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Facility Location Planning using the Analytic Hierarchy Process

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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

- 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

The Analytic Hierarchy Process

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.

The Analytic Hierarchy Process

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

The Analytic Hierarchy Process

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

goal

Selectingbest Location

criteria

Costs

Market

Transport

Berlin

Frankfurt

alternatives

The Analytic Hierarchy Process

Structuring a hierarchy:

inital costs

costs of energy

subcriteria

to

A1

A2

A3

Alternative 1 (A1)

a11

a12

a13

Alternative 2 (A2)

a21

a22

a32

Alternative 3 (A3)

a31

a32

a33

The Analytic Hierarchy Process

Pairwise comparisons:

Pairwise Comparison Matrix A = ( aij )

Values for aij :

2,4,6,8 =>

intermediate values

reciprocals =>

reverse comparisons

The Analytic Hierarchy Process

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:

costs

Berlin

Frankfurt

Berlin

1

2

Frankfurt

1/2

1

market

Berlin

Frankfurt

Berlin

1

1/4

Frankfurt

4

1

transport

Berlin

Frankfurt

Berlin

1

1/2

Frankfurt

2

1

The Analytic Hierarchy Process

Pairwise comparisons matrix with respect to criterion costs:

Pairwise comparisons matrix with respect to criterion market:

Pairwise comparisons matrix with respect to criterion transport:

The Analytic Hierarchy Process

- 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?

The Analytic Hierarchy Process

- 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 = 5is the principal/maximum eigenvalue.

- Further, x1+4x2 must be equal 5x1, thus the principaleigenvector is
- Check for scalar=1:
- For large n approximation techniques are necessary.

The Analytic Hierarchy Process

- 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):

The Analytic Hierarchy Process

- 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 istwice more preferable than c, then a is four times more preferable than c (cardinal scale).
- 2x2 pairwise comparisons matrix is consistent by construction.

The Analytic Hierarchy Process

- 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.

The Analytic Hierarchy Process

- 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:

The Analytic Hierarchy Process

- 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.

The Analytic Hierarchy Process

- Example for n=3:
consistent max=3.00, CI=0.00

inconsistent/ max=3.05, CI=0.05

transitive

intransitive max=3.93, CI=0.80

The Analytic Hierarchy Process

- 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=*xcalculate 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).

The Analytic Hierarchy Process

- 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.

The Analytic Hierarchy Process

- 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

The Analytic Hierarchy Process

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

The Analytic Hierarchy Process

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 )

The Analytic Hierarchy Process

THANK YOU

FOR YOUR ATTENTION!

The Analytic Hierarchy Process

- 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 correspondingnormalised eigenvector of the pairwise comparisons matrix.
Why eigenvectors/eigenvalues?

The Analytic Hierarchy Process

- Let videnote the “true/objective value” of selecting an alternative or criterion i out of n. Assume all viare 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 eigenvaluen.
- Some facts of matrix theory allow to conclude that n will be the maximum/principal eigenvalue.

The Analytic Hierarchy Process

- 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.