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Analytic Hierarchy Process

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- 1. Analytic Hierarchy Process Zheng-Wen Shen 2006/04/11
- 2. Outline1. Introduction of AHP2. How the AHP works3. Example
- 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. 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. 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. AHP Steps1. Computing the vector of objective weights2. Computing the matrix of scenario scores3. Ranking the scenarios4. Checking the consistency
- 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. 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. Step 1: Computing the vector of objective weightsPreferences on ObjectivesWeights on Objectives
- 10. AHP Steps1. Computing the vector of objective weights2. Computing the matrix of scenario scores3. Ranking the scenarios4. Checking the consistency
- 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. Step 2: Computing the matrix of scenario scoresLocation scores Relative Location scores Relative scores for each objective
- 13. AHP Steps1. Computing the vector of objective weights2. Computing the matrix of scenario scores3. Ranking the scenarios4. Checking the consistency
- 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. Step 3: Ranking the scenariosWeights on ObjectivesRelative scores for each objectiveABC: .335 D: .238
- 16. AHP Steps1. Computing the vector of objective weights2. Computing the matrix of scenario scores3. Ranking the scenarios4. Checking the consistency
- 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. 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 xA perfectly consistent DM should always obtain CI = 0 but inconsistencies smaller than a given threshold are tolerated.
- 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. Example (2/7) pairwise comparison matrix A for the 3 criteria Weight Vector
- 21. Example (3/7) pairwise scenario comparison matrices for the first criterion: Score Vector
- 22. Example (4/7) pairwise scenario comparison matrices for the first criterion: Score Vector
- 23. Example (5/7) pairwise scenario comparison matrices for the first criterion: Score Vector
- 24. Example (6/7) Score Matrix S is : Global Score Vector
- 25. Example (7/7) The rank is: Scenario 1: 0.523 Scenario 2: 0.385 Scenario 3: 0.092

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