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# 20060411 Analytic Hierarchy Process (AHP)

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Analytic Hierarchy Process

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### 20060411 Analytic Hierarchy Process (AHP)

1. 1. Analytic Hierarchy Process Zheng-Wen Shen 2006/04/11
2. 2. Outline1. Introduction of AHP2. How the AHP works3. Example
3. 3. 1. Introduction of AHP Is job Salary is1 best ? important .. Is Job Location2 best ? is important.. Is Job Long term3 best ? prospect is important.. Is Job4 best ? Interest is important.. Crystal is looking for job…
4. 4. AHP Features AHPis a powerful tool that may be used to make decisions when  multiple and conflicting objectives/criteria are present,  and both qualitative and quantitative aspects of a decision need to be considered. AHP reduces complex decisions to a series of pairwise comparisons.
5. 5. 2. How the AHP works1. Computing the vector of objective weights2. Computing the matrix of scenario scores3. Ranking the scenarios4. Checking the consistencyconsider m evaluation criteria and n scenarios.
6. 6. AHP Steps1. Computing the vector of objective weights2. Computing the matrix of scenario scores3. Ranking the scenarios4. Checking the consistency
7. 7. Step 1: Computing the vector of objective weights Pairwisecomparison matrix A [m × m]. Each entry ajk of A represents the importance of criterion j relative to criterion k:  If ajk > 1, j is more important than k  if ajk < 1, j is less important than k  if ajk = 1, same importance ajk and akj must satisfy ajkakj = 1.
8. 8. Step 1: Computing the vector of objective weights The relative importance between two criteria is measured according to a numerical scale from 1 to 9. A  Anorm (Normalized)
9. 9. Step 1: Computing the vector of objective weightsPreferences on ObjectivesWeights on Objectives
10. 10. AHP Steps1. Computing the vector of objective weights2. Computing the matrix of scenario scores3. Ranking the scenarios4. Checking the consistency
11. 11. Step 2: Computing the matrix of scenario scores The matrix of scenario scores S [n × m] Each entry sij of S represents the score of the scenario i with respect to the criterion j The score matrix S is obtained by the columns sj calculated as follows:  A pairwise comparison matrix Bj is built for each criterion j.  Each entry bjih represents the evaluation of the scenario i compared to the scenario h with respect to the criterion j according to the DM’s evaluations.  From each matrix Bj a score vectors sj is obtained (as in Step 1).
12. 12. Step 2: Computing the matrix of scenario scoresLocation scores Relative Location scores Relative scores for each objective
13. 13. AHP Steps1. Computing the vector of objective weights2. Computing the matrix of scenario scores3. Ranking the scenarios4. Checking the consistency
14. 14. Step 3: Ranking the scenarios Once the weight vector w and the score matrix S have been computed, the AHP obtains a vector v of global scores by multiplying S and w  v = S · w. The i-th entry vi of v represents the global score assigned by the AHP to the scenario i The scenario ranking is accomplished by ordering the global scores in decreasing order.
15. 15. Step 3: Ranking the scenariosWeights on ObjectivesRelative scores for each objectiveABC: .335 D: .238
16. 16. AHP Steps1. Computing the vector of objective weights2. Computing the matrix of scenario scores3. Ranking the scenarios4. Checking the consistency
17. 17. Step 4: Checking the consistency When many pairwise comparisons are performed, inconsistencies may arise.  criterion 1 is slightly more important than criterion 2  criterion 2 is slightly more important than criterion 3  inconsistency arises if criterion 3 is more important than criterion 1
18. 18. Step 4: Checking the consistency The Consistency Index (CI) is obtained:  x is the ratio of the j-th element of the vector A · w to the corresponding element of the vector w  CI is the average of the xA perfectly consistent DM should always obtain CI = 0 but inconsistencies smaller than a given threshold are tolerated.
19. 19. 3. Example (1/7) Small example, m = 3 criteria and n = 3 scenarios. 0 S3 S2 S1 Criterion 1 0 S1 S3 S2 Criterion 2 0 S3 S2 S1 Criterion 3
20. 20. Example (2/7) pairwise comparison matrix A for the 3 criteria Weight Vector
21. 21. Example (3/7) pairwise scenario comparison matrices for the first criterion: Score Vector
22. 22. Example (4/7) pairwise scenario comparison matrices for the first criterion: Score Vector
23. 23. Example (5/7) pairwise scenario comparison matrices for the first criterion: Score Vector
24. 24. Example (6/7) Score Matrix S is : Global Score Vector
25. 25. Example (7/7) The rank is:  Scenario 1: 0.523  Scenario 2: 0.385  Scenario 3: 0.092