Presentation on theme: "Analytical Hierarchy Process (AHP) - by Saaty"— Presentation transcript:
1 Analytical Hierarchy Process (AHP) - by Saaty • Another way to structure decision problem• Used to prioritize alternatives• Used to build an additive value function• Attempts to mirror human decision process• Easy to use• Well accepted by decision makers– Used often - familiarity– Intuitive• Can be used for multiple decision makers• Very controversial!
2 What do we want to accomplish? • Learn how to conduct an AHP analysis• Understand the how it works• Deal with controversy– Rank reversal– Arbitrary ratings• Show what can be done to make it useableBottom Line: AHP can be a useful tool. . . but it can’t be used indiscriminately!
3 AHP Procedure – Build the Hierarchy • Very similar to hierarchical value structure– Goal on top (Fundamental Objective)– Decompose into sub-goals (Means objectives)– Further decomposition as necessary– Identify criteria (attributes) to measure achievement of goals (attributes and objectives)• Alternatives added to bottom– Different from decision tree– Alternatives show up in decision nodes– Alternatives affected by uncertain events– Alternatives connected to all criteria
4 Building the Hierarchy • Note: Hierarchy corresponds to decision maker values– No right answer– Must be negotiated for group decisions• Example: Buying a carAffinityDiagram
5 AHP Procedure – Judgments and Comparisons • Numerical Representation• Relationship between two elements that share a common parent in the hierarchy• Comparisons ask 2 questions:– Which is more important with respect to the criterion?– How strongly?• Matrix shows results of all such comparisons• Typically uses a 1-9 scale• Requires n(n-1)/2 judgments• Inconsistency may arise
7 Example - Pairwise Comparisons • Consider following criteriaPurchase CostMaintenance CostGas Mileage• Want to find weights on these criteria• AHP compares everything two at a time(1) ComparePurchase CosttoMaintenance Cost– Which is more important?Say purchase cost– By how much? Say moderately3
8 Example - Pairwise Comparisons (2) ComparePurchase CosttoGas Mileage– Which is more important?Say purchase cost– By how much? Say more important5(3) CompareMaintenance CosttoGas Mileage– Which is more important?Say maintenance cost– By how much? Say more important3
9 Example - Pairwise Comparisons • This set of comparisons gives the following matrix:PMG1351/31/5• Ratings mean that P is 3 times more important than Mand P is 5 times more important than G• What’s wrong with this matrix?The ratings are inconsistent!
10 Consistency • Ratings should be consistent in two ways: (1) Ratings should be transitive– That means thatIf A is better than Band B is better than Cthen A must be better than C(2) Ratings should be numerically consistent– In car example we made 1 more comparison than we neededWe know that P = 3M and P = 5G3M = 5GM = (5/3)G
11 Consistency And Weights • So consistent matrix for the car example would look like:PMG1351/31/55/33/5– Note that matrixhas Rank = 1– That means thatall rows are multiplesof each other• Weights are easy to compute for this matrix– Use fact that rows are multiples of each other– Compute weights by normalizing any column• We get
12 Weights for Inconsistent Matrices • More difficult - no multiples of rows• Must use some averaging technique• Method 1 - Eigenvalue/Eigenvector Method– Eigenvalues are important tools in several math, science and engineering applications- Changing coordinate systems- Solving differential equations- Statistical applications– Defined as follows: for square matrix A and vector x,Eigenvalue of A when Ax = x, x nonzerox is then the eigenvector associated with – Compute by solving the characteristic equation:det(I – A) = | I – A | = 0
13 Weights for Inconsistent Matrices – Properties:- The number of nonzero Eigenvalues for a matrix is equal to its rank (a consistent matrix has rank 1)- The sum of the Eigenvalues equals the sum of thediagonal elements of the matrix (all 1’s forconsistent matrix)– Therefore: An nx n consistent matrix has one Eigenvalue with value n– Knowing this will provide a basis of determining consistency– Inconsistent matrices typically have more than 1 eigen value- We will use the largest, , for the computationmax
14 Weights for Inconsistent Matrices • Compute the Eigenvalues for the inconsistent matrixPMG1351/31/5= Aw = vector of weights– Must solve: Aw = w by solving det(I – A) = 0– We get:max= 3.039find the Eigen vector for and normalizeDifferent than before!
15 Measuring Consistency • Recall that for consistent 3x3 comparison matrix, = 3• Compare with from inconsistent matrix• Use test statistic:maxmax• From Car Example:C.I. = (3.039–3)/(3-1) =Smaller the value of CI, higher the consistency
16 Weights for Inconsistent Matrices (continued) • Method 2: Geometric Mean– Definition of the geometric mean:– Procedure:(1) Normalize each column(2) Compute geometric mean of each row– Limitation: lacks measure of consistency
17 Weights for Inconsistent Matrices (continued) • Car example with geometric meansPMG1351/31/5PMG.220.127.116.11.18.104.22.168.08Normalized= [(.65)(.69)(.56)]1/3w= 0.630.67p= [(.22)(.23)(.33)]1/3w= 0.26Normalized0.28M= [(.13)(.08)(.11)]1/3w= 0.050.05G