importance

3 Moderate importance of one over another

5 Strong or essential importance

7 Very strong or demonstrated importance

9 Extreme importance

2,4,6,8 Intermediate values

Use Reciprocals for Inverse Comparisons

Fundamental Scale of Absolute Numbers

Corresponding to Verbal Comparisons


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0.470.05

0.10

0.15 0.24

Which Drink is Consumed More in the U.S.?


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An Example of Estimation Using Judgments

Coffee Wine Tea Beer Sodas Milk Water

DrinkConsumptionin the U.S.

Coffee

WineTea

Beer

Sodas

Milk

Water

1

1/91/5

1/2

1

1

2

9

12

9

9

9

9

5

1/31

3

4

3

9

2

1/91/3

1

2

1

3

1

1/91/4

1/2

1

1/2

2

1

1/91/3

1

2

1

3

1/2

1/91/9

1/3

1/2

1/3

1

The derived scale based on the judgments in the matrix is:Coffee Wine Tea Beer Sodas Milk Water

.177 .019 .042 .116 .190 .129 .327

with a consistency ratio of .022.

The actual consumption (from statistical sources) is:

.180 .010 .040 .120 .180 .140 .330

Estimating which Food has more Protein


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A B C D E F GFood Consumptionin the U.S.

A: Steak

B: PotatoesC: Apples

D: Soybean

E: Whole Wheat Bread

F: Tasty Cake

G: Fish

1 9

1

9

11

6

1/21/3

1

4

1/41/3

1/2

1

5

1/31/5

1

3

1

1

1/41/9

1/6

1/3

1/5

1

The resulting derived scale and the actual values are shown below:

Steak Potatoes Apples Soybean W. Bread T. Cake Fish

Derived .345 .031 .030 .065 .124 .078 .328

Actual .370 .040 .000 .070 .110 .090 .320

(Derived scale has a consistency ratio of .028.)

(Reciprocals)

WEIGHT COMPARISONS


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Weight Radio Typewriter Large

Attache

Case

Projector Small

Attache

Eigenvector Actual

Relative

Weights

Radio 1 1/5 1/3 1/4 4 0.09 0.10Typewriter 5 1 2 2 8 0.40 0.39

LargeAttache

Case

3 1/2 1 1/2 4 0.18 0.20

Projector 4 1/2 2 1 7 0.29 0.27

Small

Attache

Case

1/4 1/8 1/4 1/7 1 0.04 0.04

WEIGHT COMPARISONS

DISTANCE COMPARISONS


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Comparison

of Distances

from

Philadelphia

Cairo Tokyo Chicago San

Francisco

London Montreal Eigen-

vector

Distance to

Philadelph

ia in miles

Relative

Distance

Cairo 1 1/2 8 3 3 7 0.263 5,729 0.278

Tokyo 3 1 9 3 3 9 0.397 7,449 0.361Chicago 1/8 1/9 1 1/6 1/5 2 0.033 660 0.032

San

Francisco

1/3 1/3 6 1 1/3 6 0.116 2,732 0.132

London 1/3 1/3 5 3 1 6 0.164 3,658 0.177

Montreal 1/7 1/9 1/2 1/6 1/6 1 0.027 400 0.019

DISTANCE COMPARISONS


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Relative Electricity Consumption (Kilowatt Hours) of Household Appliances

.003.01611/51/91/91/81/91/9Hair-dryer

.028.028511/51/91/61/71/9Radio

.047.0619511/41/51/51/7Iron

.120.11099411/21/51/8Dish-

washer

.167.131865211/41/5TV

.242.2619755411/2Refrig-

erator

.392.3939978521Electric

Range

Actual

Relative

Weights

Eigen-vectorHair

DryerRadioIron

Dish

WashTVRefrig

Elec.

Range

Annual

Electric

Consumption


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Relative coin sizes.

0.259346.180.2635Cent

0.338452.160.3821.6Quarter

0.190254.340.1711.62.1Dime

0.212283.380.1821.521.1Cent

Actual

relative

size

Size in mm2Priorities5CentsQuarterDime


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0.0270.0301.0000.3400.2450.1110.111My Web

0.0660.0482.9431.0000.2350.1240.118AOL

0.1070.1014.0764.2641.0000.1520.142MSN

0.2250.3679.0008.0736.5661.0000.552Yahoo

0.4850.4639.0008.4917.0571.8111.000Google

Actual

Percentage

PrioritiesMy WebAOLMSNYahooGooglePercentage of

Individuals that use

different search

engines


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.056247.04911/41/61/7Washington

D.C.

.139610.125411/31/5St. Louis

.2491049.2676311/3New Orleans

.5562446.5587531L.A.

Relative

Values

Actual

Distance

In Miles

PrioritiesWashington

D.C.

St. LouisNew

Orleans

L.A.Distance

From Pittsburgh

Relative Distances from Pittsburgh


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0.198113208420000.1934751Audi - A6

0.188679245400000.18740211Lexus - ES

0.141509434300000.15164721.31Acura - TL

0.226415094480000.2303491.21.251.51BMW - 5

0.245283019520000.2371281.251.31.611Mercedez - E

Relative ValuesActual CostPrioritiesAudi - A6Lexus - ESAcura - TLBMW - 5Mercedez - EModel

Nonlinearity of the Priorities


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RELATIVE VISUAL BRIGHTNESS-I

C1 C2 C3 C4

C1 1 5 6 7

C2 1/5 1 4 6

C3 1/6 1/4 1 4

C4 1/7 1/6 1/4 1

Nonlinearity of the Priorities


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RELATIVE VISUAL BRIGHTNESS -II

C1 C2 C3 C4

C1 1 4 6 7

C2 1/4 1 3 4

C3 1/6 1/3 1 2

C4 1/7 1/4 1/2 1

RELATIVE BRIGHTNESS EIGENVECTOR

Th I S L f O ti


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The Inverse Square Law of Optics

I IIC1 .62 .63

C2 .23 .22C3 .10 .09

C4 .05 .06

Square of Reciprocal

Normalized normalized of previous Normalized

Distance distance distance column reciprocal

9 0.123 0.015 67 0.61

15 0.205 0.042 24 0.22

21 0.288 0.083 12 0.11

28 0.384 0.148 7 0.06

There are two Kinds of


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Fundamentally Different NumbersThe behavior and arithmetic of such numbers are different

CountingAbsolute Scales

Numbers Derived fromMeasurements; MetricTopology-closeness(Scale Numbers)

MeasuringRatio Scale

Interval Scale

Numbers Derived fromComparisons; Order Topology(Dominance Numbers)

Numbers that have the samevalue no matter how many other

things there are

The numerical values of such numbersdepend on what other things a giventhing is compared with. They are oftenreferred to as relative numbers. Theratio of two absolute numbers or ratioscale numbers is absolute. So is theratio of interval scale differences.

The lesser of two elements is used asa unit in each comparison. No uniqueunit for the derived scale whichdefines an absolute scale in relativeform like probabilities.

Need a unit except for

absolute scales defined

in terms of equivalence

classes


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Meaning and Usefulness

Scale Numbers are:

Concrete (independentfrom what and how manyobjects are involved andtherefore unchangeable)

Object oriented Objective

Dominance Numbers are:

Abstract (dependent and

changeable) Perception oriented

Subjective

Measurement in physics is concrete because its objects aretangible and independent, Dominance numbers are essential tomake decisions relative to human values that are abstract andintangible. It is necessary to compare criteria to obtain theirpriorities.One can create dominance numbers from scale numbers but onecannot create scale numbers from dominance numbers becausethey are changeable and depend on the objects being compared.Creating dominance numbers from ratios of measurement is not

always meaningful.

Extending the 1-9 Scale to 1-


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Extending the 1 9 Scale to 1

The 1-9 AHP scale does not limit us if we know how

to use clustering of similar objects in each group and

use the largest element in a group as the smallest one inthe next one. It serves as a pivot to connect the two.

We then compare the elements in each group on the 1-9 scale get the priorities, then divide by the weight of

the pivot in that group and multiply by its weight from

the previous group. We can then combine all thegroups measurements as in the following example

comparing a very small cherry tomato with a very large

watermelon.


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.07 .28 .65

Unripe Cherry Tomato Small Green Tomato Lime

.08 .22 .70

Lime

1=.08

.08

.65 .651x =

Grapefruit

2.75=.08

.22

.65 1.792.75 =

Honeydew

8.75=.08

.70

.65 5.698.75x =

.10 .30 .60

Honeydew

1=.10

.10

5.69 5.691 =

Sugar Baby Watermelon

3=.10

.30

5.69 17.073 =

Oblong Watermelon

6=.10

.60

5.69 34.146 =

This means that 34.14/.07 = 487.7 unripe cherry tomatoes are equal to the oblong watermelon

Clustering & ComparisonColor


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64

How intensely more green is X than Y relative to its size?

Honeydew Unripe Grapefruit Unripe Cherry Tomato

Unripe Cherry Tomato Oblong Watermelon Small Green Tomato

Small Green Tomato Sugar Baby Watermelon Large Lime


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Comparing a Dog-Catcher w/ President


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GoalSatisfaction with School

Learning Friends School Vocational College MusicLife Training Prep. Classes

School

A

School

C

School

B

School Selection


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L F SL VT CP MC

Learning 1 4 3 1 3 4 .32

Friends 1/4 1 7 3 1/5 1 .14

School Life 1/3 1/7 1 1/5 1/5 1/6 .03

Vocational Trng. 1 1/3 5 1 1 1/3 .13

College Prep. 1/3 5 5 1 1 3 .24

Music Classes 1/4 1 6 3 1/3 1 .14

Weights

Comparison of Schools with Respectto the Six Characteristics


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LearningA B C

Priorities

A 1 1/3 1/2 .16

B 3 1 3 .59

C 2 1/3 1 .25

FriendsA B C

Priorities

A 1 1 1 .33

B 1 1 1 .33

C 1 1 1 .33

School LifeA B C

Priorities

A 1 5 1 .45

B 1/5 1 1/5 .09

C 1 5 1 .46

Vocational Trng.

A B CPriorities

A 1 9 7 .77

B 1/9 1 1/5 .05

C 1/7 5 1 .17

College Prep.A B CPriorities

A 1 1/2 1 .25

B 2 1 2 .50

C 1 1/2 1 .25

Music Classes

A B CPriorities

A 1 6 4 .69

B 1/6 1 1/3 .09

C 1/4 3 1 .22

Composition and SynthesisImpacts of School on Criteria


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Impacts of School on Criteria

CompositeImpact ofSchools

A

B

C

.32 .14 .03 .13 .24 .14L F SL VT CP MC

.16 .33 .45 .77 .25 .69 .37

.59 .33 .09 .05 .50 .09 .38

.25 .33 .46 .17 .25 .22 .25

The School Example Revisited Composition & Synthesis:Impacts of Schools on Criteria

Ideal Mode


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Distributive Mode(Normalization: Dividing eachentry by the total in its column)

A

B

C

.32 .14 .03 .13 .24 .14L F SL VT CP MC

.16 .33 .45 .77 .25 .69 .37

.59 .33 .09 .05 .50 .09 .38

.25 .33 .46 .17 .25 .22 .25

CompositeImpact ofSchools

A

B

C

.32 .14 .03 .13 .24 .14L F SL VT CP MC

.27 1 .98 1 .50 1 .65 .34

1 1 .20 .07 .50 .13 .73 .39

.42 1 1 .22 .50 .32 .50 .27

Composite Normal-Impact of izedSchools

Ideal Mode(Dividing each entry by the

maximum value in its column)

The Distributive mode is useful when the

uniqueness of an alternative affects its rank.The number of copies of each alternativealso affects the share each receives inallocating a resource. In planning, thescenarios considered must be comprehensiveand hence their priorities depend on how manythere are. This mode is essential for rankingcriteria and sub-criteria, and when there isdependence.

The Ideal mode is useful in choosing a best

alternative regardless of how many othersimilar alternatives there are.

A Complete Hierarchy to Level of Objectives

At what level should the Dam be kept: Full or Half-FullFocus:


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Financial Political Envt Protection Social Protection

Congress Dept. of Interior Courts State Lobbies

Clout Legal PositionPotentialFinancial

Loss

Irreversibilityof the Envt

Archeo-logical

Problems

CurrentFinancial

Resources

Farmers Recreationists Power Users Environmentalists

Irrigation Flood Control Flat Dam White Dam Cheap Power ProtectEnvironment

Half-Full Dam Full Dam

Focus:

DecisionCriteria:

DecisionMakers:

Factors:

GroupsAffected:

Objectives:

Alternatives:

Evaluating Employees for Raises


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GOAL

Dependability(0.075)

Education(0.200)

Experience(0.048)

Quality(0.360)

Attitude(0.082)

Leadership(0.235)

Outstanding(0.48) .48/.48 = 1

Very Good

(0.28) .28/.48 = .58

Good(0.16) .16/.48 = .33

Below Avg.

(0.05) .05/.48 = .10

Unsatisfactory(0.03) .03/.48 = .06

Outstanding(0.54)

Above Avg.

(0.23)

Average(0.14)

Below Avg.

(0.06)

Unsatisfactory(0.03)

Doctorate(0.59) .59/.59 =1

Masters

(0.25).25/.59 =.43Bachelor(0.11) etc.

High School(0.05)

>15 years(0.61)

6-15 years

(0.25)

3-5 years(0.10)

1-2 years

(0.04)

Excellent(0.64)

Very Good

(0.21)

Good(0.11)

Poor

(0.04)

Enthused(0.63)

Above Avg.

(0.23)

Average(0.10)

Negative

(0.04)

Final Step in Absolute Measurement


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Rate each employee for dependability, education, experience, quality ofwork, attitude toward job, and leadership abilities.

Esselman, T.Peters, T.Hayat, F.Becker, L.Adams, V.Kelly, S.Joseph, M.

Tobias, K.Washington, S.OShea, K.Williams, E.Golden, B.

Outstand Doctorate >15 years Excellent Enthused Outstand 1.000 0.153Outstand Masters >15 years Excellent Enthused Abv. Avg. 0.752 0.115Outstand Masters >15 years V. Good Enthused Outstand 0.641 0.098Outstand Bachelor 6-15 years Excellent Abv. Avg. Average 0.580 0.089Good Bachelor 1-2 years Excellent Enthused Average 0.564 0.086Good Bachelor 3-5 years Excellent Average Average 0.517 0.079Blw Avg. Hi School 3-5 years Excellent Average Average 0.467 0.071

Outstand Masters 3-5 years V. Good Enthused Abv. Avg. 0.466 0.071V. Good Masters 3-5 years V. Good Enthused Abv. Avg. 0.435 0.066Outstand Hi School >15 years V. Good Enthused Average 0.397 0.061Outstand Masters 1-2 years V. Good Abv. Avg. Average 0.368 0.056V. Good Bachelor .15 years V. Good Average Abv. Avg. 0.354 0.054

Dependability Education Experience Quality Attitude Leadership Total Normalized0.0746 0.2004 0.0482 0.3604 0.0816 0.2348

The total score is the sum of the weighted scores of the ratings. Themoney for raises is allocated according to the normalized total score. Inpractice different jobs need different hierarchies.


80/115


81/115


82/115


83/115


84/115

Protect rights and maintain high Incentive to Rule of Law Bring China to Help trade deficit with China

BENEFITS

Should U.S. Sanction China? (Feb. 26, 1995)


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make and sell products in China (0.696) responsible free-trading 0.206) (0.098)

Yes 0.729 No 0.271

$ Billion Tariffs make Chinese productsmore expensive (0.094)

Retaliation(0.280)

Being locked out of big infrastructurebuying: power stations, airports (0.626)

COSTS

Yes 0.787 No 0.213

Long Term negative competition(0.683)

Effect on human rights andother issues (0.200)

Harder to justify China joining WTO(0.117)

RISKS

Yes 0.597 No 0.403

Result:Benefits

Costs x Risks; YES

.729

.787 x .597= 1.55 NO

.271

.213 x .403= 3.16

YesNo

.80

.20YesNo

.60

.40YesNo

.50

.50

YesNo

.70

.30YesNo

.90

.10YesNo

.75

.25

Yes

No

.70

.30

Yes

No

.30

.70

Yes

No

.50

.50


86/115

8

7

6

5

4

3

2

1

Benefits/Costs*Risks

Experiments

0 6 18 30 42 54 66 78 90 102 114 126 138 150 162 174 186 198 210

No

Yes

..

.

.....

..

....

.....

.

.....

......

..

.. .

.

Whom to Marry - A Compatible Spouse


87/115

Flexibility Independence Growth Challenge Commitment Humor Intelligence

Psychological Physical Socio-Cultural Philosophical Aesthetic

Communication& Problem Solving

Family & Children

Temper

Security

Affection

Loyalty

Food

Shelter

Sex

Sociability

Finance

Understanding

World View

Theology

Housekeeping

Sense of Beauty& Intelligence

Campbell Graham McGuire Faucet

Marry Not MarryCASE 1:

CASE 2:

Value of Yen/Dollar Exchange : Rate in 90 Days

Relative Interest

Rate

Forward Exchange

Rate Biases

Official Exchange

Market Inter ention

Relative Degree of Confi-

dence in U S Econom

Size/Direction of U.S.

C rrent Acco nt

Past Behavior of

E change Rate


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Rate.423

Rate Biases.023

Market Intervention.164

dence in U.S. Economy.103

Current AccountBalance .252

Exchange Rate.035

FederalReserveMonetary

Policy.294

Size ofFederalDeficit

.032

Bank ofJapan

MonetaryPolicy.097

ForwardRate

Premium/Discount

.007

Size ofForward

RateDifferential

.016

Consistent

.137

Erratic

.027

RelativeInflationRates

.019

RelativeReal

Growth

.008

RelativePoliticalStability

.032

Size ofDeficit

orSurplus

.032

AnticipatedChanges

.221

Relevant

.004

Irrelevant

.031

Tighter.191

Steady.082

Easier.021

Contract.002

No Chng..009

Expand.021

Tighter.007

Steady.027

Easier.063

High.002

Medium.002

Low.002

Premium.008

Discount.008

Strong.026

Mod..100

Weak.011

Strong.009

Mod..009

Weak.009

Higher.013

Equal.006

Lower.001

More.048

Equal.003

Lower.003

More.048

Equal.022

Less.006

Large.016

Small.016

Decr..090

No Chng..106

Incr..025

High.001

Med..001

Low.001

High.010

Med..010

Low.010

Probable Impact of Each Fourth Level Factor

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

SharpDecline0.1330

ModerateDecline0.2940

NoChange0.2640

ModerateIncrease0.2280

SharpIncrease0.0820

Expected Value is 139.90 yen/$

BenefitsFocus

Best Word Processing Equipment


89/115

Time Saving Filing Quality of Document Accuracy

Training

Required Screen Capability

Service

Quality

Space

Required

Printer

Speed

Lanier(.42)

Syntrex(.37)

Qyx(.21)

Criteria

Features

Alternatives

Capital Supplies Service Training

Lanier.54

Syntrex.28

Oyx.18

CostsFocus

Criteria

Alternatives

Best Word Processing Equipment Cont.


90/115

Benefit/Cost Preference Ratios

Lanier Syntrex Qyx.42.54

.37

.28

.21

.18= 0.78 = 1.32 = 1.17

Best Alternative

Group Decision Makingand the

G t i M


91/115

Geometric MeanSuppose two people compare two apples and provide the judgments for the larger

over the smaller, 4 and 3 respectively. So the judgments about the smaller relative

to the larger are 1/4 and 1/3.Arithmetic mean

4 + 3 = 7

1/7 1/4 + 1/3 = 7/12

Geometric mean

4 x 3 = 3.46

1/

4 x 3 =

1/4 x 1/3 = 1/

4 x 3 = 1/3.46

That the Geometric Mean is the unique way to combine group judgments is a

theorem in mathematics.


92/115

0.05

0 47


93/115

0.47

0.10

0.15 0.24

Why Is the Eigenvector Necessary1

1) Consistent matrix: ;

k k

Aw nw A n A

= =


94/115

max

1

1) Consistent matrix: ; .2) Perturbed matrix: .

13) Transitivity lim / , (1,...,1). By Cesaro sumability

this converges to the same limit as / .

4) Necessi

mk T k T

kk

k T k

Aw nw A n AAw w

A e e A e w em

A e e A e

=

=

=

ty : Priority must be invariant with respect to whatever process of

synthesis one chooses. We start with an initial priority vector (1,...,1) and then

derive a first estimate of priorities from it. We then use this vector it to weightthe alternatives or weight squares of differences and derive a new priority vector,

and so on. For uniqueness we must have , where indicates the

composition pri

A x cx=o o

nciple used and indicates proportionality of the new vector

and the old vector . For additive composition we have . More generally,

x and because initially , , the principak k k k

c cx

x Ax cx

A x c x e A e c e w

=

= = = l right eigenvector

of . Thus it is necessary to use the eigenvector to derive a priority vector.A


95/115


96/115

...

...

...

...

1 n

1 1 1 1 n 1 1

n n 1 n n n n

A A

w w w w w wA

w n n

w w w w w wA

= = =

M M M M M

Let A1, A

2,, A

n, be a set of stimuli. The

quantified judgments on pairs of stimuliAi,A

j, are

represented by an n by n matrix A = (aij) ij = 1


97/115

12 1

12 2

1 2

1 ...

1/ 1 ...

1/ 1/ ... 1

n

n

n n

a a

a a

A

a a

=

M M M M

ija

jiai

represented by an n-by

-n matrix A = (aij), ij = 1,

2, . . ., n. The entries aij

are defined by the

following entry rules. If aij

= a, then aji

= 1 /a,

a 0. If Ai is judged to be of equal relative

intensity to Aj

then aij

= 1, aji

= 1, in particular, aii

= 1 for all i.

Aw=nw

Aw=cw

How to go from

to


98/115

Clearly in the first formula n is a simple eigenvalue and all other

eigenvalues are equal to zero.

A forcing perurbation of eigenvalues theorem:

If is a simple eigenvalue of A, then for small > 0, there is aneigenvalue () of A() with power series expansion in :

()= + (1)+ 2 (2)+

and corresponding right and left eigenvectors w () and v () suchthat w()= w+ w(1)+ 2 w(2)+

v()= v+ v(1)+ 2 v(2)+

Aw=cw

Aw=maxw

to

and then to

n


99/115

w=wa ijij

n

1=j

max

1=win

1=i

0

0,

10

01,

43

21

=

=

= IIA


100/115

0256)4)(1(

43

21)(

2===

=

IA

IA

2

3352

335

2

1

=

+=


101/115

max

1 1

max

1 1

max max

1 1

max for max

min for min

Thus for a row stochastic matrix we have 1=min max 1,thus =1.

n nj

ij ij i

j j i

n nj

ij ij i

j j i

n n

ij ij

j j

wa a w

w

wa a w

w

a a

= =

= =

= =

=

=

=

w)wv)-/(wAv(=w jjTjj11

Tj

n

2j=

1

Sensitivity of the Eigenvector


102/115

The eigenvector w1

is insensitive to perturbation inA, if 1) the number of terms is small

(i.e. n is small), 2) if the principal eigenvalue 1

is separated from the other eigenvalues ,

here assumed to be distinct (otherwise a slightly more complicated argument can also be

made and is given below) and, 3) if none of the products vjT wj of left and righteigenvectors is small and if one of them is small, they are all small. Howerver, v

1T w

1,

the product of the normalized left and right principal eigenvectors of a consistent matrix

is equal to n which as an integer is never very small. If n is relatively small and the

elements being compared are homogeneous, none of the components ofw1

is arbitrarily

small and correspondingly, none of the components of v1T

is arbitrarily small. Theirproduct cannot be arbitrarily small, and thus w is insensitive to small perturbations of the

consistent matrix A. The conclusion is that n must be small, and one must compare

homogeneous elements.

When the eigenvalues have greater multiplicity than one, the corresponding left and right

eigenvectors will not be unique. In that case the cosine of the angle between them whichis given by corresponds to a particular choice of and . Even when and correspond to a

simple they are arbitrary to within a multiplicative complex constant of unit modulus,

but in that case | viT w

i| is fully determined. Because both vectors are normalized, we

always have | viT w

i|


103/115

p y ynetwork;

Why make comparisons to derive priorities;

Why reciprocals and why homogeneous groups of

elements;

Why the fundamental scale 1-9, what does it mean toassign a number for a judgment;

Why allow inconsistency;

What is the minimum number of judgments needed

and why use redundant judgments;

Wh the rinci al ri ht ei envector;

Why absolute numbers and absolute scales;

Why weight and add for synthesis;

Why the distributive and ideal modes;


104/115

Why the distributive and ideal modes;

Why the supermatrix and what does raising it

to powers do;

Why stochastic supermatrix;

Why weight the components;

Why BOCR why add benefits and

opportunities and subtract costs and risks ;

Why the ideal mode;

How to allocate resources the need for ratio

scales;

Some Answers

(Only to be a little helpful) One structures a hierarchy from a goal download to criteria, subcriteria and goals, involving actors


105/115

O e st uctu es a e a c y o a goa dow oad to c te a, subc te a a d goa s, vo v g acto s

and stakeholders and terminating in alternatives at the bottom. The ideas to go gradually from the

general to the particular. In a network, elements are put in clusters or components with their

connections indicating influence.

Comparisons are more scientific in deriving scales because they use a unit and estimate multiples of

that unit rather than simply assigning numbers by guessing.Reciprocals are needed because if one element is five times more important than another then the

other is a forteori one fifth as important as the first. One deals with homogeneous clusters to make

the comparisons possible, closer and more accurate.

The scale 1-9 helps us quantify our feelings and judgments in comparing elements.

Human judgment expressed in the form of paired comparisons is naturally inconsistent. A modicum

of inconsistency enables us to improve our understanding by focusing on the most inconsistentjudgments.

The minimum number of judgments needed to connect n elements is n-1. Redundant judgments

improve the validity of the derived priority vector.

Ratios and ratio scales give us information on both the rank order of the elements and on their

relative values. It also makes possible proportionate resource allocation.

Weighting and adding follows from simple operations we do all the time and is no different forpriorities. Suppose the goal has two components of values 0.6 and 0.4, and assume that one has a 0.2

share in the first component and a 0.7 share in the second. The total share with respect to the goal is

0.6 x 0.2 + 0.4 x 0.7 = 0.4.

The supermatrix is the framework for organizing the priorities derived from paired comparisons.

Raising it to powers gives the overall influence of each element on all the other elements.

ASSIGNING NUMBERS vs.

PAIRED COMPARISONS


106/115

PAIRED COMPARISONS

A number assigned directly to anobject is at best an ordinal andcannot be justified.

When we compare two objects orideas we use the smaller as a unit

and estimate the larger as a multipleof that unit.

If the objects are homogeneous and if


107/115

If the objects are homogeneous and ifwe have knowledge and experience,

paired comparisons actually derivemeasurements that are likely to be closeand that indicate magnitude on anabsolute scale.

WHY IS AHP EASY TO USE?I d k f d h


108/115

WHY IS AHP EASY TO USE? It does not take for granted themeasurements on scales, but asks thatscale values be interpreted accordingto the objectives of the problem.

It relies on elaborate hierarchicstructures to represent decision prob-

lems and is able to handle problems ofrisk, conflict, and prediction.

It can be used to make directresource allocation benefit/cost


109/115

t ca be used to a e d ectresource allocation, benefit/costanalysis, resolve conflicts, design

and optimize systems.

It is an approach that describeshow good decisions are made ratherthan prescribes how they should be

made.

WHY THE AHP IS POWERFUL

IN CORPORATE PLANNING


110/115

IN CORPORATE PLANNING1. Breaks down criteria into manage-

able components.

2. Leads a group into making a specific

decision for consensus or tradeoff.

3. Provides opportunity to examine

disagreements and stimulatediscussion and opinion.

4. Offers opportunity to change

criteria, modify judgments.


111/115

, y j g

5. Forces one to face the entire

problem at once.

6. Offers an actual measurement

system. It enables one toestimate relative magnitudes and

derive ratio scale prioritiesaccurately.

7. It organizes, prioritizes andsynthesizes complexity within a


112/115

y p yrational framework.

8. Interprets experience in a relevantway without reliance on a black

box technique like a utility function.

9. Makes it possible to deal with

conflicts in perception and injudgment.

Table 1 Comparison of Group Decision Making Methods

AnalysisStructureProblem AbstractionGroup MaintenanceMethod


113/115

Medium

High

Very High

Very High

High

Very High

Low

Low

High

Low

High

High

Low

High

Very High

Medium

Medium

Medium

High

High

Very High

Medium

Medium

High

Structuring and Measuring

Baysian Analysis

MAUT/MAVTAHP

NA

NANA

NA

NA

Low

Low

Low

Medium

Medium

Medium

Medium

High

NA

NANA

NA

NA

Low

Low

Low

Medium

Medium

Very High

Very High

Medium

NA

NANA

NA

High

Low

Low

Low

Low

Low

Low

Low

Low

NA

NANA

NA

High

Low

Low

Low

High

High

High

Low

High

Low

LowMedium

Very High

Very High

NA

High

High

Medium

Low

Low

Low

High

Medium

HighLow

High

High

NA

Medium

Medium

Medium

Medium

Medium

Medium

Medium

Medium

MediumLow

Medium

Medium

Low

Medium

Medium

High

Medium

Low

Low

High

Low

MediumLow

Low

Medium

Low

Medium

Medium

Medium

Medium

Low

Low

Medium

Structuring

Analogy, AssociationBoundary Examination

Brainstorming/Brainwriting

Morphological Connection

Why-What's Stopping

Ordering and Ranking

Voting

Nominal Group Technique

Delphi

Disjointed Incrementalism

Matrix Evaluation

Goal Programming

Conjoint Analysis

Outranking

Breadth and

Depth of

Analysis

(What if)

Faithfulness

of

Judgments

DepthBreadthDevelopment

of

Alternatives

ScopeLearningLeadership

Effectiveness

NA = Not Applicable

Table1 (cont'd)

Validity of

th O t

Applicability

t

Psycho-

h i l

Applicability

t I t ibl

Scientific

d

Consideration

f Oth

Prioritizing

G

Cardinal

S ti

Applicability, Validity, and TruthfulnessFairnessMethod


114/115

Medium

Medium

High

NA

Medium

High

Low

Medium

Very High

Medium

Medium

Very High

High

High

High

Low

Medium

High

NA

High

Very High

High

High

High

Structuring and Measuring

Baysian Analysis

MAUT/MAVT

AHP

NA

NA

NA

NA

NA

Low

Low

Low

Medium

Medium

Low

LowMedium

NA

NA

NA

NA

NA

NA

NA

NA

NA

NA

NA

NANA

NA

NA

NA

NA

NA

NA

NA

NA

Low

Low

NA

NAMedium

NA

NA

NA

NA

NA

NA

NA

NA

Low

Low

Medium

MediumMedium

NA

NA

NA

NA

NA

Medium

Medium

Medium

Low

Low

Medium

MediumMedium

NA

NA

NA

NA

NA

NA

NA

NA

Medium

Medium

Low

NALow

NA

NA

NA

NA

NA

Low

NA

NA

NA

NA

NA

NAHigh

NA

NA

NA

NA

NA

Low

NA

NA

NA

NA

High

HighHigh

Structuring

Analogy/Association

Boundary Examination

Brainstorming/Brainwriting

Morphological Connection

Why-What's Stopping

Ordering and Ranking

Voting

Nominal Group Technique

Delphi

Disjointed Incrementalism

Matrix EvaluationGoal Programming

Conjoint Analysis

Outranking

the Outcome

(Prediction)

to

Conflict

Resolution

physical

Applicability

to Intangiblesand

Mathematical

Generality

of Other

Actors &

Stakeholders

Group

Members

Separation

of

Alternatives

NA = Not Applicable

03/27/2006 09:58 PM

Dear Prof. Saaty:

Recently, I am thinking my future research fields all along. It is a really difficult decision. I want to devote

myself to MCDM. I also know that Herbert A. Simon got the Nobel prize in 1978 for his contribution to

organizational decision making, and Nash for his contribution to game theory and its applications. I wonder

if there is any possibility for the research on MCDM to make the same great contribution.


115/115

regards, Sun Yonghong

Sun Yonghong

-----Original Message-----From: [email protected] [mailto:[email protected]]

Sent: Tuesday, March 07, 2006 8:24 PM

To: Sun Yonghong

Subject: Re:

Dear Mr. Sun,

I assume you are just learning the subject because I think these are old papers that are not written

electronically. Would it be ok to send you other papers on the subject or should I try to find them and scan

them? I have generalized the AHP to the Analytic Network Process with dependence and feedback (ANP)

you may want something on it too. I teach a graduate course on the ANP starting tomorrow.

Kind regards,

Tom Saaty

Documents

Course Slides 1 Summer School 2007