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?

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

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

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

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]

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

Ranking of priorities S R F S R F Row sums Normalized Row sums

Preference Style Reliability Fuel Economy 0.121

Ranking alternatives 1. Style Style Civic Saturn Escort 1/1 1/44/1 1/ /1 1/14/1 1/ /4 1/4 1/11/5 1.7 Miata6/1 4/1 5/1 1/116 CivicSaturnEscortMiata Normalized row sum Row sum 

Ranking alternatives 2. Reliability Reliability Civic Saturn Escort 1/1 2/15/1 1/1 9 1/2 1/1 3/1 2/ /5 1/3 1/11/ Miata1/1 1/2 4/1 1/16.5 CivicSaturnEscortMiata Normalized row sum Row sum 

Fuel Economy (quantitative information) Civic Saturn Escort Miata Miles/gallon Normalized Ranking alternatives 3. Fuel Economy

- Civic Saturn Escort Miata Civic Saturn Escort Miata Civic Saturn Escort Miata 0.248

Overall Ranking of alternatives Style Reliability Fuel Economy Civic Escort Miata Saturn * = Best

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

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

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

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

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 Sum10/37/48 Normalized Weights (rows) not consistent

Ranking of priorities Matrix A S R F S R F Row sums Normalized Row sums

Ranking of priorities Matrix A 2 S R F S R F Row sums A 2 Row sums A Row sums Diff. in sums

Ranking of priorities Matrix A 3 S R F S R F Row sums A 3 Row sums A Row sums Diff. in sums

Preference Style Reliability Fuel Economy Selecting a New Car 1.0 Style Reliability Fuel Economy 0.122

Ranking alternatives 1. Style Style Civic Saturn Escort 1/1 1/44/1 1/ /1 1/14/1 1/ /4 1/4 1/11/5 1.7 Miata6/1 4/1 5/1 1/116 CivicSaturnEscortMiata Normalized row sum Row sum  Matrix A

Ranking alternatives 1. Style Style Civic Saturn Escort Miata CSEM Miata Norm. row sum Row sum  Matrix A A 2 - A row sum

Ranking alternatives 1. Style Style Civic Saturn Escort Miata CSEM Miata Norm. row sum Row sum  Matrix A A 3 - A 2 row sum

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 Eigenvector

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

Fuel Economy (quantitative information) Civic Saturn Escort Miata Miles/gallon Normalized Ranking alternatives 3. Fuel Economy

- Civic Saturn Escort Miata Civic Saturn Escort Miata Civic Saturn Escort Miata Selecting a New Car 1.0 Style Reliability Fuel Economy

Overall Ranking of alternatives Style Reliability Fuel Economy Civic Escort Miata Saturn * = Best

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

Cost vs. Benefits MIATA$18K CIVIC$12K SATURN$15K ESCORT$9K Cost Normalized Cost Cost/Benefits Ratio 54K1.0 

Complex decisions Many levels of criteria and sub-criteria

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