20060411ahp 0411-130118075335-phpapp01
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The Analytic Hierarchy Process (AHP) is a decision-making tool that breaks down complex decisions into a series of pairwise comparisons. It allows both qualitative and quantitative factors to be considered. The AHP computes weights for objectives, scores scenarios against each objective, and combines them to produce global scores and a ranking of scenarios. It also checks for consistency in the pairwise comparisons to validate the results. An example illustrates how AHP works through its four steps: computing objective weights, scoring scenarios, ranking by global scores, and consistency checking.
![Step 1: Computing the vector of
objective weights
? Pairwise comparison matrix A [m ¡Á 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.](https://image.slidesharecdn.com/20060411ahp-0411-130118075335-phpapp01-211222053731/85/20060411ahp-0411-130118075335-phpapp01-7-320.jpg)
![Step 2: Computing the matrix of
scenario scores
? The matrix of scenario scores S [n ¡Á 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 bj
ih 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).](https://image.slidesharecdn.com/20060411ahp-0411-130118075335-phpapp01-211222053731/85/20060411ahp-0411-130118075335-phpapp01-11-320.jpg)
20060411ahp 0411-130118075335-phpapp01
- 2. Outline 1. Introduction of AHP 2. How the AHP works 3. Example
- 3. 1. Introduction of AHP Salary is important .. Location is important.. Long term prospect is important.. Interest is important.. Is job 1 best ? Is Job 2 best ? Is Job 3 best ? Is Job 4 best ? Crystal is looking for job¡
- 4. AHP Features ? AHP is 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 works 1. Computing the vector of objective weights 2. Computing the matrix of scenario scores 3. Ranking the scenarios 4. Checking the consistency consider m evaluation criteria and n scenarios.
- 6. AHP Steps 1. Computing the vector of objective weights 2. Computing the matrix of scenario scores 3. Ranking the scenarios 4. Checking the consistency
- 7. Step 1: Computing the vector of objective weights ? Pairwise comparison matrix A [m ¡Á 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 weights Preferences on Objectives Weights on Objectives
- 10. AHP Steps 1. Computing the vector of objective weights 2. Computing the matrix of scenario scores 3. Ranking the scenarios 4. Checking the consistency
- 11. Step 2: Computing the matrix of scenario scores ? The matrix of scenario scores S [n ¡Á 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 bj ih 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 scores Location scores Relative Location scores Relative scores for each objective
- 13. AHP Steps 1. Computing the vector of objective weights 2. Computing the matrix of scenario scores 3. Ranking the scenarios 4. 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 scenarios Relative scores for each objective Weights on Objectives A B C: .335 D: .238
- 16. AHP Steps 1. Computing the vector of objective weights 2. Computing the matrix of scenario scores 3. Ranking the scenarios 4. 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 x ? A 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. Criterion 1 0 S3 S2 S1 Criterion 2 0 S3 S2 S1 Criterion 3 0 S3 S2 S1
- 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