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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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Current State of Corporate Decision Making
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
Current State of Corporate Decision Making
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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 Apple B Apple C
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
Apple A Apple B Apple C
SizeComparison
Apple A Apple B Apple C
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.
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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
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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
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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
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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.
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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
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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.
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0.05
0 47
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0.47
0.10
0.15 0.24
Why Is the Eigenvector Necessary1
1) Consistent matrix: ;
k k
Aw nw A n A
= =
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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
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...
...
...
...
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
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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
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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
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w=wa ijij
n
1=j
max
1=win
1=i
0
0,
10
01,
43
21
=
=
= IIA
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0256)4)(1(
43
21)(
2===
=
IA
IA
2
3352
335
2
1
=
+=
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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
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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|
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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;
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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
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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
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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
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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
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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
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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
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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.
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, 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
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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
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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
8/14/2019 Course Slides 1 Summer School 2007
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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.
8/14/2019 Course Slides 1 Summer School 2007
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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