An Illustrated Guide to the
ANALYTIC HIERARCHY PROCESS
Oliver Meixner & Rainer Haas
Institute of Marketing & Innovation
University of Natural Resources and Life Sciences, Vienna
http://www.wiso.boku.ac.at/mi/
Version Aug. 2017 1
In remembrance & honour of
Thomas Saaty
1926 – 2017
https://www.forevermissed.com/thomas-saaty
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Do your decision conferences turn out like this?
TOO BAD!
WE WANT
WE WANT
PROGRAM A !!
PROGRAM B !!
COME ON IN
THE WATER IS
FINE!
sea of indecision
or does this happen?
3
DO YOUR RECOMMENDATIONS BUT BOSS...
TURN OUT LIKE THIS? THAT WAS MY
BEST GUESS!
GUESS AGAIN
MAYBE YOU NEED A
NEW APPROACH
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... another way of decision making
I THINK I ‘LL TRY THE
ANALYTIC HIERARCHY
PROCESS (AHP) !!!
OKAY TELL US
ABOUT AHP
PROF. DR THOMAS L.
SAATY DEVELOPED
THE PROCESS IN
THE EARLY 1970’S
AND...
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THE PROCESS HAS BEEN USED TO
ASSIST NUMEROUS CORPORATE AND
GOVERNMENT DECISION MAKERS.
Some examples of decision problems:
- choosing a telecommunication system
- formulating a drug policy
- choosing a product marketing strategy
- ...
Let’s show
how it works PROBLEMS ARE DECOMPOSED
INTO A HIERARCHY OF CRITERIA
AND ALTERNATIVES
Problem
Criterion 1 Criterion 2 ... Criterion n
Criterion 1.1 ...
...
Alternative 1 Alternative 2 ... Alternative n
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!"#$%I
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AN IMPORTANT PART OF THE PROCESS
IS TO ACCOMPLISH THESE THREE STEPS
• STATE THE OBJECTIVE:
– SELECT A NEW CAR
• DEFINE THE CRITERIA:
– STYLE, RELIABILITY, FUEL ECONOMY
• PICK THE ALTERNATIVES:
– CIVIC COUPE, SATURN COUPE, FORD ESCORT,
RENAULT CLIO
WHAT ABOUT COST?
(BE QUIET, WE’LL TALK ABOUT THAT LATER)
SKEPTIC-GATOR 11
THIS INFORMATION IS THEN ARRANGED IN A
HIERARCHICAL TREE
OBJECTIVE
CRITERIA
Select a
new car
Fuel
Style Reliability
Economy
Civic Civic Civic
ALTERNATIVES Saturn Saturn Saturn
Escort Escort Escort
Clio Clio Clio
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THE INFORMATION
IS THEN
BOTH QUALITATIVE
SYNTHESIZED TO
AND QUANTITATIVE
DETERMINE
CRITERIA CAN BE
RELATIVE
COMPARED USING
RANKINGS OF
INFORMED
ALTERNATIVES
JUDGMENTS TO
DERIVE WEIGHTS
AND PRIORITIES
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HOW DO YOU DETERMINE THE RELATIVE
IMPORTANCE OF THE CRITERIA?
Here’s one way !
STYLE
RELIABILITY
FUEL ECONOMY
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Hmm, I think reliability is the most
important followed by style and fuel
HERE’S ANOTHER WAY economy is least importeant so I will
make the following judgements ....
USING JUDGMENTS TO
DETERMINE THE RANKING
OF THE CRITERIA
1. RELIABILITY IS 2 TIMES AS IMPORTANT AS STYLE
2. STYLE IS 3 TIMES AS IMPORTANT AS FUEL ECONOMY
3. RELIABILITY IS 4 TIMES AS IMPORTANT AS FUEL ECONOMY
he’s not very consistent here ... that’s o.k.
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Pairwise Comparisons A
B
USING PAIRWISE COMPARISONS, THE RELATIVE IMPORTANCE
OF ONE CRITERION OVER ANOTHER CAN BE EXPRESSED
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Pairwise Comparisons A
B
USING PAIRWISE COMPARISONS, THE RELATIVE IMPORTANCE
OF ONE CRITERION OVER ANOTHER CAN BE EXPRESSED
1 equal 3 moderate 5 strong 7 very strong 9 extreme
STYLE RELIABILITY FUEL ECONOMY
STYLE 1/1 1/2 3/1
RELIABILITY 1/1 4/1
FUEL ECONOMY 1/1
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Pairwise Comparisons A
B
USING PAIRWISE COMPARISONS, THE RELATIVE IMPORTANCE
OF ONE CRITERION OVER ANOTHER CAN BE EXPRESSED
1 equal 3 moderate 5 strong 7 very strong 9 extreme
STYLE RELIABILITY FUEL ECONOMY
STYLE 1/1 1/2 3/1
RELIABILITY 2/1 1/1 4/1
FUEL ECONOMY 1/3 1/4 1/1
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How do you turn this MATRIX
into ranking of criteria?
STYLE RELIABILITY FUEL ECONOMY
STYLE 1/1 1/2 3/1
RELIABILITY 2/1 1/1 4/1
FUEL ECONOMY 1/3 1/4 1/1
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HOW DO YOU GET A RANKING OF PRIORITIES FROM A
PAIRWISE MATRIX?
AND THE
SURVEY SAYS
EIGENVECTOR !!
ACTUALLY...
PROF. DR. THOMAS L. SAATY , (UNIVERSITY OF PITTSBURGH),
DEMONSTRATED MATHEMATICALLY THAT THE
EIGENVECTOR SOLUTION WAS THE BEST APPROACH.
REFERENCE : THE ANALYTIC HIERARCHY PROCESS, 1990, THOMAS L. SAATY
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HERE’S HOW TO SOLVE FOR THE EIGENVECTOR:
1. A SHORT COMPUTATIONAL WAY TO OBTAIN THIS RANKING
IS TO RAISE THE PAIRWISE MATRIX TO POWERS THAT ARE
SUCCESSIVELY SQUARED EACH TIME.
2. THE ROW SUMS ARE THEN CALCULATED AND NORMALIZED.
3. THE COMPUTER IS INSTRUCTED TO STOP WHEN THE
DIFFERENCE BETWEEN THESE SUMS IN TWO CONSECUTIVE
CALCULATIONS IS SMALLER THAN A PRESCRIBED VALUE.
SHOW ME AN
SAY WHAT! EXAMPLE
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IT’S MATRIX ALGEBRA TIME !!!
STYLE RELIABILITY FUEL ECONOMY
STYLE 1/1 1/2 3/1
RELIABILITY 2/1 1/1 4/1
FUEL ECONOMY 1/3 1/4 1/1
FOR NOW, LET’S REMOVE THE NAMES AND
CONVERT THE FRACTIONS TO DECIMALS :
1.0000 0.5000 3.0000
2.0000 1.0000 4.0000
0.3333 0.2500 1.0000
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STEP 1: SQUARING THE MATRIX
1.0000 0.5000 3.0000
THIS TIMES 2.0000 1.0000 4.0000
0.3333 0.2500 1.0000
1.0000 0.5000 3.0000
THIS 2.0000 1.0000 4.0000
0.3333 0.2500 1.0000
I.E. (1.0000 * 1.0000) + (0.5000 * 2.0000) +(3.0000 * 0.3333) = 3.0000
3.0000 1.7500 8.0000
RESULTS
IN THIS 5.3332 3.0000 14.0000
1.1666 0.6667 3.0000
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STEP 2 : NOW, LET’S COMPUTE OUR FIRST EIGENVECTOR
(TO FOUR DECIMAL PLACES)
FIRST, WE SUM THE ROWS
3.0000 + 1.7500 + 8.0000 = 12.7500 0.3194
5.3332 + 3.0000 + 14.0000 = 22.3332 0.5595
1.1666 + 0.6667 + 3.0000 = 4.8333 0.1211
SECOND, WE SUM THE ROW TOTALS 39.9165 1.0000
FINALLY, WE NORMALIZE BY DIVIDING
THE ROW SUM BY THE ROW TOTALS
(I.E. 12.7500 DIVIDED BY 39.9165 EQUALS 0.3194)
0.3194
THE RESULT IS OUR EIGENVECTOR
( A LATER SLIDE WILL EXPLAIN THE 0.5595
MEANING IN TERMS OF OUR EXAMPLE)
0.1211
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THIS PROCESS MUST BE ITERATED UNTIL THE EIGENVECTOR
SOLUTION DOES NOT CHANGE FROM THE PREVIOUS ITERATION
(REMEMBER TO FOUR DECIMAL PLACES IN OUR EXAMPLE)
CONTINUING OUR EXAMPLE,
AGAIN, STEP 1: WE SQUARE THIS MATRIX
3.0000 1.7500 8.0000
5.3332 3.0000 14.0000
1.1666 0.6667 3.0000
27.6653 15.8330 72.4984
WITH THIS RESULT 48.3311 27.6662 126.6642
10.5547 6.0414 27.6653
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AGAIN STEP 2 : COMPUTE THE EIGENVECTOR (TO FOUR DECIMAL PLACES)
27.6653 + 15.8330 + 72.4984 = 115.9967 0.3196
48.3311 + 27.6662 + 126.6642 = 202.6615 0.5584
10.5547 + 6.0414 + 27.6653 = 44.2614 0.1220
TOTALS 362.9196 1.0000
COMPUTE THE DIFFERENCE OF THE
PREVIOUS COMPUTED EIGENVECTOR
TO THIS ONE:
0.3194 0.3196 = - 0.0002
0.5595 0.5584 = 0.0011
0.1211 0.1220 = - 0.0009
TO FOUR DECIMAL PLACES THERE’S NOT MUCH DIFFERENCE
HOW ABOUT ONE MORE ITERATION?
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I SURRENDER !!
DON’T MAKE ME COMPUTE
ANOTHER EIGENVECTOR
OKAY,OKAY
ACTUALLY, ONE MORE
ITERATION WOULD SHOW
NO DIFFERENCE TO FOUR
DECIMAL PLACES
LET’S NOW LOOK AT
THE MEANING OF THE
EIGENVECTOR
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HERE’S OUR PAIRWISE
MATRIX WITH THE NAMES
STYLE RELIABILITY FUEL ECONOMY
STYLE 1/1 1/2 3/1
RELIABILITY 2/1 1/1 4/1
FUEL ECONOMY 1/3 1/4 1/1
AND THE COMPUTED EIGENVECTOR GIVES US THE RELATIVE
RANKING OF OUR CRITERIA
STYLE 0.3196 THE SECOND MOST IMPORTANT CRITERION
RELIABILITY 0.5584 THE MOST IMPORTANT CRITERION
FUEL ECONOMY 0.1220 THE LEAST IMPORTANT CRITERION
NOW BACK TO THE HIEARCHICAL TREE...
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HERE’S THE TREE
WITH THE CRITERIA
WEIGHTS
OBJECTIVE
CRITERIA
Select a new
car
1.00
Style Reliability Fuel Economy
.3196 .5584 .1220
Civic Civic Civic
ALTERNATIVES Saturn Saturn Saturn
Escort Escort Escort
Clio Clio Clio
WHAT ABOUT THE ALTERNATIVES?
I’M GLAD YOU ASKED...
SKEPTIC-GATOR 29
IN TERMS OF STYLE, PAIRWISE COMPARISONS
DETERMINES THE PREFERENCE
OF EACH ALTERNATIVE OVER ANOTHER
STYLE
CIVIC SATURN ESCORT CLIO
CIVIC 1/1 1/4 4/1 1/6
SATURN 4/1 1/1 4/1 1/4
ESCORT 1/4 1/4 1/1 1/5
CLIO 6/1 4/1 5/1 1/1
AND...
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IN TERMS OF RELIABILITY, PAIRWISE COMPARISONS
DETERMINES THE PREFERENCE
OF EACH ALTERNATIVE OVER ANOTHER
RELIABILITY
CIVIC SATURN ESCORT CLIO
CIVIC 1/1 2/1 5/1 1/1
SATURN 1/2 1/1 3/1 2/1
ESCORT 1/5 1/3 1/1 1/4
CLIO 1/1 1/2 4/1 1/1
ITS MATRIX ALGEBRA TIME!!!
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COMPUTING THE EIGENVECTOR DETERMINES THE RELATIVE
RANKING OF ATERNATIVES UNDER EACH CRITERION
(ACTUALLY, WE SQUARED THE MATRIX MORE OFTEN TO GET TO THESE RESULTS)
RANKING STYLE RANKING RELIABILITY
3 CIVIC .1159 1 CIVIC .3786
2 SATURN .2468 2 SATURN .2902
4 ESCORT .0600 4 ESCORT .0742
1 CLIO .5773 3 CLIO .2571
WHAT ABOUT FUEL ECONOMY?
ANOTHER GOOD QUESTION...
SKEPTIC-GATOR 32
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AS STATED EARLIER,
AHP CAN COMBINE BOTH QUALITATIVE
AND QUANITATIVE INFORMATION
FUEL ECONOMY INFORMATION IS OBTAINED FOR EACH
ALTERNATIVE:
FUEL ECONOMY
(MILES/GALLON)
CIVIC 34 34 / 113 = .3009
SATURN 27 27 / 113 = .2389
ESCORT 24 24 / 113 = .2124
CLIO 28 28 / 113 = .2478
113 1.0000
NORMALIZING THE FUEL ECONOMY INFO
ALLOWS US TO USE IT WITH OTHER RANKINGS
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HERE’S THE TREE
WITH ALL THE
WEIGHTS
OBJECTIVE
CRITERIA
Select a new
car
1.00
Style Reliability Fuel Economy
.3196 .5584 .1220
Civic .1159 Civic .3786 Civic .3009
ALTERNATIVES Saturn .2468 Saturn .2902 Saturn .2389
Escort .0600 Escort .0742 Escort .2124
Clio .5773 Clio .2571 Clio .2478
OKAY, NOW WHAT ? I THINK WE’RE READY
FOR THE ANSWER...
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A LITTLE MORE MATRIX ALGEBRA GIVES US THE SOLUTION:
RELI- FUEL CRITERIA
STYLE ABILITY ECONOMY RANKING
CIVIC .1159 .3786 .3009 0.3196 STYLE
SATURN .2468 .2902 .2389
ESCORT .0600 .0742 .2124
* 0.5584 RELIABILITY
CLIO .5773 .2571 .2478 0.1220 FUEL ECONOMY
I.E. FOR THE CIVIC (.1159 * .3196) + (.3786 * .5584) + (.3009 * .1220) = .2851
Civic .2851 AND THE WINNER IS !!!
Saturn .2700 THE CLIO IS THE
= HIGHEST RANKED CAR
Escort .0865
Clio .3583
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IN SUMMARY, THE ANALYTIC HIERARCHY PROCESS
PROVIDES A LOGICAL FRAMEWORK TO DETERMINE
THE BENEFITS OF EACH ALTERNATIVE
1. Clio .3583
2. Civic .2851
3. Saturn .2700
4. Escort .0865
WHAT ABOUT COSTS?
WELL, I’LL TELL YOU...
SKEPTIC-GATOR
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ALTHOUGH COSTS COULD HAVE BEEN INCLUDED,
IN MANY COMPLEX DECISIONS, COSTS SHOULD BE
SET ASIDE UNTIL THE BENEFITS OF THE
ALTERNATIVES ARE EVALUATED
OTHERWISE THIS COULD HAPPEN...
YOUR PROGRAM COST TOO MUCH I
DON’T CARE ABOUT ITS BENEFITS
DISCUSSING COSTS
TOGETHER WITH BENEFITS
CAN SOMETIMES BRING FORTH
MANY POLITICAL AND
EMOTIONAL RESPONSES
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WAYS TO HANDLE BENEFITS AND
COSTS INCLUDE THE FOLLOWING:
1. GRAPHING BENEFITS AND COSTS OF EACH ALTERNATIVE
. .
CHOSE ALTERNATIVE WITH LOWEST
BENEFITS . COST AND HIGHEST BENEFIT
.
COSTS
2. BENEFIT TO COST RATIOS
3. LINEAR PROGRAMMING
4. SEPARATE BENEFIT AND COST HIERARCHICAL TREES
AND THEN COMBINE THE RESULTS
IN OUR EXAMPLE...
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LET’S USE BENEFIT TO COST RATIOS
(AGAIN, WE HAVE QUANTITATIVE INFORMATION HERE)
NORMALIZED
COST $ COSTS
1. CLIO 18,000 .3333
2. CIVIC 12,000 .2222
3. SATURN 15,000 .2778
4. ESCORT 9,000 .1667
54,000 1.0000
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LET’S USE BENEFIT TO COST RATIOS
NORMALIZED
COST $ COSTS BENEFIT - COST RATIOS
1. CLIO 18,000 .3333 .3583 / .3333 = 1.0750
2. CIVIC 12,000 .2222 .2851 / .2222 = 1.2831
3. SATURN 15,000 .2778 .2700 / .2778 = .9719
4. ESCORT 9,000 .1667 .0865 / .1667 = .5189
54,000 1.0000
(REMEMBER THE BENEFITS WERE DERIVED
EARLIER FROM THE AHP)
AND...
THE CIVIC IS THE WINNER WITH THE HIGHEST BENEFIT TO COST RATIO
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AHP CAN BE USED FOR VERY
COMPLEX DECISIONS
MANY LEVELS OF CRITERIA GOAL
AND SUBCRITERIA CAN
BE INCLUDED
HERE ARE SOME EXAMPLES
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AHP CAN BE USED FOR A WIDE VARIETY
OF APPLICATIONS
STRATEGIC PLANNING
RESOURCE ALLOCATION
SOURCE SELECTION
BUSINESS/PUBLIC POLICY
PROGAM SELECTION
AND MUCH MUCH MORE...
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