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1 Chapter 10 Multicriteria Decision-Marking Models

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5 10.3 AHP (Analytical Hierarchy Process)

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6 AHP (Analytical Hierarchy Process) keys in the scoring method, identifying weights of factors (i.e., objectives) ratings of each alternative for each factor weight microwaverefersstoves manuf. cap./cost 4438 market demand 5842 profit margin 3695 (long-term) prof./growth 5367 transp. costs 2924 useful life 1156 /

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7 AHP (Analytical Hierarchy Process) 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? weightmicrowaverefersstoves manuf. cap./cost4438 market demand5842 profit margin3695 (long-term) prof./growth5367 transp. costs2924 useful life1156

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8 AHP (Analytical Hierarchy Process) 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 Priorities Alternative 3Alternative 2Alternative 1 weight factor ………… useful life ………… transp. costs ………… (long-term) prof./growth ………… profit margin ………… market demand ………… manuf. cap./cost sum to 1......

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

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10 AHP (Analytical Hierarchy Process) 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

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

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

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…………Options …………Sound …………Price ClarityLuciditySharp weightsFactors Alternatives 13 Example 10-4 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

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14 Example 10-4: Example 10-4: To Find the Relative Importance of Criteria Totals Options Sound Price OptionsSoundPriceCriterion Table 10-2: Pairwise Comparison Table for the Stereo System Selection Problem 84.3331.5833 11/31/4 311/3 431 Totals Options Sound Price Average %OptionsSoundPriceCriterion Table 10-3: Normalized Pairwise Comparison Table for the Stereo System Selection Problem 1.0 0.1579 0.2105 1.0 0.1199 0.2721 0.6079 1.0 0.125 0.375 0.5 1.0 0.0769 0.2308 0.69230.6316 …………Options …………Sound …………Price ClarityLuciditySharp weightsFactors Alternatives

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15 Example 10-4: Example 10-4: To Find the Relative Importance of Brands in Price 3.33381.75Totals 131/2Clarity 1/311/4Lucidity 241Sharp ClarityLuciditySharpCriterion Table 10-6: Pairwise Comparison Matrix Price 1.0 Totals 0.32920.300.3750.2857Options 0.12260.100.1250.1429Sound 0.55710.60.50.5714Price Average %OptionsSoundPriceCriterion Table 10-7: Proportion Percentage Matrix for Price …………Options …………Sound …………Price ClarityLuciditySharp weightsFactors Alternatives Price

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16 Example 10-4: Example 10-4: To Find the Relative Importance of Brands in Sound 1.58334.57Totals 134Clarity 1/312Lucidity 1/41/21Sharp ClarityLuciditySharpCriterion Pairwise Comparison Matrix Sound 1.0 Totals 0.62320.63160.66670.5714Options 0.23950.21050.22220.2857Sound 0.13730.15790.11110.1429Price Average %OptionsSoundPriceCriterion Proportion Percentage Matrix for Sound …………Options …………Sound …………Price ClarityLuciditySharp weightsFactors Alternatives

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17 Example 10-4: Example 10-4: To Find the Relative Importance of Brands in Options 461.75Totals 111/2Clarity 111/4Lucidity 241Sharp ClarityLuciditySharpCriterion Pairwise Comparison Matrix Options 1.0 Totals 0.23410.250.16670.2857Options 0.18650.250.16670.1429Sound 0.57940.50.66670.5714Price Average %OptionsSoundPriceCriterion Proportion Percentage Matrix for Options …………Options …………Sound …………Price ClarityLuciditySharp weightsFactors Alternatives

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18 Example 10-4: Example 10-4: To Find the Overall Importance of the Brands overall importance: by weighted score of brands 0.39840.16210.4456 Weighted score 0.23410.18650.57940.1199Options 0.62320.23950.13730.2721Sound 0.32020.12260.55710.6079Price ClarityLuciditySharpweight (%)Criterion Weights of Factors, Priorities of Brands in Factors, and Weighted Score of Brands 0.1621 = (0.6079)(0.1226)+(0.2721)(0.2395)+(0.1199)(0.1865)

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19 Inconsistency in Pairwise Comparison

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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? ABC A121 B1/213 C11/31 these pairwise comparisons are not consistent

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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

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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)

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23 Example on Consistency Index 84.3331.5833Totals 11/31/4c 311/3b 431a cbaCriterion 1.0 Totals 0.11990.1250.07690.1579c 0.27210.3750.23080.2105b 0.60790.50.69230.6316a Average %cbaCriterion 1234 5 6

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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

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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.

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26 Chapter 10: Homework for AHP Problem 16

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27 Idea of AHP to Determine the Relative Importance simple to compare for two items for relative importance of factors 1/3 1 useful life 1 Transp. costs 1 (long-term) prof./growth 1 profit margin 1 market demand 1 manuf. cap./cost weight useful lifetransp. costs (long-term) prof./growth profit margin market demand manuf. cap./cost row sum reflects importance of a factor 3 see their relationship? Priorities Alternative 3Alternative 2Alternative 1 weight factor …………useful life …………Transp. costs …………(long-term) prof./growth …………profit margin …………market demand …………manuf. cap./cost sum to 1 determining the relative importance (i.e., weights) of the factors

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Idea of AHP to Determine the Relative Importance 1/3 1 useful life 1 Transp. costs 1 (long-term) prof./growth 1 profit margin 1 market demand 1 manuf. cap./cost weight useful lifetransp. costs (long-term) prof./growth profit margin market demand manuf. cap./cost row sum reflects importance of a factor 3 see their relationship? simple to compare for two items for (relative) priorities of alternatives in each factor Priorities Alternative 3Alternative 2Alternative 1 weight factor …………useful life …………Transp. costs …………(long-term) prof./growth …………profit margin …………market demand …………manuf. cap./cost sum to 1 determining the relative importance (i.e., the priorities) of the factors for the alternative

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