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Chapter 10 Multicriteria Decision-Marking Models

Chapter 10 Multicriteria Decision-Marking Models. 投票. 怎樣做會令教學的效果好一點？ 用英文？ 說話慢一點？ 一般用中文，重點用英文？ 中文說一片，重點用英文再說一片？. 2. 投票方法. 多輪舉手投票，每輪每人一票 每輪去掉票數最少的選項，直至剩下一選項為止 若兩個或以上的選項票數最少時，只考慮該等選項，仍以多輪舉手投票，每輪每人一票去掉票數最少的選項. 3. 投票結果. 4. 10.3 AHP (Analytical Hierarchy Process) 層級分析法. 5.

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Chapter 10 Multicriteria Decision-Marking Models

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  1. Chapter 10Multicriteria Decision-Marking Models 1

  2. 投票 怎樣做會令教學的效果好一點？ 用英文？ 說話慢一點？ 一般用中文，重點用英文？ 中文說一片，重點用英文再說一片？ 2

  3. 投票方法 多輪舉手投票，每輪每人一票 每輪去掉票數最少的選項，直至剩下一選項為止 若兩個或以上的選項票數最少時，只考慮該等選項，仍以多輪舉手投票，每輪每人一票去掉票數最少的選項 3

  4. 投票結果 4

  5. 10.3 AHP (Analytical Hierarchy Process)層級分析法 5

  6. AHP (Analytical Hierarchy Process)層級分析法 keys in the scoring method, identifying weights of factors (i.e., objectives) ratings of each alternative for each factor 選擇/選項 目標 6

  7. AHP (Analytical Hierarchy Process)層級分析法 drawback of the scoring or weighting methods: subjective hard to simultaneously compare multiple items questions: Is there any method with more analytical basis? easy to compare, e.g., each comparison is only between two options? 7

  8. AHP (Analytical Hierarchy Process)層級分析法 AHP each comparison between two factors or two altrenatives more analytical approach to get the weights normalized weights of factors normalized priorities of each alternative for factors factor Priorities weight Alternative 1 Alternative 2 Alternative 3 manuf. cap./cost … … … … sum to 1 sum to 1 market demand … … … … profit margin … … … … (long-term) prof./growth … … … … transp. costs … … … … useful life … … … … sum to 1 . . . 8

  9. AHP (Analytical Hierarchy Process)層級分析法 AHP normalized weights of factors normalized priorities of each alternative for factors factor Priorities weight Alternative 1 Alternative 2 Alternative 3 manuf. cap./cost … … … … sum to 1 market demand … … … … profit margin … … … … (long-term) prof./growth … … … … transp. costs … … … … useful life … … … … sum to 1 question: how to determine those weights and priorities? determining the relative importance (i.e., the priorities) of the factors for the alternative determining the relative importance (i.e., weights) of the factors 9

  10. AHP (Analytical Hierarchy Process)層級分析法 ideas of AHP to determine the relative importance simple to compare for two alternatives combining the pairwise comparisons into overall comparisons for weights of all factors, and for priorities of alternatives in each factor 每次都只比較兩樣東西，或是兩目標，或是兩選項，最終將這些兩兩相比轉化成所有選項的比較。 10

  11. Idea of AHP to Determine the Relative Importance Table 10-1 Preference Scale for the Pairwise Comparisons Verbal Statement of the preference Numerical Value Equally preferred Equally to moderately preferred Moderately preferred Moderately to strongly preferred Strongly preferred Strongly to very strongly preferred Very strongly preferred Very strongly to extremely preferred Extremely preferred 1 2 3 4 5 6 7 8 9 11

  12. Idea of AHP to Determine the Relative Importance simple to compare for two items for relative importance of factors manuf. cap./cost market demand profit margin (long-term) prof./growth transp. costs useful life weight manuf. cap./cost 1 row sum reflects the importance of a factor 3 market demand 1 profit margin 1 (long-term) prof./growth 1 Transp. costs 1/3 1 useful life 1 see their relationship? 12

  13. decision: stereo system to purchase brands (i.e., alternatives): Sharp, Lucidity, Clarity criteria (i.e., factors, objectives): sound, price, options to find the relative importance of criteria to find the relative importance of brands in each criterion Example 10-4 Alternatives Factors weights Sharp Lucidity Clarity Price … … … … Sound … … … … Options … … … … 13

  14. Example 10-4: To Find the Relative Importance of Criteria Table 10-2: Pairwise Comparison Table for the Stereo System Selection Problem Criterion Price Sound Options Price 1 3 4 Sound 1/3 1 3 Alternatives Factors weights Sharp Lucidity Clarity Options 1/4 1/3 1 Price … … … … Sound … … … … Totals 1.5833 4.333 8 Options … … … … Table 10-3: Normalized Pairwise Comparison Table for the Stereo System Selection Problem Criterion Price Sound Options Average % Price 0.6316 0.6923 0.5 0.6079 Sound 0.2105 0.2308 0.375 0.2721 Options 0.1579 0.0769 0.125 0.1199 Totals 1.0 1.0 1.0 1.0 14

  15. Example 10-4: To Find the Relative Importance of Brands in Price Table 10-7: Proportion Percentage Matrix for Price Criterion Price Sound Options Average % Price 0.5714 0.5 0.6 0.5571 Sound 0.1429 0.125 0.10 0.1226 Options 0.2857 0.375 0.30 0.3292 Totals 1.0 1.0 1.0 1.0 這步驟是說，以Price作目標，你較喜歡那個選項。留意，喜歡與否，只與喜好有關，沒特別說是較便宜還是較貴。 Table 10-6: Pairwise Comparison Matrix Price Criterion Sharp Lucidity Clarity Sharp 1 4 2 Lucidity 1/4 1 1/3 Alternatives Factors weights Sharp Lucidity Clarity Clarity 1/2 3 1 Price … … … … Sound … … … … Totals 1.75 8 3.333 Options … … … … 15

  16. Example 10-4: To Find the Relative Importance of Brands in Sound Proportion Percentage Matrix for Sound Criterion Price Sound Options Average % Price 0.1429 0.1111 0.1579 0.1373 Sound 0.2857 0.2222 0.2105 0.2395 Options 0.5714 0.6667 0.6316 0.6232 Totals 1.0 1.0 1.0 1.0 Pairwise Comparison Matrix Sound Criterion Sharp Lucidity Clarity Sharp 1 1/2 1/4 Lucidity 2 1 1/3 Alternatives Factors weights Sharp Lucidity Clarity Clarity 4 3 1 Price … … … … Sound … … … … Totals 7 4.5 1.5833 Options … … … … 16

  17. Example 10-4: To Find the Relative Importance of Brands in Options Proportion Percentage Matrix for Options Criterion Price Sound Options Average % Price 0.5714 0.6667 0.5 0.5794 Sound 0.1429 0.1667 0.25 0.1865 Options 0.2857 0.1667 0.25 0.2341 Totals 1.0 1.0 1.0 1.0 Pairwise Comparison Matrix Options Criterion Sharp Lucidity Clarity Sharp 1 4 2 Lucidity 1/4 1 1 Alternatives Factors weights Sharp Lucidity Clarity Clarity 1/2 1 1 Price … … … … Sound … … … … Totals 1.75 6 4 Options … … … … 17

  18. Example 10-4: To Find the Overall Importance of the Brands overall importance: by weighted score of brands Weights of Factors, Priorities of Brands in Factors, and Weighted Score of Brands Criterion weight (%) Sharp Lucidity Clarity Price 0.6079 0.5571 0.1226 0.3202 Sound 0.2721 0.1373 0.2395 0.6232 Options 0.1199 0.5794 0.1865 0.2341 Weighted score 0.4456 0.1621 0.3984 0.1621 = (0.6079)(0.1226)+(0.2721)(0.2395)+(0.1199)(0.1865) 18

  19. Inconsistency in Pairwise Comparison 19

  20. Possibility of Inconsistency in a Pairwise Comparison Matrix the pairwise comparisons may not be consistent any method to check whether a pairwise comparison matrix is consistent or not? these pairwise comparisons are not consistent 20

  21. Random Index to Check the Consistency of an AHP general idea (with detail given later) consistency index, CI: an index calculated from a pairwise comparison matrix random index, RI: an index calculated from randomly generated pairwise comparison matrices The pairwise comparison matrix is inconsistent if CI/RI > some critical value 21

  22. Consistency Index for a Pairwise Comparison Matrix procedure (for an n-dimensional comparison matrix) 1 Construct a pairwise comparison matrix P 2 Find the normalized weights or priorities  for P 3 Calculate  = P 4 Calculate ratios for each element of  and of , i.e., calculate i/i for i = 1, …, n 5 Calculate average ratio, A = (1/1 + … + n/n)/n 6 Calculate CI = (An)/(n1) 22

  23. Example on Consistency Index Criterion a b c a 1 3 4 b 1/3 1 3 c 1/4 1/3 1 Totals 1.5833 4.333 8 Criterion a b c Average % 5 1 2 3 6 4 a 0.6316 0.6923 0.5 0.6079 b 0.2105 0.2308 0.375 0.2721 c 0.1579 0.0769 0.125 0.1199 Totals 1.0 1.0 1.0 1.0 23

  24. Random Index and the Criterion of Consistency RI, the consistency index for a pairwise matrix where each pairwise comparison is randomly generated RI as a function of n in Table 10-5 Table 10-5 Random Index Values for the Comparison of n items • consistent if CI/RI < 0.1 • consistent because CI = 0.0371, RI = 0.58 for n = 3 24

  25. Final Remarks Ours is a simplified version of AHP. For example: AHP is more for a decision problem with hierarchical decisions; The theory of AHP is related to the maximum eigenvalue and the corresponding eigenvector of a pairwise comparison matrix, something that we have skipped. 25

  26. Chapter 10: Homework for AHP Problem 16 26

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