The Analytic Hierarchy Process. Danny Hahn

The Analytic Hierarchy Process Danny Hahn

The Analytic Hierarchy Process (AHP) A Decision Support Tool developed in the 1970s by Thomas L. Saaty, an American mathematician, currently University Chair, Quantitative Group, Katz Graduate School of Business, University of Pittsburgh. A theory and methodology for modeling problems in the economic, social and management sciences. A problem solving framework used for: Determining the best of several alternatives Setting Priorities Allocating Resources Requires a pair-wise determination of the relative importance of each of the criteria. Expert Choice is one commercial software tool based on the AHP. Slide 2

The Process Break down an unstructured situation into its component parts. Arrange the parts or variables into a hierarchic order. Assign numerical values to subjective judgments on the relative importance of each variable. Synthesize the judgments to determine which variables have the highest priority and should be acted upon to influence the outcome of the situation. Slide 3

The Hierarchy Goal Factor 1 Factor 2 Factor 3 Option 1 Option 2 Slide 4

Scale of Relative Importance 1 Two factors are Equally Important 3 One factor is Slightly more Important than the other 5 One is Strongly more Important 7 One is Very strongly more Important 9 One is Absolutely more Important 2, 4, 6, 8 Intermediate Values of one criteria over the other Saaty s book, The Analytic Hierarchy Process, provides background and theory on why he chose the 1 9 scale and his validation of this judgment with measurable science. Slide 5

Step by Step Example Buy the Right Car Determine the Criteria (factors) Price (lower price is better) Body Style Miles per Gallon (more MPG is better) Interior Quality Engine Size Design the Hierarchy Use an analytic process to help make a decision Slide 6

The Car Decision Hierarchy Buy the Right Car Price Body Style MPG Interior Quality Engine Size X-Treem Yaawhee Zoomer Slide 7

My Preferences (My Judgments) Body style is more important than Price. I would pay more for the Body Style I want Price is more important than MPG. I would not pay extra for more MPG Interior Quality is more important than Price. I would pay more for better Interior Quality Engine Size is more important than Price. Body Style is more important than MPG. Etc. (see next slide) Slide 8

Pair-wise Comparison of Criteria More Important Equal More Important 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 Price Body Style Price MPG Price Interior Quality Price Engine Size Body Style MPG Body Style Interior Quality Body Style Engine Size MPG Interior Quality MPG Engine Size Interior Quality Engine Size Slide 9

Matrix Review Price Body Style MPG Interior Quality Engine Size Price 1 1/4 3 1/5 1/5 Body Style 4 1 5 3 1/3 MPG 1/3 1/5 1 1/5 1/3 Interior Quality 5 1/3 5 1 5 Engine Size 5 3 3 1/5 1 An n x n matrix is a square matrix where n is the number of rows and columns. In this case n = 5. An element is equally important when compared to itself therefore the main diagonal must be a 1. By convention, the comparison of strength is always of an activity appearing in the column on the left against an activity appearing in the row on top. Body Style is 5 times more important than MPG The reverse comparisons (B to A) produce the reciprocal of the basic comparison. This is called a reciprocal matrix. MPG is 1/5 as important as Body Style Slide 10

Convert Criteria Comparisons to a Matrix Convert the pair-wise comparisons from Slide 9 to a matrix. Prioritizing the 5 Criteria Price Body Style MPG Interior Quality Engine Size Price 1 1/4 3 1/5 1/5 Body Style 4 1 5 3 1/3 MPG 1/3 1/5 1 1/5 1/3 Interior Quality 5 1/3 5 1 5 Engine Size 5 3 3 1/5 1 Column Sum 15.333 4.783 17.000 4.600 6.867 Normalize the matrix by dividing each value by the column sum (e.g. 1 / 15.33 = 0.065). Then compute the average value for each row. Price Body Style MPG Interior Quality Engine Size Average Price 0.065 0.052 0.176 0.043 0.029 0.073 Body Style 0.261 0.209 0.294 0.652 0.049 0.293 MPG 0.022 0.042 0.059 0.043 0.049 0.043 Interior Quality 0.326 0.070 0.294 0.217 0.728 0.327 Engine Size 0.326 0.627 0.176 0.043 0.146 0.264 Looking at the average value of each row, notice that 33% of my objective weight is on Interior Quality, 29% on Body Style, 26% on Engine Size. These are the weights of the criteria. Slide 11

The Decision Candidates The cars under consideration X-Treem Yaawhee Zoomer Price $25,000 $27,000 $29,000 Body Style 4-door Mid-size 5-door SportWagon 4-door Full-size MPG 19 22 17 Interior Quality Standard Deluxe Above Average Engine Size 3.8 Liter V-6 2.8 Liter 4 Cyl. 5.0 Liter V-8 The next step is to evaluate all three cars on each of the five criteria as shown in the Hierarchy Chart on Slide 7 Slide 12

Evaluate Price for Each Car X-Treem = $25,000, Yaawhee = $27,000, Zoomer = $29,000 Price of three cars. Lower Cost is better. X is slightly more important than Y. X strongly more important than Z. Y is slightly more important than Z. X-Treem Yaawhee Zoomer X-treem 1 3 5 Yaawhee 1/3 1 3 Zoomer 1/5 1/3 1 Column Sum 1.533 4.333 9.000 Normalize the Price matrix by dividing each value by the column sum (e.g. 1 / 1.533 = 0.652). Then compute the average value for each row. X-Treem Yaawhee Zoomer Average X-treem 0.652 0.692 0.556 0.633 Yaawhee 0.217 0.231 0.333 0.260 Zoomer 0.130 0.077 0.111 0.106 Slide 13

Evaluate Body Style for Each Car X-Treem = Mid-sized, Yaawhee = Wagon, Zoomer = Full-sized Body Style of three cars. I prefer Full-size, then Wagon, then Mid-size. Z is slightly more important than Y. Z is strongly more important than X. Y is slightly more important than X. X-Treem Yaawhee Zoomer X-treem 1 1/3 1/5 Yaawhee 3 1 1/3 Zoomer 5 3 1 Column Sum 9.000 4.333 1.533 Normalized Body Style Matrix X-Treem Yaawhee Zoomer Average X-treem 0.111 0.077 0.130 0.106 Yaawhee 0.333 0.231 0.217 0.260 Zoomer 0.556 0.692 0.652 0.633 Slide 14

Evaluate MPG for Each Car X-Treem = 19 MPG, Yaawhee = 22 MPG, Zoomer = 17 MPG MPG of three cars. Higher MPG is better. Y is slightly better than X. Y is strongly better than Z. X is slightly better than Z. X-Treem Yaawhee Zoomer X-treem 1 1/3 3 Yaawhee 3 1 5 Zoomer 1/3 1/5 1 Column Sum 4.333 1.533 9.000 Normalized MPG matrix X-Treem Yaawhee Zoomer Average X-treem 0.231 0.217 0.333 0.260 Yaawhee 0.692 0.652 0.556 0.633 Zoomer 0.077 0.130 0.111 0.106 Slide 15

Evaluate Interior Quality X-Treem = Standard, Yaawhee = Deluxe, Zoomer = Above Average Interior Quality of three cars. I prefer Deluxe, then Above Avg, then Standard. Y is slightly better than Z. Y is strongly better than X. Z is slightly better than X. X-Treem Yaawhee Zoomer X-treem 1 1/5 1/3 Yaawhee 5 1 3 Zoomer 3 1/3 1 Column Sum 9.000 1.533 4.333 Normalized Interior Quality Matrix X-Treem Yaawhee Zoomer Average X-treem 0.111 0.130 0.077 0.106 Yaawhee 0.556 0.652 0.692 0.633 Zoomer 0.333 0.217 0.231 0.260 Slide 16

Evaluate Engine Size X-Treem = 3.8 liter V-6, Yaawhee = 2.8 liter 4 Cyl, Zoomer = 5.0 liter V-8 Engine Size of three cars. I prefer V6, then V8, then 4-Cylinder X is slightly better than Z. X is strongly better than Y. Z is slightly better than Y. X-Treem Yaawhee Zoomer X-Treem 1 5 3 Yaawhee 1/5 1 1/3 Zoomer 1/3 3 1 1.533 9.000 4.333 Normalize the Engine Size matrix and compute the average of each row X-Treem Yaawhee Zoomer Average X-Treem 0.652 0.556 0.692 0.633 Yaawhee 0.130 0.111 0.077 0.106 Zoomer 0.217 0.333 0.231 0.260 Slide 17

Compute the Final Result Relative Scores for each Objective. Collect all the computed average values from each normalized matrix and multiply the original criteria weights. X-Treem Yaawhee Zoomer Criteria Weight Price 0.633 0.260 0.106 0.073 Body Style 0.106 0.260 0.633 0.293 MPG 0.260 0.633 0.106 0.043 Interior Quality 0.106 0.633 0.260 0.327 Engine Size 0.633 0.106 0.260 0.264 Use these relative scores for each objective and multiply by the original weights of the criteria: X-Treem = 0.633(.073) +.106(.293) +.260(.043) +.106(.327) +.633(.264) = 0.290 Yaawhee = 0.260(.073) +.260(.293) +.633(.043) +.633(.327) +.106(.264) = 0.357 Zoomer = 0.106(.073) +.633(.293) +.106(.043) +.260(.327) +.260(.264) = 0.351 The winner is the Yaawhee. Slide 18

What Happened?? This is not the result I expected from my original preferences! I would not buy a 4-cylinder sport wagon. Should I have given Engine Size a wider separation in importance? I forgot to consider consistency. If A is bigger than B, and B is bigger than C, then A must be bigger than C. Perfect consistency would be if A is 2 times bigger than B, and B is 3 times bigger than C, then A must be 6 times bigger than C. Or if Body Style is 4 times more important than Price, and Price is 3 times more important than MPG, then Body Style must be 12 times more important than MPG. Determining the Consistency Index and the Consistency Ratio should have been done on the initial pair-wise comparisons. This was the very first matrix that defined the relative priorities of the criteria (Slide 11). Slide 19

Consistency Index and Consistency Ratio There are at least 2 methods to evaluate the weights for errors in judgment: logarithmic least squares, and Saaty s eigenvector method. Additionally there are several techniques available to estimate Saaty s eigenvector method. Important terms needed to understand Saaty s method: Lambda max. = the maximum eigenvalue (Perron root) of the matrix = Lmax = λmax C.I. = Consistency Index = (λmax n) / (n 1) R.I. = Random Index. For each matrix of size n, Saaty s team generated random matrices and computed their mean C.I. value and called it the Random Index. These values are shown in the next slide. C.R. = Consistency Ratio = (C.I.) / (R.I.). A value less than or equal to 0.1 is acceptable. Larger values require the decision maker to reduce the inconsistencies by revising judgments. Slide 20

Random Index Random Consistency Index Table n 1 2 3 4 5 6 7 8 9 10 Random Index 0 0 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49 This table represents a composite of two different experiments performed by Saaty and his colleagues at the Oak Ridge National Laboratory and at the Wharton School of the University of Pennsylvania. 500 random reciprocal n x n matrices were generated for n = 3 to n = 15 using the 1 to 9 scale. The maximum eigenvalue was determined by raising each random matrix to increasing powers and normalizing the result until the process converged. The consistency index was then computed on each matrix for n = 1 through n = 15. Only n = 1 through n = 10 is presented here. Slide 21

Step by Step on Original Matrix Prioritizing the 5 Criteria Price Body Style MPG Interior Quality Engine Size Price 1 1/4 3 1/5 1/5 Body Style 4 1 5 3 1/3 MPG 1/3 1/5 1 1/5 1/3 Interior Quality 5 1/3 5 1 5 Engine Size 5 3 3 1/5 1 Column Sum 15.333 4.783 17.000 4.600 6.867 1. Add a new column (5 th Root) and compute the 5 th root of the product of the values in each row. 2. Sum this 5 th Root column. 3. Add another column (Priority Vector) and divide each value from Step 1 by the sum in Step 2. 4. Add a new row (Priority Row) under Column Sum row and multiply the Column Sum vector by the Priority Vector. 5. Lambda Max = the sum of the values computed in Step 4. 6. C.I. = (Lambda Max 5) / (4) 7. C.R. = (C.I.) / (R.I.) = Step 6 divided by 1.12 from the Random Index Table for n = 5 Slide 22

Determine My Consistency (Inconsistency?) Back to my original pair-wise comparisons Price Body Style MPG Interior Quality Engine Size 5th Root of Product Priority Vector Price 1 1/4 3 1/5 1/5 0.496 0.079 Body Style 4 1 5 3 1/3 1.821 0.288 MPG 1/3 1/5 1 1/5 1/3 0.339 0.054 Interior Quality 5 1/3 5 1 5 2.108 0.334 Engine Size 5 3 3 1/5 1 1.552 0.246 Sum row 15.333 4.783 17.000 4.600 6.867 6.315 1.000 Priority row 1.204 1.379 0.911 1.536 1.687 Compute the n-th root of the product of the values in each row. (n is the number of criteria = 5) e.g. 0.496 = the fifth root of (1*1/4*3*1/5*1/5) This technique is called the geometric mean. Priority Vector is the nth root divided by the sum of the nth root values. e.g. 0.079 = (0.496 / 6.315) Sum row = sum of each column Priority row = (sum row value)*(priority vector) LambdaMax = 6.717 = sum of Priority Row Consistency Index (CI) = 0.429 = (LambdaMax -n) / (n-1) = (6.717-5) / (4) Consistency Ratio (CR) = 0.383 = (CI) / (Random Index) = 0.429 / 1.12 CR should be less than 0.10 (up to 0.20 is tolerable) 38.3% is too inconsistent Slide 23

Reconsider My Judgments More Important Equal More Important 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 Price Body Style Price MPG Price Interior Quality Price Engine Size Body Style MPG Body Style Interior Quality Body Style Engine Size MPG Interior Quality MPG Engine Size Interior Quality Engine Size Engine Size is slightly more important to me than Interior Quality, not vice versa. This was a mistake on my initial judgment matrix. After making this change my Consistency Ratio moved from 0.383 to 0.145. Saaty states if this value is more than 10%, the judgments may be somewhat random and should perhaps be revised. Engine Size is strongly more important than MPG. I really just changed my mind here to be more consistent. My Consistency Ratio then became 0.108. Slide 24

Convert Revised Judgments to Matrix Convert the pair-wise comparisons to a matrix. Shaded items are changes from the original. Prioritizing the 5 Criteria for New Judgements Price Body Style MPG Interior Quality Engine Size Price 1 1/4 3 1/5 1/5 Body Style 4 1 5 3 1/3 MPG 1/3 1/5 1 1/5 1/5 Interior Quality 5 1/3 5 1 1/3 Engine Size 5 3 5 3 1 Column Sum 15.333 4.783 19.000 7.400 2.067 Normalize the matrix by dividing each value by the column sum (e.g. 1 / 15.33 = 0.065). Then compute the average value for each row. Price Body Style MPG Interior Quality Engine Size Average Price 0.065 0.052 0.158 0.027 0.097 0.080 Body Style 0.261 0.209 0.263 0.405 0.161 0.260 MPG 0.022 0.042 0.053 0.027 0.097 0.048 Interior Quality 0.326 0.070 0.263 0.135 0.161 0.191 Engine Size 0.326 0.627 0.263 0.405 0.484 0.421 Looking at the average value of each row, notice that 42% of my objective weight is now on Engine Size, 26% on Body Style, 19% on Interior Quality. These are the new weights of the criteria. Slide 25

Re- Determine My Consistency Price Body Style MPG Interior Quality Engine Size 5th Root of Product Priority Vector Price 1 1/4 3 1/5 1/5 0.496 0.073 Body Style 4 1 5 3 1/3 1.821 0.268 MPG 1/3 1/5 1 1/5 1/5 0.306 0.045 Interior Quality 5 1/3 5 1 1/3 1.227 0.180 Engine Size 5 3 5 3 1 2.954 0.434 Column total 15.333 4.783 19.000 7.400 2.067 6.803 1.000 Priority row 1.118 1.280 0.854 1.334 0.897 Compute the n-th root of the product of the values in each row. (n is the matrix size = 5) e.g. 0.496 = the fifth root of (1*1/4*3*1/5*1/5) In Excel this formula is =Power(number,power) This technique is called the geometric mean. Priority Vector is the nth root divided by the sum of the nth root values. e.g. 0.073 = (0.496 / 6.803) Column Total = sum of each column Priority row = (sum row value)*(priority vector) LambdaMax = 5.483 = sum of Priority Row Consistency Index (CI) = 0.121 = (LambdaMax -n) / (n-1) = (5.483-5) / (4) Consistency Ratio (CR) = 0.108 = (CI) / (Random Index) = 0.121 / 1.12 CR should be less than 0.10 (up to 0.20 is tolerable) I am now more consistent but still not perfect. Slide 26

Re-Compute Using New Criteria Weights Note the average scores comparing the criteria against each car remain unchanged. Only the Criteria Weights have changed. X-Treem Yaawhee Zoomer Criteria Weight Price 0.633 0.260 0.106 0.080 Body Style 0.106 0.260 0.633 0.260 MPG 0.260 0.633 0.106 0.048 Interior Quality 0.106 0.633 0.260 0.191 Engine Size 0.633 0.106 0.260 0.421 Use the same relative scores for each objective and multiply by the revised weights of the criteria: X-Treem = 0.633(.080) +.106(.260) +.260(.048) +.106(.191) +.633(.421) = 0.377 Yaawhee = 0.260(.080) +.260(.260) +.633(.048) +.633(.191) +.106(.421) = 0.284 Zoomer = 0.106(.080) +.633(.260) +.106(.048) +.260(.191) +.260(.421) = 0.337 The winner is the X-Treem, the Mid-sized V-6. This is the result I really expected. Slide 27

Summary 1. Define the problem and specify the solution desired. Lay out the elements of a problem as a hierarchy. Structure the hierarchy from the top levels to the level at which decisions to solve the problem is possible. 2. Do paired comparisons among the elements of a level as required by the criteria of the next higher level. Give a judgment that indicates the dominance as a whole number. Enter that number and its reciprocal in the appropriate position in the matrix. An element on the left is examined regarding its dominance over an element at the top of the matrix. 3. These comparisons produce priorities and finally, through synthesis, to overall priorities. Check for consistency. 4. Repeat steps 2 and 3 for all levels in the hierarchy. Slide 28

New Developments The example just presented used the Geometric Mean technique for approximating an eigenvector. This technique is described in Saaty s book, The Analytic Hierarchy Process, written in 1980, and is also the technique presented in Appendix D.9 of the INCOSE Systems Engineering Handbook, Version 2a, dated 2004. In 2001, Saaty wrote another book, Decision Making for Leaders, that in some respects differs from his original technique. He recomputed the Random Consistency Index. (Comparison on next slide.) For the approximation procedure to obtain Lambda Max, he states that the geometric mean method (using the n th root of the products) should only be used for a matrix of size n = 3. Otherwise the row average method should be used. The consistency ratio should be 5% or less for n = 3; 9% or less for n = 4; and 10% or less for n > 4. Both the Geometric Mean and the Row Average techniques for approximating the eigenvector of a reciprocal matrix are described in Saaty s 1980 book and in the reference sited in the INCOSE SE Handbook (IEEE Transactions on Engineering Management, August 1983). The INCOSE SE Handbook only presents the Geometric Mean technique. Slide 29

Random Consistency Index Changes Random Consistency Index Table - 1980 n 1 2 3 4 5 6 7 8 9 10 Random Index 0 0 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49 Random Consistency Index Table - 2001 n 1 2 3 4 5 6 7 8 9 10 Random Index 0 0 0.52 0.89 1.11 1.25 1.35 1.40 1.45 1.49 In Saaty s 2001 book he notes these values were recently recalculated. What if I re-compute my consistency using these new developments? Slide 30

Re-Compute Consistency Prioritized New Judgments Price Body Style MPG Interior Quality Engine Size Price 1 1/4 3 1/5 1/5 Body Style 4 1 5 3 1/3 MPG 1/3 1/5 1 1/5 1/5 Interior Quality 5 1/3 5 1 1/3 Engine Size 5 3 5 3 1 Column Sum 15.33 4.78 19.00 7.40 2.07 Normalize the matrix above by dividing each entry by its column sum Add a column to sum each row and then take the average. Price Body Style MPG Interior Quality Engine Size Row Sum Priority Vector (Row sum average) Price 0.07 0.05 0.16 0.03 0.10 0.40 0.08 Body Style 0.26 0.21 0.26 0.41 0.16 1.30 0.26 MPG 0.02 0.04 0.05 0.03 0.10 0.24 0.05 Interior Quality 0.33 0.07 0.26 0.14 0.16 0.96 0.19 Engine Size 0.33 0.63 0.26 0.41 0.48 2.11 0.42 Column sum 1.000 1.000 1.000 1.000 1.000 5.000 1.000 Slide 31

Row Average Technique - Continued Multiply original non-normalized matrix by Priority Vector Total each Row Price Body Style MPG Interior Quality Engine Size Row Totals Price 0.08 0.07 0.15 0.04 0.08 0.42 Body Style 0.32 0.26 0.25 0.57 0.14 1.54 MPG 0.03 0.05 0.05 0.04 0.08 0.25 Interior Quality 0.40 0.09 0.25 0.19 0.14 1.07 Engine Size 0.40 0.78 0.25 0.57 0.42 2.42 e.g. For the first row of the matrix on Slide 31 --- 1 *.08, ¼ *.26, 3 *.05, 1/5 *.19, and 1/5 *.42 Slide 32

Estimate the Eigenvector Take column of Row Totals and divide by the Priority Vector 0.42 0.08 5.21 1.54 0.26 5.92 0.25 divide by 0.05 equals 5.01 1.07 0.19 5.61 2.42 0.42 5.76 Now average the result to obtain Lambda Max (5.21 + 5.92 + 5.01 + 5.61 + 5.76) / 5 Lambda Max 5.50 Consistency Index 0.13 = (LambdaMax -n) / (n-1) = (5.50-5) / (4) Consistency Ratio 0.12 = (CI) / (Random Index) = 0.13 / 1.11...note new RI value used here Techniques compared: Row Average Geometric Mean Lambda Max 5.50 5.48 Consistency Index.13.12 Consistency Ratio.12.11 The Row Average technique produces a consistency ratio that is slightly worse than the Geometric Mean technique (0.11). Slide 33

Summary Consistency in the pair-wise comparisons of your criteria in very important. My first attempt would have led to an incorrect decision. Revising my judgments changed my consistency ratio from 38% to 11%, where the goal is 10% or less. These more consistent judgments changed the results of my decision. Using Saaty s recommendations from his 2001 book instead of his original 1980 book produced a larger inconsistency (12%) of my judgments of the pair-wise comparisons. This implies I should go back to my judgments (pair-wise comparisons) of the criteria and reconsider their relative importance to me. Slide 34

References Golden, Bruce L., Wasil, Edward A, and Harker, Patrick T. (editors): The Analytic Hierarchy Process - Applications and Studies, Springer-Verlag, Berlin, 1989. INCOSE Systems Engineering Handbook, Version 2a, Appendix D.9, International Council on Systems Engineering, INCOSE-TP- 2003-016-02, Version 2a, 1 June 2004. Saaty, Thomas L.: Decision Making for Leaders, RWS Publications, Pittsburgh, 2001. Saaty, Thomas L.: Priority Setting in Complex Problems, IEEE Transactions on Engineering Management, Vol. EM-30, No. 3, August 1983. Saaty, Thomas L.: The Analytic Hierarchy Process, McGraw-Hill, Inc., New York, 1980. Slide 35