The Analytic Hierarchy Process. Danny Hahn

(1)

The Analytic Hierarchy Process

(2)

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.

(3)

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

(4)

The Hierarchy

Goal

Factor 1

Factor 2

Factor 3

(5)

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

(6)

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

(7)

The Car Decision Hierarchy

Buy the

Right Car

Price

Body

Style

Interior

Quality

X-Treem

Yaawhee

Engine

Size

MPG

Zoomer

(8)

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.

(9)

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 | | | | | | | | | | | | | | | | | MPG

Body Style | | | | | | | | | | | | | | | | | MPG

MPG | | | | | | | | | | | | | | | | | Interior Quality

MPG | | | | | | | | | | | | | | | | | Engine Size

(10)

Matrix Review

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

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

(11)

Convert Criteria Comparisons to a Matrix

Convert the pair-wise comparisons from Slide 9 to a matrix.

Prioritizing the 5 Criteria

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.

(12)

The Decision Candidates

The next step is to evaluate all three cars on each of the five criteria as shown in the Hierarchy Chart on Slide 7

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

(13)

Evaluate Price for Each Car

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 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

(14)

Evaluate Body Style for Each Car

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 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 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

(15)

Evaluate MPG for Each Car

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 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 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

(16)

Evaluate Interior Quality

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 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 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

(17)

Evaluate Engine Size

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 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 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

(18)

Compute the Final Result

The winner is the Yaawhee.

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

(19)

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).

(20)

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.

(21)

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.

(22)

Step by Step on Original Matrix

1. Add a new column (5th _{Root) and compute the 5}th _{root of the product of the}

values in each row.

2. Sum this 5th _{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

Prioritizing the 5 Criteria

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

(23)

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)

(24)

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

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.

(25)

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 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)

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)

(27)

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

(28)

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.

(29)

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 nth_{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).

(30)

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.

(31)

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

(32)

Row Average Technique - Continued

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

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

(33)

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

(34)

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.

(35)

The Analytic Hierarchy Process. Danny Hahn