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Analytic Hierarchy Process Multiple-criteria decision-making Real world decision problems –multiple, diverse criteria –qualitative as well as quantitative.

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Presentation on theme: "Analytic Hierarchy Process Multiple-criteria decision-making Real world decision problems –multiple, diverse criteria –qualitative as well as quantitative."— Presentation transcript:

1 Analytic Hierarchy Process Multiple-criteria decision-making Real world decision problems –multiple, diverse criteria –qualitative as well as quantitative information Comparing apples and oranges? Spend on defence or agriculture? Open the refrigerator - apple or orange?

2 AHP Information is decomposed into a hierarchy of alternatives and criteria Information is then synthesized to determine relative ranking of alternatives Both qualitative and quantitative information can be compared using informed judgements to derive weights and priorities

3 Example: Car Selection Objective –Selecting a car Criteria –Style, Reliability, Fuel-economyCost? Alternatives –Civic Coupe, Saturn Coupe, Ford Escort, Mazda Miata

4 Hierarchical tree - Civic - Saturn - Escort - Miata - Civic - Saturn - Escort - Miata - Civic - Saturn - Escort - Miata

5 Ranking of criteria Weights? AHP –pair-wise comparison matrix a ij = a i /a j = weight of row (i) criterion to relative to weight of column (j) criterion a ij = [1:Equal, 3:Moderate, 5:Strong, 7:Very strong, 9:Extreme]

6 Ranking of criteria Pair-wise relative importance StyleReliabilityFuel Economy Style Reliability Fuel Economy 1/11/23/1 2/11/14/1 1/31/41/1

7 Ranking of priorities S R F S R F 1 0.5 3 2 14 0.333 0.251.0 Row sums 4.5 7 1.58333 13.08333 Normalized Row sums 0.344 0.535 0.121 1.0

8 Preference Style 0.344 Reliability 0.535 Fuel Economy 0.121

9 Ranking alternatives 1. Style Style Civic Saturn Escort 1/1 1/44/1 1/6 5.417 4/1 1/14/1 1/4 9.25 1/4 1/4 1/11/5 1.7 Miata6/1 4/1 5/1 1/116 CivicSaturnEscortMiata 0.167 0.286 0.053 0.494 Normalized row sum 32.3671.0 Row sum 

10 Ranking alternatives 2. Reliability Reliability Civic Saturn Escort 1/1 2/15/1 1/1 9 1/2 1/1 3/1 2/1 6.5 1/5 1/3 1/11/41.783 Miata1/1 1/2 4/1 1/16.5 CivicSaturnEscortMiata 0.378 0.273 0.075 0.273 Normalized row sum Row sum 23.7831.0 

11 Fuel Economy (quantitative information) Civic Saturn Escort Miata 34 27 24 28 113 Miles/gallon Normalized 0.301 0.239 0.212 0.248 1.0 Ranking alternatives 3. Fuel Economy

12 - Civic 0.167 - Saturn 0.286 - Escort 0.053 - Miata 0.494 - Civic 0.378 - Saturn 0.273 - Escort 0.075 - Miata 0.273 - Civic 0.301 - Saturn 0.239 - Escort 0.212 - Miata 0.248

13 Overall Ranking of alternatives Style Reliability Fuel Economy Civic Escort Miata Saturn 0.167 0.378 0.301 0.286 0.273 0.239 0.053 0.075 0.212 0.494 0.273 0.248 * 0.344 0.535 0.121 = 0.296 0.273 0.084 0.346 Best

14 AHP Eigenvector Method Objective –Eliminates inconsistency (errors) in pair-wise comparisons Applies –To ranking (weights) of criteria –To ranking (scores) of alternatives under each criteria Approach –Iterative

15 Ranking of priorities Eigenvector [Ax = x] Iterate 1. Take successively higher powers of matrix A = {a ij = a i /a j } 2. Normalize the row sums Continue until difference between successive row sums is less than a pre-specified value

16 Car Selection Example: Hierarchical tree - Civic - Saturn - Escort - Miata - Civic - Saturn - Escort - Miata - Civic - Saturn - Escort - Miata

17 Ranking of criteria Pair-wise relative importance Matrix A StyleReliabilityFuel Economy Style Reliability Fuel Economy 1/11/23/1 2/11/14/1 1/31/41/1

18 Ranking of criteria Errors in pair-wise matrix A StyleReliabilityFuel Economy Style Reliability Fuel Economy 1/11/23/1 2/11/14/1 1/31/41/1 Style Reliability Fuel Economy 0.30.2860.375 0.60.5710.5 0.10.1430.125 Sum10/37/48 Normalized Weights (rows) not consistent

19 Ranking of priorities Matrix A S R F S R F 1 0.5 3 2 14 0.333 0.251.0 Row sums 4.5 7 1.583 13.083 Normalized Row sums 0.344 0.535 0.121 1.0

20 Ranking of priorities Matrix A 2 S R F S R F 3 1.75 8 5.333 3 14 1.167 0.667 3 Row sums 12.75 23.333 4.833 39.917 A 2 Row sums 0.319 0.559 0.121 1.0 A Row sums 0.344 0.535 0.121 1.0 Diff. in sums - 0.025 0.024 0

21 Ranking of priorities Matrix A 3 S R F S R F 9.167 5.25 24 16 9.167 42 3.5 2 9.167 Row sums 38.417 67.167 14.667 120.25 A 3 Row sums 0.319 0.559 0.122 1.0 A Row sums 0.319 0.559 0.121 1.0 Diff. in sums 0 0.001

22 Preference Style 0.319 Reliability 0.559 Fuel Economy 0.122 Selecting a New Car 1.0 Style 0.319 Reliability 0.559 Fuel Economy 0.122

23 Ranking alternatives 1. Style Style Civic Saturn Escort 1/1 1/44/1 1/6 5.417 4/1 1/14/1 1/4 9.25 1/4 1/4 1/11/5 1.7 Miata6/1 4/1 5/1 1/116 CivicSaturnEscortMiata 0.167 0.286 0.053 0.494 Normalized row sum 32.3671.0 Row sum  Matrix A

24 Ranking alternatives 1. Style Style Civic Saturn Escort 4 2.167 9.833 1.196 17.196 10.5 4 25.25 1.967 41.717 2.7 1.363 4 0.504 8.567 Miata29.25 10.75 50 4 94 CSEM Miata 0.106 0.258 0.053 0.582 Norm. row sum 161.481.0 Row sum  Matrix A 2 -0.061 -0.028 0 0.088 A 2 - A row sum

25 Ranking alternatives 1. Style Style Civic Saturn Escort 22.3 10.408 40.479 4.371 77.558 44.613 20.804 93.083 9.667 168.27 12.175 5.054 22.771 2.095 42.095 Miata108.75 46.563 230 21.563 406.88 CSEM Miata 0.112 0.242 0.061 0.586 Norm. row sum 694.791.0 Row sum  Matrix A 3 0.006 -0.016 0.008 0.004 A 3 - A 2 row sum

26 Ranking alternatives 1. Style Style Civic Saturn Escort 1/1 1/44/1 1/6 4/1 1/14/1 1/4 1/4 1/4 1/11/5 Miata6/1 4/1 5/1 1/1 CivicSaturnEscortMiata 0.1160 0.2470 0.0600 0.5770 Eigenvector

27 Ranking alternatives 2. Reliability Reliability CivicSaturnEscortMiata Civic Saturn Escort 1/1 2/15/1 1/1 1/2 1/1 3/1 2/1 1/5 1/3 1/11/4 Miata1/1 1/2 4/1 1/1 Eigenvector 0.3790 0.2900 0.0740 0.2570

28 Fuel Economy (quantitative information) Civic Saturn Escort Miata 34 27 24 28 113 Miles/gallon Normalized 0.301 0.239 0.212 0.248 1.0 Ranking alternatives 3. Fuel Economy

29 - Civic 0.116 - Saturn 0.247 - Escort0.060 - Miata 0.577 - Civic 0.379 - Saturn 0.290 - Escort 0.074 - Miata 0.257 - Civic 0.301 - Saturn 0.239 - Escort 0.212 - Miata 0.248 Selecting a New Car 1.0 Style 0.3196 Reliability 0.5584 Fuel Economy 0.1220

30 Overall Ranking of alternatives Style Reliability Fuel Economy Civic Escort Miata Saturn 0.116 0.379 0.301 0.247 0.290 0.239 0.060 0.074 0.212 0.577 0.257 0.248 * 0.3196 0.5584 0.1220 = 0.306 0.272 0.094 0.328 Best

31 Handling Costs Dangers of including Cost as another criterion –political, emotional responses? Separate Benefits and Costs hierarchical trees Costs vs. Benefits evaluation –Alternative with best benefits/costs ratio

32 Cost vs. Benefits MIATA$18K0.3330.9840 CIVIC$12K0.2221.3771 SATURN$15K0.27780.9791 ESCORT$9K0.16670.5639 Cost Normalized Cost Cost/Benefits Ratio 54K1.0 

33 Complex decisions Many levels of criteria and sub-criteria

34 Application areas –strategic planning –resource allocation –source selection, program selection –business policy –etc., etc., etc.. AHP software (ExpertChoice) –computations –sensitivity analysis –graphs, tables Group AHP

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