Description ofAnalytic Hierarchy Process

methodologyD.I.I., Universita di Siena

DITTY project

Ferrara, 13 December 2005

DITTY – p. 1

OutlineAnalytic Hierarchy Process

Implementation

Example

Conclusions

DITTY – p. 2

OutlineAnalytic Hierarchy Process

Implementation

Example

Conclusions

DITTY – p. 3

AHPThe Analytic Hierarchy Process(AHP), [T.Saaty (1980)], is a powerful tool thatmay be used to make decisions when

multiple and conflictingobjectives/criteria are present,

and both qualitative and quantitativeaspects of a decision need to beconsidered.

DITTY – p. 4

Literature review

Thousands of AHP applications havebeen proposed in the literature.

However, there exist relatively fewapplications of AHP to environmental ornatural resource problems.

DITTY – p. 5

How the AHP works

The AHP considers a set of evaluationcriteria, and a set of alternativescenarios among which the bestdecision is to be made.

It generates a weight for each evaluationcriterion and scenario according to theinformation provided by the DM.

The AHP combines the objective andscenario evaluations determining aranking of the scenarios.

DITTY – p. 6

Features

AHP is effective in dealing with complexdecision making because it reducescomplex decisions to a series of pairwisecomparisons.

DITTY – p. 7

Features

AHP is effective in dealing with complexdecision making because it reducescomplex decisions to a series of pairwisecomparisons.

AHP reduces the bias in the decisionmaking process because it also checksthe consistency of the DM’s evaluations.

DITTY – p. 7

FeaturesAHP may be considered as a tool that isable to translate the pairwise relativeevaluations (both qualitative andquantitative) made by the DM into amulticriteria ranking.

DITTY – p. 8

FeaturesAHP may be considered as a tool that isable to translate the pairwise relativeevaluations (both qualitative andquantitative) made by the DM into amulticriteria ranking.

The AHP is simple because there is noneed of building a complex expertsystem with the DM’s knowledgeembedded in it.

DITTY – p. 8

Concepts

Three major concepts behind the AHP:

The AHP is analytic

The AHP structures the problem as ahierarchy

The AHP helps in thedecision-making process

DITTY – p. 9

Concepts

Three major concepts behind the AHP:

The AHP is analytic

The AHP structures the problem as ahierarchy

The AHP helps in thedecision-making process

The AHP is analytic because it convertsDM’s inputs into numbers.

DITTY – p. 9

Concepts

Three major concepts behind the AHP:

The AHP is analytic

The AHP structures the problem as ahierarchy

The AHP helps in thedecision-making process

The AHP is hierarchical becausereduces complex decision-makingproblems into pairwise comparisions.

DITTY – p. 9

Concepts

Three major concepts behind the AHP:

The AHP is analytic

The AHP structures the problem as ahierarchy

The AHP helps in thedecision-making process

The AHP incorporates DM’s inputs anddefines a process for decision-making.

DITTY – p. 9

OutlineAnalytic Hierarchy Process

Implementation

Example

Conclusions

DITTY – p. 10

Implementation

The AHP consists of these steps:

1. Computing the vector of objectiveweights

2. Computing the matrix of scenarioscores

3. Ranking the scenarios

4. Checking the consistency

We consider m evaluation criteria and nscenarios.

DITTY – p. 11

Implementation

1. Computing the vector of objectiveweights

2. Computing the matrix of scenarioscores

3. Ranking the scenarios

4. Checking the consistency

DITTY – p. 12

Step 1

Pairwise comparison matrix A [m × m].

Each entry ajk of A represents theimportance of criterion j relative tocriterion k:

If ajk > 1, j is more important than k

if ajk < 1, j is less important than k

if ajk = 1, same importance

ajk and akj must satisfy ajkakj = 1.DITTY – p. 13

Step 1

The relative importance between twocriteria is measured according to anumerical scale from 1 to 9.

Value Interpretation

1 j and k are equally important

3 j is slightly more important than k

5 j is strongly more important than k

7 j is very strongly more important than k

9 j is absolutely more important than k

DITTY – p. 14

Step 1

The normalized pairwise comparisonmatrix Anorm is derived by A by makingequal to 1 the sum of the entries on eachcolumn.

DITTY – p. 15

Step 1

The normalized pairwise comparisonmatrix Anorm is derived by A by makingequal to 1 the sum of the entries on eachcolumn.

Finally, the objective weight vector w isbuilt by averaging the entries on eachrow of Anorm.

DITTY – p. 15

Implementation

1. Computing the vector of objectiveweights

2. Computing the matrix of scenarioscores

3. Ranking the scenarios

4. Checking the consistency

DITTY – p. 16

Step 2

The matrix of scenario scores S [n × m].

Each entry sij of S represents the scoreof the scenario i with respect to thecriterion j.

The score matrix S is obtained by thecolumns sj calculated as follows:

DITTY – p. 17

Step 2

A pairwise comparison matrix Bj is builtfor each criterion j.

Each entry bjih represents the evaluation

of the scenario i compared to thescenario h with respect to the criterion jaccording to the DM’s evaluations.

From each matrix Bj a score vectors sj

is obtained (as in Step 1).

DITTY – p. 18

Implementation

1. Computing the vector of objectiveweights

2. Computing the matrix of scenarioscores

3. Ranking the scenarios

4. Checking the consistency

DITTY – p. 19

Step 3

Once the weight vector w and the scorematrix S have been computed, the AHPobtains a vector v of global scores bymultiplying S and w, v = S · w.

The i-th entry vi of v represents theglobal score assigned by the AHP to thescenario i.

DITTY – p. 20

Step 3

Once the weight vector w and the scorematrix S have been computed, the AHPobtains a vector v of global scores bymultiplying S and w, v = S · w.

The i-th entry vi of v represents theglobal score assigned by the AHP to thescenario i.

The scenario ranking is accomplished byordering the global scores in decreasingorder.

DITTY – p. 20

Implementation

1. Computing the vector of objectiveweights

2. Computing the matrix of scenarioscores

3. Ranking the scenarios

4. Checking the consistency

DITTY – p. 21

Step 4

When many pairwise comparisons areperformed, inconsistencies may arise.

Consider the case

criterion 1 is slightly more importantthan criterion 2

criterion 2 is slightly more importantthan criterion 3

inconsistency arises if criterion 3 ismore important than criterion 1

DITTY – p. 22

Step 4

The Consistency Index (CI) is obtained:

x is the ratio of the j-th element of thevector A · w to the correspondingelement of the vector w

CI is the the average of the x

A perfectly consistent DM should alwaysobtain CI = 0, but inconsistenciessmaller than a given threshold aretolerated.

DITTY – p. 23

OutlineAnalytic Hierarchy Process

Implementation

Example

Conclusions

DITTY – p. 24

Example

Small example, m = 3 criteria and n = 3scenarios.

DITTY – p. 25

Example

The DM first builds the following pairwisecomparison matrix A for the 3 criteria:

A =

1 3 5

1/3 1 3

1/5 1/3 1

to which corresponds the weight vectorw = [0.633 0.261 0.106]T .

DITTY – p. 26

Example

Then, the DM builds the followingpairwise scenario comparison matricesfor the first criterion:

B1 =

1 3 7

1/3 1 5

1/7 1/5 1

to which correspond the score vectors1 = [0.643 0.283 0.074]T .

DITTY – p. 27

Example

Then, the DM builds the followingpairwise scenario comparison matricesfor the second criterion:

B2 =

1 1/5 1

5 1 5

1 1/5 1

to which correspond the score vectors2 = [0.143 0.714 0.143]T .

DITTY – p. 28

Example

Then, the DM builds the followingpairwise scenario comparison matricesfor the third criterion:

B3 =

1 5 9

1/5 1 3

1/9 1/3 1

to which correspond the score vectors3 = [0.748 0.180 0.072]T .

DITTY – p. 29

Example

Hence, the score matrix S is

S = [s1s2s3] =

0.643 0.143 0.748

0.283 0.714 0.180

0.074 0.143 0.072

and the global score vector isv = S · w = [0.523 0.385 0.092]T .

DITTY – p. 30

Example

The rank is:

scenario 1 (0.523)

scenario 2 (0.385)

scenario 3 (0.092)

Note that the first scenario results to bethe most preferable, though it is theworst of the three with respect to thesecond criterion.

DITTY – p. 31

OutlineAnalytic Hierarchy Process

Implementation

Example

Conclusions

DITTY – p. 32

Main advantages

Flexible tool

multiple and conflictingobjectives/criteria

qualitative and quantitative aspects ofa decision

DITTY – p. 33

Main advantages

Flexible tool

multiple and conflictingobjectives/criteria

qualitative and quantitative aspects ofa decision

Driven by the Decision Maker

reduces complex decisions to a seriesof pairwise comparisons

computations guided by the DM’sexperience DITTY – p. 33

Main advantages

Robust

capture both subjective and objectiveaspects of a decision

incorporates a technique for checkingthe consistency of the DM’sevaluations

DITTY – p. 34

Main advantages

Robust

capture both subjective and objectiveaspects of a decision

incorporates a technique for checkingthe consistency of the DM’sevaluations

Easy to implement

few matrix manipulations are required

DITTY – p. 34

Drawback

The AHP may require a large number ofevaluations by the DM, especially forlarge problems.

Every evaluation requires the DM toexpress how well two scenarios orcriteria compare to each other...

DITTY – p. 35

Drawback

The AHP may require a large number ofevaluations by the DM, especially forlarge problems.

Every evaluation requires the DM toexpress how well two scenarios orcriteria compare to each other......but the number of pairwisecomparisons grows quadratically withthe number of criteria and scenarios.

DITTY – p. 35

DrawbackAn example

With n = 10 scenarios and m = 4objectives.

Pairwise comparisons required:

Weight vector 4 · 3/2 = 6

Score matrix 4 · (10 · 9/2) = 180

DITTY – p. 36

DrawbackHowever, it may be completely orpartially automated for deciding somepairwise comparisons.

DITTY – p. 37

DrawbackHowever, it may be completely orpartially automated for deciding somepairwise comparisons.

Bjab = 1 + 8(

xja−x

jb

max xji−min x

ji

)

DITTY – p. 37

DrawbackHowever, it may be completely orpartially automated for deciding somepairwise comparisons.

Bjab = 1 + 8(

xja−x

jb

max xji−min x

ji

)

Different behaviours may beimplemented.

DITTY – p. 37