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Facility Location Planning using
the Analytic Hierarchy Process
Specialisation Seminar
„Facility Location Planning“
Wintersemester 2002/2003
presented by
Johanna Lind and Anna Schurba
The Analytic Hierarchy Process
Table of contents
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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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Facility location planning using the AHP
The Analytic Hierarchy Process
Introduction: What is the AHP?
The Analytic Hierarchy Process developed by T. L. Saaty
(1971) is one of practice relevant techniques of the
hierarchical additive weighting methods for multicriteria
decision problems.
• Decomposing a decision into smaller parts
• Pairwise comparisons on each level
• Synthesising judgements
The method has been applied in many areas.
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Facility location planning using the AHP
The Analytic Hierarchy Process
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 multicriteria decision problems.
Applying in quite a simple way
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Facility location planning using the AHP
The Analytic Hierarchy Process
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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Facility location planning using the AHP
The Analytic Hierarchy Process
Developing the Hierarchy
Structuring a hierarchy:
goal
Selecting
best Location
Costs
inital
costs
Market
costs of
energy
Berlin
6
Transport
criteria
subcriteria
Frankfurt
alternatives
Facility location planning using the AHP
The Analytic Hierarchy Process
Pairwise Comparison Matrix
Pairwise comparisons:
to
A1
A2
A3
Alternative 1 (A1)
a11
a12
a13
Alternative 2 (A2)
a21
a22
a32
Alternative 3 (A3)
a31
a32
a33
Pairwise Comparison Matrix A = ( aij )
Values for aij :
Numerical values
7
Verbal 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 => intermediate
values
reciprocals => reverse
comparisons
Facility location planning using the AHP
The Analytic Hierarchy Process
Pairwise Comparisons
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
Pairwise
comparisons of
the criteria:
8
costs market transport
costs
1
1/2
1/3
market
2
1
1/3
transport
3
3
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Facility location planning using the AHP
The Analytic Hierarchy Process
Pairwise Comparisons Matrix
costs
Pairwise comparisons matrix
with respect to criterion costs:
Berlin
Frankfurt
Pairwise comparisons matrix
with respect to criterion market:
market
9
Frankfurt
1
2
1/2
1
Berlin
Frankfurt
Berlin
1
1/4
Frankfurt
4
1
transport
Pairwise comparisons matrix with
respect to criterion transport:
Berlin
Berlin
Frankfurt
Berlin
1
1/2
Frankfurt
2
1
Facility location planning using the AHP
The Analytic Hierarchy Process
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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Facility location planning using the AHP
The Analytic Hierarchy Process
Synthesising Judgements (2)
• Eigenvalues of A are all scalars  satisfying det(I - A)=0.
• For a 2x2 matrix one should solve a quadratic equation:
1 4
A
 
3
2


  1  4 
I  A  
,

3


2


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
1
•
x  scalar*  .
1
Check for scalar=1:
1 4 1 1  4  5
Ax  
 1  3  2  5  x.
3
2

  
  
• For large n approximation techniques are necessary.
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Facility location planning using the AHP
The Analytic Hierarchy Process
Synthesising Judgements (3)
• To compute a good estimate of the principal eigenvector 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):
c
m
t
c
1
1/ 2
1/ 3
m
2
1
1/ 3
t
3
3
1
sum 6.00 4.50 1.67
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c
c

m
t
0.17 0.11 0.20
m 0.33 0.22 0.20
t
0.50 0.67 0.60
(norm alised)
c 0.16
 m 0.25
t 0.59
Facility location planning using the AHP
The Analytic Hierarchy Process
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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Facility location planning using the AHP
The Analytic Hierarchy Process
Estimating Consistency (2)
• Pairwise comparisons nxn matrix (for n>2) is consistent if
e.g.  1 2 4
aik  aij a jk
i, j, k  1,...,n
1 / 2 1 2
1 / 4 1 / 2 1 
• 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.
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Facility location planning using the AHP
The Analytic Hierarchy Process
Estimating Consistency (3)
• 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:
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n
3
4
5
6
7
8
9
10
RI(n)
0.58
0.90
1.12
1.24
1.32
1.41
1.45
1.49
Facility location planning using the AHP
The Analytic Hierarchy Process
Estimating Consistency (4)
• 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.
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Facility location planning using the AHP
The Analytic Hierarchy Process
Estimating Consistency (5)
• Example for n=3:
2 4
 1
1 / 2 1 2


1 / 4 1 / 2 1
2 8
 1
1 / 2 1 2


1 / 8 1 / 2 1
2 1 / 4
 1
1 / 2 1
2 


 4 1 / 2 1 
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consistent
max=3.00, CI=0.00
inconsistent/
transitive
max=3.05, CI=0.05
intransitive
max=3.93, CI=0.80
Facility location planning using the AHP
The Analytic Hierarchy Process
Estimating Consistency (6)
• 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:
0.17 0.11 0.20 0.16
 0.33 0.22 0.20 * 0.25

 

0.50 0.67 0.60 0.59

 0.48
 0.16
0.77 1.82 
 3.00 3.08 3.08
0.25 0.59 
max=3.05, CI=0.025, CR=0.025 / 0.58=0.043 (acceptable).
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Facility location planning using the AHP
The Analytic Hierarchy Process
Overall Priority Ranking
• The overall priority of an alternative is computed by multiplying 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.
Berlin
Frankfurt
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Costs
0.16
0.67
0.33
Criteria
Market
0.25
0.20
0.80
Overall
Transport
0.59
0.33
0.67
0.35
0.65
Facility location planning using the AHP
The Analytic Hierarchy Process
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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Facility location planning using the AHP
The Analytic Hierarchy Process
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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Facility location planning using the AHP
The Analytic Hierarchy Process
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 equation
det(I - A) of the nth order )
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Facility location planning using the AHP
The Analytic Hierarchy Process
THANK YOU
FOR YOUR ATTENTION!
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Facility location planning using the AHP
The Analytic Hierarchy Process
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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Facility location planning using the AHP
The Analytic Hierarchy Process
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.
vj
• Thus,
aij *  1 i, j  1,...,n
vi
• Sum over j:
n

j 1
aij
vj
vi
n
i  1,...,n
n
or

j 1
aij v j  n vi
i  1,...,n
• 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.
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Facility location planning using the AHP
The Analytic Hierarchy Process
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
vj
aij *
will not be
1
for all i, j .
vi
• 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.
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Facility location planning using the AHP