presented by Johanna Lind and Anna Schurba

Facility Location Planning using the Analytic Hierarchy Process

Specialisation Seminar

„Facility Location Planning“

Wintersemester 2002/2003

The Analytic Hierarchy Process

Facility location planning using the AHP2

Table of contents

• Introduction• Key steps of the method• Step 1 – Developing a hierarchy• Step 2 - Pairwise comparisons and Pairwise comparisons matrix• Step 3 - Synthesising judgements and Estimating consistency• Step 4 – Overall priority ranking• Summary

• Appendix

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Introduction: What is the AHP?

The Analytic Hierarchy Process developed by T. L. Saaty(1971) is one of practice relevant techniques of thehierarchical additive weighting methods for multicriteriadecision problems.

The method has been applied in many areas.

• Decomposing a decision into smaller parts

• Synthesising judgements

• Pairwise comparisons on each level

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Introduction: Why the AHP?

FLP-problems involve an extensive decision function for a

firm/ company since a multiplicity of criteria and

requests are to be considered.

• How to weight these decision criteria appropriately in order to archieve an optimal facility location?

• Problem: There are not only quantitative but also qualitative factors that have to be measured.

The AHP is a comprehensive and flexible tool for complex multi-criteria decision problems.

Applying in quite a simple way

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Key Steps of the Method

Three key steps of the AHP:

1. Decomposing the problem into a hierarchy – one overall goal on the top level, several decision alternatives on the bottom level and several criteria contributing to the goal

2. Comparing pairs of alternatives with respect to each criterion and pairs of criteria with respect to the achievement of the overall goal

3. Synthesising judgements and obtaining priority rankings of the alternatives with respect to each criterion and the overall priority ranking for the problem

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Developing the Hierarchy

Structuring a hierarchy:

Costs Market Transport

FrankfurtBerlin

Selectingbest Location

goal

criteria

alternatives

subcriteriainital costs

costs of energy

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Pairwise Comparison Matrix

Pairwise comparisons:

Pairwise Comparison Matrix A = ( aij )

a33a32a31Alternative 3 (A3)

a32a22a21Alternative 2 (A2)

a13a12a11Alternative 1 (A1)

A3A2A1to

Values for aij :

Numerical valuesVerbal judgement of

preferences

1 equally important

3 weakly more important

5 strongly more important

7 very strongly more important

9 absolutely more important

2,4,6,8 =>

reciprocals =>

intermediate values

reverse comparisons

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Pairwise Comparisons

costs market transport

costs 1 1/2 1/3

market 2 1 1/3

transport 3 3 1

Pairwise comparisons of the criteria:

For all i and j it is necessary that:

(a) aii = 1 A comparison of criterion i with itself: equally important

(b) aij = 1/ aji aji are reverse comparisons and must be the reciprocals of aij

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Pairwise Comparisons Matrix

Pairwise comparisons matrix with respect to criterion costs:

11/2Frankfurt

21Berlin

FrankfurtBerlincosts

14Frankfurt

1/41Berlin

FrankfurtBerlinmarket

12Frankfurt

1/21Berlin

FrankfurtBerlintransport

Pairwise comparisons matrix with respect to criterion market:

Pairwise comparisons matrix with respect to criterion transport:

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Synthesising Judgements (1)

• Relative priorities of criteria with respect to the overall goal and those of alternatives w.r.t. each criterion are calculated from the corresponding pairwise comparisons matrices.

• A scalar is an eigenvalue and a nonzero vector x is the corresponding eigenvector of a square matrix A if Ax = x.

• To obtain the priorities, one should compute the principal (maximum) eigenvalue and the corresponding eigenvector of the pairwise comparisons matrix.

• It can be shown that the (normalised) principal eigenvector is the priorities vector. The principal eigenvalue is used to estimate the degree of consistency of the data.

• In practice, one can compute both using approximation.

Why approximation?

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Synthesising Judgements (2)

• Eigenvalues of A are all scalars satisfying det(I - A)=0.

• For a 2x2 matrix one should solve a quadratic equation:

det(I - A)=(–1)(–2)–12=2–3–10=(–5)(+2)=0, therefore = 5 is the principal/maximum eigenvalue.

• Further, x1+4x2 must be equal 5x1, thus the principal eigenvector is

• Check for scalar=1:

• For large n approximation techniques are necessary.

,23

41

23

41

AIA

.1

1*

scalarx

.5

5

23

41

1

1

23

41xAx

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Synthesising Judgements (3)

• To compute a good estimate of the principal eigen-vector of a pairwise comparisons matrix, one can either— normalise each column and then average

over each row or

— take the geometric average of each row and normalise the numbers.

• Applying the first method for the example matrix (criteria):

59.0

25.0

16.0

)(60.067.050.0

20.022.033.0

20.011.017.0

67.150.400.6

133

3/112

3/12/11

t

m

c

normalisedt

m

c

tmc

sum

t

m

c

tmc

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Estimating Consistency (1)

• The AHP does not build on “perfect rationality” of judgements, but allows for some degree of inconsistency instead.

• Difference between transitivity and consistency:— transitivity (e.g., in the utility theory): if a is preferred

to b, b is preferred to c, then a is preferred to c(ordinal scale).

— consistency: if a is twice more preferable than b, b is

twice more preferable than c, then a is four times more preferable than c (cardinal scale).

• 2x2 pairwise comparisons matrix is consistent by construction.

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• Pairwise comparisons nxn matrix (for n>2) is consistent if

e.g.

• For n>2 a consistent pairwise comparisons matrix can be generated by filling in just one row or column of the matrix and then computing other entries.

• It can be shown that the principal eigenvalue max of such a matrix will be n (number of items compared).

• If more than one row/column are filled in manually, some inconsistency is usually observed.

• Deviation of max from n is a measure of inconsistency in the pairwise comparisons matrix.

Estimating Consistency (2)

nkjiaaa jkijik ,...,1,,

12/14/1

212/1

421

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• Consistency Index is defined as follows:

CI = (max – n) / (n – 1)

(Deviation max from n is a measure of inconsistency.)

• Random Index (RI) is the average consistency index of 100 randomly generated (inconsistent) pairwise comparisons matrices. These values have been tabulated for different values of n:

Estimating Consistency (3)

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• Consistency Ratio is the ratio of the consistency index to the corresponding random index:

CR=CI / RI(n)

• CR of less than 0.1 (“10% of average inconsistency” of randomly generated pairwise comparisons matrices) is usually acceptable.

• If CR is not acceptable, judgements should be revised. Otherwise the decision will not be adequate.

Estimating Consistency (4)

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• Example for n=3:

consistent max=3.00, CI=0.00

inconsistent/ max=3.05, CI=0.05transitive

intransitive max=3.93, CI=0.80

Estimating Consistency (5)

12/14/1

212/1

421

12/18/1

212/1

821

12/14

212/1

4/121

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• To compute an estimate of max for a pairwise comparisons matrix: — multiply the normalised matrix with the priorities vector,

(principal eigenvector of the matrix), i.e., obtain A*x; — divide the elements in the resulting vector by the

corresponding elements of the vector of priorities and take the average, i.e., from the equivalence A*x=*x calculate an approximate value of scalar .

• For the matrix from the example:

max=3.05, CI=0.025, CR=0.025 / 0.58=0.043 (acceptable).

Estimating Consistency (6)

08.308.300.359.0

82.1

25.0

77.0

16.0

48.0

59.0

25.0

16.0

*

60.067.050.0

20.022.033.0

20.011.017.0

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• The overall priority of an alternative is computed by mul-tiplying its priorities w.r.t each criterion with the priority of the corresponding criterion and summing up the numbers:

Priority Alternative i = (Priority Alternative i w.r.t. Criterion j)*

*(Priority Criterion j)

• Priority(Berlin)=0.67*0.16+0.20*0.25+0.33*0.59=0.35. Priority(Frankfurt)=0.65, thus Frankfurt should be selected.

Overall Priority Ranking

OverallCosts Market Transport0.16 0.25 0.59

Berlin 0.67 0.20 0.33 0.35Frankfurt 0.33 0.80 0.67 0.65

Criteria

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Summary (1)

• Identification of levels: goal, criteria, (subcriteria) and alternatives

• Developing a hierarchy of contributions of each level to another

• Pairwise comparisons of criteria/ alternatives with each other

• Determining the priorities of the alternatives/ criteria/ (subcriteria) from pairwise comparisons (=>creating a vector of priorities)

• Analyse of deviation from a consistency (=> Measurement of inconsistency)

• Overall priority ranking and decision

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Summary (2)

Advantages of the AHP:

• The AHP has been developed with consideration of the way a human mind works: Breaking the decision problem into levels => Decision maker can focus on smaller sets of decisions . (Miller‘s Law: Humans can only compare 7+/-2 items at a time)

• AHP does not need perfect rationality of judgements. Degree of inconsistency can be assessed.

• AHP is in the position to include and measure also the qualitative factors as well.

Important for modelling of a mathematical decision process based on numbers

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Summary (3)

Remarks concerning the exact solution of the priorities

vector:

For a large number of alternatives/ criteria: Approximation methods or Software package Expert Choice

( difficulties with solving an equationdet(I - A) of the nth order )

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THANK YOU

FOR YOUR ATTENTION!

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Appendix (1)

• Relative priorities of criteria with respect to the overall goal and those of alternatives w.r.t. each criterion are calculated from the corresponding pairwise comparisons matrices.

• To obtain the priorities, one should compute the principal (maximum) eigenvalue and the corresponding normalised eigenvector of the pairwise comparisons matrix.

Why eigenvectors/eigenvalues?

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Appendix (2)

• Let vi denote the “true/objective value” of selecting an alternative or criterion i out of n. Assume all vi are known.

• Then the entry aij for a pair i,j in the pairwise comparisons nxn matrix will be equal vi/vj.

• Thus,• Sum over j:

• The last formula in matrix notation: Av=nv.• In matrix theory such vector v of “true values” is called an

eigenvector of matrix A with eigenvalue n.• Some facts of matrix theory allow to conclude that n will be the

maximum/principal eigenvalue.

njiv

va

i

jij ,...,1,1*

nivnvaorninv

va ijij

n

ji

jij

n

j

,...,1,...,111

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Appendix (3)

• Consider a case with the “true values” unknown.

• aij will be obtained from subjective judgements and therefore will deviate from the “true ratios” vi/vj, thus

• Sum of n these terms will deviate from n.

• So Av=nv will no longer hold.

• Therefore, compute the principal eigenvector and the corresponding eigenvalue. If the principal eigenvalue does not equal n, then A does not contain the “true ratios”.

• Deviation of the principal eigenvalue max from n is thus a measure of inconsistency in the pairwise comparisons matrix.

.,1* jiallforbenotwillv

va

i

jij