1. Lecture 08 Analytic Hierarchy Process (Module 1)
Industrial Systems Engineering Dept.- IU
Office: Room 508

2. Learning Objectives Students will be able to:
Use the multifactor evaluation process in making decisions that involve a number of factors, where importance weights can be assigned.
Understand the use of the analytic hierarchy process in decision making.
Contrast multifactor evaluation with the analytic hierarchy process.

3. Module Outline M1.1 Introduction M1.2 Multifactor Evaluation Process
M1.3 Analytic Hierarchy Process

4. Introduction
Multifactor decision making involves individuals subjectively and intuitively considering various factors prior to making a decision.
Multifactor evaluation process (MFEP) is a quantitative approach that gives weights to each factor and scores to each alternative.
Analytic hierarchy process (AHP) is an approach designed to quantify the preferences for various factors and alternatives.
Example: Buying a computer or laptop, smart phone: have to consider some factor: cost, brand name, specifications, …
Give the weights for each of these factor (cost 60%, specification 30% and brand name 10%)
Give the scores of each factor for the alternatives you want to buy.

5. Multifactor Evaluation Process
Example: Steve. M.: considering employment with 3 companies.
determined 3 factors important to him, assigned each factor a weight.
Steve evaluated the various factors on a 0 to 1 scale for each of these jobs.
Factor
Importance
(weight)
AA Co.
EDS,
LTD.
PW,
Inc.
Salary
0.3
0.7
0.8
0.9
Career Advancement
0.6
Location
0.1
Score Table
Weights should sum to 1

6. Evaluation of AA Co. Factor Factor Factor Weighted
Weight Evaluation Evaluation X =
Factor Factor Factor Weighted
Name Weight Evaluation Evaluation
Salary
Career
Location
Total

7. Comparison of Results Decision is AA Co: Highest weighted evaluation
Factor
AA Co.
EDS,LTD.
PW,Inc.
Salary
0.21
0.24
0.27
Career
0.54
0.42
0.36
Location
0.06
0.08
0.09
Weighted Evaluation
0.81
0.74
0.72
Decision is AA Co: Highest weighted evaluation

8. The Analytic Hierarchy Process (AHP)
Founded by Saaty in 1980.
It is a popular and widely used method
for multi-criteria decision making.
Allows the use of qualitative, as well as
quantitative criteria in evaluation.
Wide range of applications exists:
Selecting a car for purchasing
Deciding upon a place to visit for vacation
Deciding upon an MBA program after graduation. …
Dr. Thomas L. Saaty
Distinguished Prof. at U. of Pittsburgh
In many situations one may not be able to assign weights to the different decision factors. Therefore one must rely on a technique that will allow the estimation of the weights.
What is a solution?
One such process, The Analytical Hierarchy Process (AHP), involves pairwise comparisons between the various factors.
Method for ranking decision alternatives and selecting the best one when the decision maker has multiple objectives, or criteria

9. AHP-General Idea
Develop an hierarchy of decision criteria and define the alternative courses of actions.
AHP algorithm is basically composed of two steps:
1. Determine the relative weights of the decision criteria
2. Determine the relative rankings (priorities) of alternatives
Both qualitative and quantitative information can be compared by using informed judgments to derive weights and priorities.

10. Steps Step 0: Construction of Hierarchy Structure
(including: Goal, Factors, Criteria, and Alternatives)
Step 1: Calculation of Factor Weight
Step 1-1: Pairwise Comparison Matrix
Step 1-2: Eigenvalue and Eigenvector (Priority vector)
Step 1-3:Consistency Test
Consistency Index
Consistency Ratio
Step 2:Calculation of Level Weight
Step 3: Calculation of Overall Ranking

11. Hierarchy Tree More General Goal More Specific Alternatives C1 C2 C3
Sub-criteria at the lowest level
Level 0
Level 1 (factors)
Level 2 (criteria)
Level ..
Tom Saaty suggests that hierarchies be limited to six levels and nine items per level.
This is based on the psychological result that people can consider 7 +/- 2 items simultaneously (Miller, 1956).

12. Pairwise Comparisons Size Apple A Apple B Apple C
Resulting
Priority
Eigenvector
Relative Size
of Apple
Apple A / A
Apple B 1/ / B
Apple C 1/ / / C
A compare with A = 1
consider the apple A and B about the size. A = 2B => B= ½A
Explain Eigenvector later
Criteria #1
Criteria #2 1
Intensity of
Importance 2 3 4 5 6 8 7 9

13. Pairwise Comparison Matrix
Ranking of Criteria and Alternatives
Pairwise Comparison Matrix
a33
a32
a31 A3
a22
a21 A2
a13
a12
a11 A1 to
Pairwise Comparison Matrix A = ( aij )
(a) aii = 1
A comparison of criterion i with itself: equally important
(b) aij = 1/ aji
aji are reverse comparisons and must be the reciprocals of aij
Ranking Scale for Criteria and Alternatives
Values for aij :
Numerical values
Verbal judgment of preferences 1
equally important 3
weakly more important 5
strongly more important 7
very strongly more important 9
absolutely more important
If compare through verbal judgment, the judgments will be transform to numerical values.
intermediate values
2,4,6,8 =>
reverse comparisons
reciprocals =>

14. Example 1: Car Selection (1/15)
Objective
Selecting a car
Criteria
Style, Reliability, Fuel-economy Cost?
Alternatives
Civic Coupe, Saturn Coupe, Ford Escort, Mazda Miata
In this example, Cost is not a factor. It will be consider later. 14 2

15. Example 1: Car Selection (2/15)
Hierarchy tree
Civic Saturn Escort Miata 3

16. Example 1: Car Selection (3/15)
Ranking of Criteria
Style
Reliability
Fuel Economy
1/1 1/2 3/1
2/1 1/1 4/1
1/3 1/4 1/1 16 4

17. Example 1: Car Selection (4/15) Example 1: Car Selection (4/15)
Ranking of Priorities
Consider [Ax = x] where
A is the comparison matrix of size n×n, for n criteria, also called the priority matrix.
x is the Eigenvector of size n×1, also called the priority vector.
 is the Eigenvalue,   > n.
To find the ranking of priorities, namely the Eigen Vector X:
1) Normalize the column entries by dividing each entry by the sum of the column.
2) Take the overall row averages.
Normalize is to transform the values to proportion (sum will be 1)
Pairwise Comp. Matrix
Norm. Pairwise Comp. Matrix
Priority vector
Row
Averages
Normalized
Column Sums
0.3196
0.5584
0.1220 A= X=
Column sums 5

18. Example 1: Car Selection (5/15)
Ranking of Priorities (cont.)
Criteria weights
Style ≈ .3
Reliability ≈ .6
Fuel Economy ≈ .1
Second most important criterion
First important criterion
The least important criterion
Here is the tree of criteria with the criteria weights 18 7

19. Checking for Consistency
Example 1: Car Selection (6/15)
Checking for Consistency
Consistency Ratio (CR): measure how consistent the judgments have been relative to large samples of purely random judgments.
AHP evaluations are based on the asumption that the decision maker is rational, i.e., if A is preferred to B and B is preferred to C, then A is preferred to C.
Suppose we judge apple A to be twice as large as apple B and apple B to be three times as large as apple C.
To be perfectly consistent, apple A must be six times as large as apple C.
If the CR is greater than 0.1 the judgments are untrustworthy because they are too close for comfort to randomness and the exercise is valueless or must be repeated.

20. Calculation of Consistency Ratio
Example 1: Car Selection (7/15)
Calculation of Consistency Ratio
The next stage is to calculate , Consistency Index (CI) and the Consistency Ratio (CR).
Consider [Ax = x] where x is the Eigenvector.
A x Ax x
A x Ax x
A x Ax x
0.90
1.60
0.35
0.30
0.60
0.10
0.30
0.60
0.10
=  = = =
0.90/0.30
1.60/0.60
0.35/0.10
3.00
2.67
3.50
Consistency Vector = =
Consistency index (CI) is found by
Note: This is just an approximate method to determine value of λ

21. Example 1: Car Selection (8/15)
Consistency Index
reflects the consistency of one’s judgement
Random Index (RI)
the CI of a randomly-generated pairwise comparison matrix
Tabulated by size of matrix (n): (given by author)
n RI
2 0.0
Prof. Saaty suggest the compare CI with the Random Consistency Index RI

22. Example 1: Car Selection (9/15)
Consistency Ratio
In practice, a CR of 0.1 or below is considered acceptable.
Any higher value at any level indicate that the judgements warrant re-examination.
In the above example:
so, the evaluations are consistent

23. Example 1: Car Selection (10/15)
Ranking Alternatives
Priority vector
Style
Civic
Saturn
Escort
Miata
Civic
/ /6
0.13
0.24
0.07
0.56
Saturn /4
Escort
1/ / /5
Miata
Miata
Reliability
Civic
Saturn
Escort
Miata
0.38
0.29
0.07
0.26
Civic
Similar with Factor Ranking, the Priority vectors of Alternatives are determined.
The corresponding CR should be determined to ensure the consistency of pair-wise judgment
Saturn 1/
Escort
1/ / /4
Miata / 8

24. Example 1: Car Selection (11/15) Example 1: Car Selection (11/15)
Ranking Alternatives (cont.)
Ranking Alternatives (cont.)
Miles/gallon
Normalized
Civic 34
.30
Fuel Economy
Saturn 27
.24
Escort 24
.21
Miata
Miata 28
113
.25
1.0
! Since fuel economy is a quantitative measure, fuel consumption ratios can be used to determine the relative ranking of alternatives; however this is not obligatory. Pairwise comparisons may still be used in some cases. 9

25. Example 1: Car Selection (12/15)
Ranking Alternatives (cont.)
Selecting a New Car
1.00
Style
0.30
Reliability
0.60
Fuel Economy
0.10
Civic
Saturn
Escort
Miata
Civic
Saturn
Escort
Miata
Civic
Saturn 0.29
Escort
Miata
Choose the largest value.
Car Style(0.3) Reliability(0.6) Fuel Economy(0.1) Total
Civic
Saturn
Escort
Miata
largest 10

26. Ranking of Alternatives (cont.)
Example 1: Car Selection (13/15)
Ranking of Alternatives (cont.)
Reliability
Style
Economy
Fuel
Civic
Escort
Miata
Saturn x
.30
.60
.10 =
.27
.08
.35
Priority matrix
Factor Weights 11

27. Including Cost as a Decision Criteria
Example 1: Car Selection (14/15)
Including Cost as a Decision Criteria
Adding “cost” as a a new criterion is very difficult in AHP. A new column and a new row will be added in the evaluation matrix. However, whole evaluation should be repeated since addition of a new criterion might affect the relative importance of other criteria as well!
Instead one may think of normalizing the costs directly and calculate the cost/benefit ratio for comparing alternatives!
Cost/Benefits
Ratio
Normalized
Cost
Cost
Benefits
CIVIC $12K
SATURN $15K
ESCORT $ 9K
MIATA $18K 13

28. Methods for including Cost Criterion
Example 1: Car Selection (15/15)
Methods for including Cost Criterion
Use graphical representations to make trade-offs.
Calculate cost/benefit ratios
Use linear programming
Use seperate benefit and cost trees and then combine the results
Miata
Miata
Civic
Civic
Saturn
Saturn
Escort
Escort

29. Complex Decisions
Many levels of criteria and sub-criteria exists for complex problems.

30. Example 2: Buying the best car
*Goal: Buying the best car
*There are three criteria:
Cost
Quality
Maintenance
Insurance
Services
*Three alternatives: Honda, Mercedes, Hyundai 30

31. The Hierarchy for problem Buying the best car
Example 2: Buying the best car
Level 0
Level 1
Criteria
Level 2
Sub-criteria
Alternatives
The Hierarchy for problem Buying the best car 31

32. Example 2: Buying the best car
Step 1: Criterion comparison
Criterion comparison
Normalize values:
Find Column vector
The process is repeated for the sub-criteria until the evaluation for all other alternatives. This example will be supported by Expert Choice software 32

33. Example 2: Buying the best car
Step 2: Determining the Consistency Ratio - CR
2.1. Determining the Consistency vector
We begin by determining the weighted sum vector. This is done by multiplying the column vector times the pairwise comparison matrix.
Column vector: Pairwise comparison matrix: 1/
1/ / X
Demo on board
Column vector = priority vector
Consistency vector
Weighted sum vector
1.948
0.690
0.366
Consistency vector =
Weighted sum vector/ Column vector 33

34. Example 2: Buying the best car
2.2. Determining  and the Consistency Index-CI
 = ( ) / 3 = 3.002
The CI is:
CI = ( ) / (3 - 1) = 0.001
2.3. Determining the Consistency Ratio-CR
with n = 3, we get RI = 0.58
CR = / 0.58 =
Since 0< CR < 0.1, we accept this result and move to the lower level. The procedure is repeated till the lowest level. 34

35. For Cost For Service For Insurance: For Quality
Continue for other levels:
For subcriteria Insurance – Service:
For Cost
For Service
For Insurance:
For Quality
And make your final evaluation (students self develop this evaluation) 35

37. Demo on board

38. Other way:

39. 1) Weights are defined for each hierarchical level...
2) ...and multiplied down to get the final lower level weights.
0.6
0.4
0.6
0.4
Multiply
0.7
0.3
0.2
0.6
0.2
0.7
0.3
0.2
0.6
0.2
0.42
0.18
0.08
0.24
0.08
Elicitation: suy luan

40. The AHP solving is computer-aided by Expert Choice (EC) software.
Notes:
In general, the evaluation scores are collected from many experts and the average scores is used in the pairwise comparison matrix.
The AHP solving is computer-aided by Expert Choice (EC) software.
- Building structure of problem !!!
- Enter judgments (Pairwise Comparisons)
- Analysis the weights
- Sensitivity Analysis
- Advantages and disadvantages
- Miscellaneous 40

41. More about AHP: Pros and Cons
It allows multi criteria decision making.
It is applicable when it is difficult to formulate criteria evaluations, i.e., it allows qualitative evaluation as well as quantitative evaluation.
It is applicable for group decision making environments
Pros
There are hidden assumptions like consistency. Repeating evaluations is cumbersome.
Difficult to use when the number of criteria or alternatives is high, i.e., more than 7.
Difficult to add a new criterion or alternative
Difficult to take out an existing criterion or alternative, since the best alternative might differ if the worst one is excluded.
Users should be trained to use AHP methodology.
Cons
Pros and Cons = advantages & disadvantage
Use cost/benefit ratio if applicable

42. Homework 08 (Due: next class)
M 1.4, M 1.10, M 1.11