Chapter 10 Multicriteria DecisionMarking Models 1 10 3

Chapter 10 Multicriteria Decision-Marking Models 1

10. 3 AHP (Analytical Hierarchy Process) 層級分析法 5

AHP (Analytical Hierarchy Process) 層級分析法 è keys in the scoring method, identifying èweights of factors (i. e. , objectives) 選擇/選項 目標 èratings of each alternative for each factor manuf. cap. /cost market demand profit margin (long-term) prof. /growth transp. costs useful life weight microwave refers stoves 4 5 3 5 2 1 4 8 6 3 9 1 3 4 9 6 2 5 8 2 5 7 4 6 6

AHP (Analytical Hierarchy Process) 層級分析法 è drawback methods: of the scoring or weighting weight microwave refers stoves èsubjective èhard 4 4 3 8 market demand 5 8 4 2 profit margin 3 6 9 5 (long-term) prof. /growth 5 3 6 7 transp. costs 2 9 2 4 useful life 1 1 5 6 to simultaneously compare multiple items è questions: èwith manuf. cap. /cost Is there any method more analytical basis? èeasy to compare, e. g. , each comparison is only between two options? 7

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

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

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

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

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

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

Example 10 -4: To Find the Relative Importance of Criteria Table 10 -2: Pairwise Comparison. Alternatives Table for the Stereo System Selection Problem Factors weights Criterion Price Sound Options Totals Table 10 -3: Criterion Price Sound Options Totals Price … 1/3 … 1/4 1. 5833 Sharp … … … Lucidity Sound 3… 1… 1/3… 4. 333 Clarity Options … 4 … 3 … 1 8 Normalized Pairwise Comparison Table for the Stereo System Selection Problem Price 0. 6316 0. 2105 0. 1579 1. 0 Sound 0. 6923 0. 2308 0. 0769 1. 0 Options 0. 5 0. 375 0. 125 1. 0 Average % 0. 6079 0. 2721 0. 1199 1. 0 14

Example 10 -4: To Find the Relative Importance of Brands in Price Table 10 -6: Factors Criterion Sharp Price Lucidity Sound Clarity Options Totals 這步驟是說， 以Price作目 Clarity 標，你較喜歡 那個選項。留 … 2 1/3 意，喜歡與否， … 只與喜好有關， 1 沒特別說是較 … 3. 333 便宜還是較貴。 Alternatives Pairwise Comparison Matrix Price weights Sharp … 1 1/4 … 1/2 … 1. 75 Sharp. Lucidity … … … 4 1 3 8 … … … Table 10 -7: Proportion Percentage Matrix for Price Criterion Price Sound Options Totals Price 0. 5714 0. 1429 0. 2857 1. 0 Sound 0. 5 0. 125 0. 375 1. 0 Options 0. 6 0. 10 0. 30 1. 0 Average % 0. 5571 0. 1226 0. 3292 1. 0 15

Example 10 -4: To Find the Relative Importance of Brands in Sound Pairwise Comparison Matrix Sound Alternatives Factors weights Criterion Sharp Lucidity Clarity Sharp 1 1/2 1/4 Price … … Lucidity 2 1 1/3 Sound … … Clarity 4 3 1 Options … … Totals 7 4. 5 1. 5833 Proportion Percentage Matrix for Sound Criterion Price Sound Options Totals Price 0. 1429 0. 2857 0. 5714 1. 0 Sound 0. 1111 0. 2222 0. 6667 1. 0 Options 0. 1579 0. 2105 0. 6316 1. 0 Average % 0. 1373 0. 2395 0. 6232 1. 0 16

Example 10 -4: To Find the Relative Importance of Brands in Options Pairwise Comparison Matrix Options Alternatives Factors Criterion Sharp Price Lucidity Sound Clarity Options Totals weights Sharp … 1 1/4 … 1/2 … 1. 75 Sharp Lucidity … … … 4 1 1 6 Clarity … … 2 … … … 1 1 … 4 Proportion Percentage Matrix for Options Criterion Price Sound Options Average % Price 0. 5714 0. 6667 0. 5794 Sound 0. 1429 0. 1667 0. 25 0. 1865 Options 0. 2857 0. 1667 0. 25 0. 2341 Totals 1. 0 17

Example 10 -4: To Find the Overall Importance of the Brands è 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 0. 4456 0. 1621 0. 3984 Weighted score 0. 1621 = (0. 6079)(0. 1226)+(0. 2721)(0. 2395)+(0. 1199)(0. 1865) 18

Inconsistency in Pairwise Comparison 19

Possibility of Inconsistency in a Pairwise Comparison Matrix è the pairwise comparisons may not be consistent è any method to check whether a pairwise comparison matrix is consistent or not? these pairwise comparisons are not consistent A B C A 1 1/2 1 B 2 1 1/3 C 1 3 1 20

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

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

Example on Consistency Index Criterion a b c Totals 3 4 a 1 1/3 1/4 1. 5833 a 0. 6316 0. 2105 0. 1579 1. 0 b 3 1 1/3 4. 333 b 0. 6923 0. 2308 0. 0769 1. 0 c 4 3 1 8 c 0. 5 0. 375 0. 125 1. 0 1 Average % 0. 6079 0. 2721 0. 1199 1. 0 2 5 6 23

Random Index and the Criterion of Consistency è RI, the consistency index for a pairwise matrix where each pairwise comparison is randomly generated è RI as a function of n in Table 10 -5 Random Index Values for the Comparison of n items öconsistent if CI/RI < 0. 1 öconsistent because CI = 0. 0371, RI = 0. 58 for n = 3 24

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

Chapter 10: Homework for AHP èProblem 16 26