AHP fundamentals

Multicriteria Decision AnalysisMCDA

Pablo Aragonés Beltrán and

Mónica García Melón

ANALYTIC HIERARCHY PROCESSAHP

MEASURING DM´S PREFERENCES (Saaty y Peniwati, 2008)

There are two ways to measure something about a property (or criteria). One is to apply an existing scale (distance in meters, weight in kg, price or costs in

Euros, etc.) Another is to compare the measuring object with another that is comparable to it

with respect to the property that is being measured. In this case an object serves as a measuring unit and the other is estimated as a multiple of that unit using the expert judgment of the DM.

If you assign a number to something when there is no measurement scale, this number is arbitrary and a meaningless measure.

According to the decision process described above, first we have to measure the importance of the criteria and in a second step measure how each alternative satisfies the criteria.

Finally we should combine those measures to obtain an aggregated measure which ranks alternatives from highest to lowest preference (or priority).

INTRODUCTION

AHP is a well known Multicriteria Decision Technique It allows to make decisions considering different points of view and criteria. It was proposed by Professor Thomas Saaty, University of Pittsburgh, in the late

70s. It is based on the idea that the complexity inherent in a decision making problem

with multiple criteria can be solved by the hierarchy of the problems. At each level of the hierarchy, pairwise comparisons are made between the

elements of the same level, based on the importance or contribution of each of them to the element of the upper level to which they are linked.

This comparison process leads to a measurement scale of priorities or weights on the elements.

Pairwise comparisons were performed using ratios of preference (in comparison alternatives), and ratios of importance (when compared criteria), which are evaluated on a numerical scale by the method proposed.

INTRODUCTION

The method allows to analyze the degree of inconsistency of the judgments of the decision maker.

According Th Saaty, AHP is a theory of relative measurement of intangible criteria. AHP provides a method for measuring Decisor judgments or preferences, which in

turn are always subjective. Although criteria can be measured on a physical scale (full scale: kg, m, Euros,

etc.), the meaning that measure has to the decision maker is always subjective. E.g. we can not tell if an object weighting 50 kg is very heavy or very light without knowing in what context is this measure, for what purpose and with what other objects you are comparing.

AXIOMAS BÁSICOS DEL AHP

The decision maker must be able to make comparisons and to establish the strength of his preferences. The intensity of these preferences must satisfy the reciprocal condition: If A is x times more preferred than B, then B is 1 / x times more preferred than A.

Reciprocal comparison

The elements of a hierarchy must be comparable. The preferences are represented by a scale of limited comparability.

Homogeneity

When preferences are expressed, it is assumed that the criteria are independent of the properties of the alternatives.Independency

For the purpose of decision making, it is assumed that the hierarchy is complete. All elements (criteria and alternatives) of the problem are taken into account by the decision maker.

Expectations

2. STEPS OF THE METHOD

AHP FUNDAMENTS

Arranging the MCDA problem in three hierarchical levels:– Level:1. Main objective.– Level 2. Criteria.– Level 3. Alternatives.

Prioritization by pairwise comparisons of the elements of the same level:Each criterion or alternative i is compared to each criterion or alternative j. The following question has to be answered:

Is criterion i equal, more or less important than criterion j ? With respect to criterion j, is alternative i equal, more or less preferred than

alternative j ?

Comparisons are made folowing one specific scale.

1

2

AHP FUNDAMENTS

Construction of the decision matrix.

Calculation of overall priorities associated with each alternative. Aggregation by weighted sum of priorities obtained in each level:

3

4p x pi ij

jj

1

n

GOAL

C1

C11

C2

C12 C21 C22 C23

A1 A2 A3

STEP 1.- Arranging the MCDA problem as a hierarchy

STEP 2.- ESTABLISHING PRIORITIES

Prioritization by pairwise comparisons of the elements of the same level:Each criterion or alternative i is compared to each criterion or alternative j. The following question has to be answered:

Is criterion i equal, more or less important than criterion j ? With respect to criterion j, is alternative i equal, more or less preferred than

alternative j ?

Comparisons are made folowing one specific scale.

2

PAIRWISE COMPARISON SCALE

Is criterion i equal, more or less important than criterion j ?

We answer with the following scale:

1 Same importance3 Moderately more important5 Strongly more important7 Very strongly more important9 Extremely more important2, 4, 6, 8 Intermediate values

If criterion (alternative) i dominates j strongly, then: aij=5 and aji=1/5

We only need to do n(n-1)/2 comparisonsbeing n nr. of elements


C1 C2 C3

C1 1 a12 a13

C2 1/ a12 1 a23

C3 1/ a13 1/ a23 1

We compare element i with element j:1 Same importance3 Moderately more important5 Strongly more important7 Very strongly more important9 Extremely more important2, 4, 6, 8 Intermediate values

Is criterion i equal, more or less important than criterion j ?

Example of pairwise comparison matrix

C1 C2 C3

C1 1 6 3

C2 1/ 6 1 1/2

C3 1/ 3 2 1

Elements are compared among them:• C1 is between strongly and very strongly

better than C2

• C1 is moderately better than C3

• C3 is between equal and moderately

better than C2



Example of pairwise comparison matrix

Homogeneity: rii = 1

Reciprocity: rij · rji = 1

Transitivity: rij · rjk = rik

PROPERTIES OF PAIRWISE COMPARISON MATRICES

How do you get a ranking of priorities from a pairwise matrix? Dr Thomas Saaty, demonstrated mathematically that the Eigenvector

solution was the best approachReference : The Analytic Hierarchy Process, 1990, Thomas L. Saaty

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.


Mathematical demonstration for the Eigenvector

Through a pairwise comparison matrix a reciprocal matrix is constructed:

aij = 1 / aij

If the decision maker is consistent (ideal DM): aij = ai / aj

a1/a1

a2 /a1

an /a1

.

.

.

a1/a2

a2 /a2

an /a2

.

.

.

...

...

...

.

.

.

a1/an

a2 /an

an /an

.

.

.

a1

a2

an

.

.

.= n

a1

a2

an

.

.

.

aij aij ajk = aik

i, j, k(transitivity)

Let A be a nxn matrix of judgements. We call Eigenvalues of A (λ1, λ2, …, λn) the solutions to the equation: det (A-λI) = 0

The principal Eigenvalue (λmax) is the maximum of the Eigenvalues. n is the dominant Eigenvaluees of [A] and [a] is the asociated Eigenvector (ideal

case)

If there is no consistency, the matrix of judgements becomes [R] a perturbation of

[A] and fulfills: [R] · [a] = max · [a] ( max dominant Eigenvalue + and [a] its Eigenvector)

THE EIGENVECTOR ASSOCIATED TO THE DOMINANT EIGENVALUE IS THE WEIGHTS VECTOR

Mathematical demonstration for the Eigenvector

e11 C1 C2 C3 priority

C1 1 r12 r13 W1

C2 r21 1 r23 W2

C3 r31 r32 1 W3

Pairwise comparison matrix

After verifying the consistency the priority vector will be the principal eigenvector of the matrix.

An example of calculation of priorities among elements

e11 C1 C2 C3 priority

C1 1 9 3 0,692

C2 1/9 1 1/3 0,077

C3 1/3 3 1 0,231

Pairwise comparison matrix


After verifying the consistency the priority vector will be the principal eigenvector of the matrix.

A C1 C2 C3 SF SFN

C1 1 2 0,33 3,33 0,238

C2 0,50 1 0,20 1,70 0,121

C3 3 5 1 9,00 0,641

Example

A2 C1 C2 C3 SF SFN

C1 3 5,66 1,07 9,73 0,230

C2 1,60 3 0,57 5,17 0,122

C3 8,50 16 3 27,50 0,649

A3 C1 C2 C3 SF SFN

C1 9,03 17 3,20 29,23 0,230

C2 4,80 9,03 1,70 15,53 0,122

C3 25,50 48 9,03 82,53 0,648

A4 C1 C2 C3 SF SFN

C1 27,13 51,07 9,61 87,81 0,230

C2 14,42 27,13 5,11 46,66 0,122

C3 76,60 144,2 27,13 247,9 0,648

w1

w2

w3


CALCULATION OF THE CONSISTENCY RATIO

Consistency Index (CI):1n

nλCI max

n 1 2 3 4 5 6 7 8 9 10

RI 0 0 0,525 0,882 1,115 1,252 1,341 1,404 1,452 1,484

λmax: Principal Eigenvalue

n: dimension of the matrix

Random Consistency Index (RI):


Consistency Ratio (CR):RICICR

CI: Consistency IndexRI: Random Consistency Index

Accepted if: CR≤0,05 with n=3

CR≤0,08 with n=4

CR≤0,10 with n≥5

CALCULATION OF λmax

Vector B Vector CMatrix A x =

C/B Vector D=Arithmetic mean

λmax = 3,004

1,9480,3670,690

0,6480,1220,230

1000,5000,3200,01500,0333,0000,21

x

3,0073,0013,003

81,948/0,6420,367/0,1200,690/0,23

3.- Multiply matrix A original by vector B (Eigenvector) to obtain vector C.4.- Divide vector C by vector B component by component to obtain vector D.5.- The aritmetic mean of vector D components is the aproximate value of λmax.

e1 e2 e3

e1 1 2 1/3

e2 1/2 1 1/5

e3 3 5 1

λmax = 3,004 CI = (3,004-3)/(3-1) = 0,002

n = 3 RI = 0,525

CR = 0,002/0,525 = 0,004 < 0,05

Example


Step 3. Construction of the decision matrix

Construction of the decision matrix.3.1 Weighting of criteria.3.2 Assessment of alternatives.

3

GOAL

C1 C2

GOAL C1 C2

C1 1 r12

C2 1/r12 1

Step 3. Construction of the decision matrix.Weighting of criteria

GOAL

C1 C2

GOAL C1 C2

C1 1 5

C2 1/5 1

If criterion i dominates criterion j , e.g. with a 5 (strongly) then, aij = 5 and aji= 1/5

C1 is strongly more important than C2


GOAL

C1 C2

Goal C1 C2

C1 1 5

C2 1/5 1

C1 C2 media geom mgeo normC1 1,000 5,000 2,236 0,833

C2 0,200 1,000 0,447 0,167


C1

C11 C12

C1 C11 C12

C11 1 r11-12

C12 1/r11-12 1

C2

C21 C22 C23

C2 C21 C22 C23

C21 1 r21-22 r21-23

C22 1/r21-22 1 r22-23

C23 1/r21-23 1/r22-23 1


C1

C11 C12

C1 C11 C12

C11 1 1/3C12 3 1

C2

C21 C22 C23

C2 C21 C22 C23

C21 1 5 1/3

C22 1/5 1 1/9

C23 3 9 1C1 C2 media geom mgeo norm

C1 1,000 0,333 0,577 0,250

C2 3,000 1,000 1,732 0,750

C1 C2 C3 media geom mgeo normC1 1,000 5,000 0,333 1,186 0,265

C2 0,200 1,000 0,111 0,281 0,063

C3 3,000 9,000 1,000 3,000 0,672


GOAL

C1 C2

C11 C12

0.883 0.167

0.250 0.750

1

Local weight

0.883 0.167

0.221 0.662

1Global weight

=

X

X C21

0.2650.044

C22

0.0630.011

C23

0.6720.112


C21 A1 A2 A3

A1 1 a12 a13

A2 1/ a12 1 a23

A3 1/ a13 1/ a23 1

C21

A1A2 A3

Assessment of alternatives for C21

¿ Is alternative i equal, …, more important than alternative j ?

Step 3. Construction of the decision matrix.Assessment of alternatives


Assessment of alternatives for C21

C21 A1 A2 A3

A1 1 6 3

A2 1/ 6 1 1/2

A3 1/ 3 2 1C21

A1 A2 A3

With respect to criterion C21:• A1 is between strongly and very strongly

better than A2

• A1 is moderately better than A3

• A3 es between equal and moderately better than A2

C1 C2 C3 media geom mgeo normC1 1,000 6,000 3,000 2,621 0,667

C2 0,167 1,000 0,500 0,437 0,111

C3 0,333 2,000 1,000 0,874 0,222


Step 3. Construction of the decision matrix.Assessment of alternatives

THE DECISION MATRIX

C11 C12 C21 C22 C23

WEIGHTS 0,221 0,662 0,044 0,011 0,112

A1 0,316 0,105 0,667 0,143 0,072

A2 0,386 0,258 0,111 0,429 0,649

A3 0,298 0,637 0,222 0,429 0,279

STEP 4. CALCULATION OF OVERALL PRIORITIES

Calculation of overall priorities associated with each alternative. Aggregation by weighted sum of priorities obtained in each level:

4p x pi ij

jj

1

n

C11 C12 C21 C22 C23

pesos 0,221 0,662 0,044 0,011 0,112

A1 0,316 0,105 0,667 0,143 0,072

A2 0,386 0,258 0,111 0,429 0,649

A3 0,298 0,637 0,222 0,429 0,279

U(A1) = 0,221 x 0,316 + 0,662 x 0,105 + 0,044 x 0,308 + 0,011 x 0,143 + 0,112 x 0,072 = 0,163U(A2) = 0,221 x 0,386 + 0,662 x 0,258 + 0,044 x 0,231 + 0,011 x 0,429 + 0,112 x 0,649 = 0,344U(A3) = 0,221 x 0,298 + 0,662 x 0,637 + 0,044 x 0,308 + 0,011 x 0,429 + 0,112 x 0,279 = 0,544

STEP 4. CALCULATION OF OVERALL PRIORITIES

GLOBAL PRIORITIES

0,163

0,344

0,544

0,000 0,100 0,200 0,300 0,400 0,500 0,600

A1

A2

A3

AHP fundamentals

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