Johanna Lind and Anna Schurba

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Facility Location Planning using the Analytic
Hierarchy Process

• Specialisation Seminar
• Facility Location Planning
• Wintersemester 2002/2003

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

inital costs
costs of energy
subcriteria
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Pairwise Comparison Matrix
Pairwise comparisons
Pairwise Comparison Matrix A ( aij )
Values for aij
2,4,6,8 gt
intermediate values
reciprocals gt
reverse comparisons
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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
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Pairwise Comparisons Matrix
Pairwise comparisons matrix with respect to
criterion costs
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?23?10(?5)(?2)0,
  therefore ? 5 is the principal/maximum
  eigenvalue.
• Further, x14x2 must be equal 5x1, thus the
  principal eigenvector is
• Check for scalar1
• For large n approximation techniques are
  necessary.

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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)

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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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Estimating Consistency (2)

• Pairwise comparisons nxn matrix (for ngt2) is
  consistent if
• e.g.
• For ngt2 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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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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Estimating Consistency (4)

• Consistency Ratio is the ratio of the consistency
  index to the corresponding random index
• CRCI / 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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Estimating Consistency (5)

• Example for n3
• consistent ?max3.00, CI0.00
• inconsistent/ ?max3.05, CI0.05
• transitive
• intransitive ?max3.93, CI0.80

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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 Ax
  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 Ax?x calculate an approximate
  value of scalar ?.
• For the matrix from the example
• ?max3.05, CI0.025, CR0.025 / 0.580.043
  (acceptable).

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Overall Priority Ranking

• 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)0.670.160.200.250.330.590.3
  5. Priority(Frankfurt)0.65, thus Frankfurt
  should be selected.

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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
  (gtcreating a vector of priorities)
• Analyse of deviation from a consistency
  (gt 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 gt Decision maker can focus on smaller
  sets of decisions .
  (Millers 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 equation
det(?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 Avnv.
• 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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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 Avnv 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.