1. presented by Johanna Lind and Anna Schurba Facility Location Planning using the Analytic Hierarchy Process Specialisation Seminar „Facility Location Planning“ Wintersemester 2002/2003

2. The Analytic Hierarchy Process Facility location planning using the AHP 2 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

3. The Analytic Hierarchy Process Facility location planning using the AHP 3 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. The method has been applied in many areas. Decomposing a decision into smaller parts Synthesising judgements Pairwise comparisons on each level

4. The Analytic Hierarchy Process Facility location planning using the AHP 4 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

5. The Analytic Hierarchy Process Facility location planning using the AHP 5 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

6. The Analytic Hierarchy Process Facility location planning using the AHP 6 Developing the Hierarchy Structuring a hierarchy: CostsMarketTransport FrankfurtBerlin Selecting best Location goal criteria alternatives subcriteria inital costs costs of energy

7. The Analytic Hierarchy Process Facility location planning using the AHP 7 Pairwise Comparison Matrix Pairwise comparisons: Pairwise Comparison Matrix A = ( a ij ) a 33 a 32 a 31 Alternative 3 (A3) a 32 a 22 a 21 Alternative 2 (A2) a 13 a 12 a 11 Alternative 1 (A1) A3A2A1to Values for a ij : Numerical values Verbal judgement of preferences 1equally important 3weakly more important 5strongly more important 7very strongly more important 9absolutely more important 2,4,6,8 => reciprocals => intermediate values reverse comparisons

8. The Analytic Hierarchy Process Facility location planning using the AHP 8 Pairwise Comparisons costsmarkettransport costs 11/2 1/3 market 21 1/3 transport 33 1 Pairwise comparisons of the criteria: For all i and j it is necessary that: (a) a ii = 1A comparison of criterion i with itself: equally important (b) a ij = 1/ a ji a ji are reverse comparisons and must be the reciprocals of a ij

9. The Analytic Hierarchy Process Facility location planning using the AHP 9 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:

10. The Analytic Hierarchy Process Facility location planning using the AHP 10 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?

11. The Analytic Hierarchy Process Facility location planning using the AHP 11 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, x 1 +4x 2 must be equal 5x 1, thus the principal eigenvector is Check for scalar=1: For large n approximation techniques are necessary.

12. The Analytic Hierarchy Process Facility location planning using the AHP 12 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):

13. The Analytic Hierarchy Process Facility location planning using the AHP 13 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.

14. The Analytic Hierarchy Process Facility location planning using the AHP 14 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)

15. The Analytic Hierarchy Process Facility location planning using the AHP 15 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)

16. The Analytic Hierarchy Process Facility location planning using the AHP 16 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)

17. The Analytic Hierarchy Process Facility location planning using the AHP 17 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 Estimating Consistency (5)

18. The Analytic Hierarchy Process Facility location planning using the AHP 18 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)

19. The Analytic Hierarchy Process Facility location planning using the AHP 19 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

20. The Analytic Hierarchy Process Facility location planning using the AHP 20 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

21. The Analytic Hierarchy Process Facility location planning using the AHP 21 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

22. The Analytic Hierarchy Process Facility location planning using the AHP 22 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 )

23. The Analytic Hierarchy Process Facility location planning using the AHP 23 THANK YOU FOR YOUR ATTENTION!

24. The Analytic Hierarchy Process Facility location planning using the AHP 24 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?

25. The Analytic Hierarchy Process Facility location planning using the AHP 25 Appendix (2) Let v i denote the “true/objective value” of selecting an alternative or criterion i out of n. Assume all v i are known. Then the entry a ij for a pair i,j in the pairwise comparisons nxn matrix will be equal v i /v j. 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.

26. The Analytic Hierarchy Process Facility location planning using the AHP 26 Appendix (3) Consider a case with the “true values” unknown. a ij will be obtained from subjective judgements and therefore will deviate from the “true ratios” v i /v j, 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.