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Analytic hierarchy process

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### Analytic hierarchy process

Dr. Yan Liu

Department of Biomedical, Industrial & Human Factors Engineering

Wright State University

Introduction

- Conflicting Objectives and Tradeoffs in Decision Problems
- e.g. higher returns vs. lower risks in investment, better performance vs. lower price of computer

- Objectives with Incomparable Attribute Scales
- e.g. maximize profits vs. minimize impacts on environments

- Multi-Attribute Decision Making (MADM)
- A study of methods and procedures that handle multiple attributes
- Usages
- Identify a single most preferred alternative
- Rank alternatives
- Shortlist a limited number of alternatives for subsequent detailed appraisal
- Distinguish acceptable from unacceptable possibilities

Introduction (Cont.)

- Types of MADM Techniques
- Multiattribute scoring model (in Chapter 4)
- Covert attributes to comparable measures
- Assign weights to these attributes and then calculate the weighted average of each consequence set as an overall score
- Compare alternatives using the overall score

- Multi-objective mathematical programming
- Tackle complex problems involving a large number of decision variables that are subject to constraints

- Analytic hierarchy process (AHP)
- Multi-attribute utility theory (MAUT)
- …

- Multiattribute scoring model (in Chapter 4)

What is AHP?

- A Process that Leads One to (Saaty, 1980)
- Structure a problem as a hierarchy or as a system with dependence loops
- Elicit judgments that reflect ideas, feelings, and emotions
- Represent those judgments with meaningful numbers
- Synthesize results
- Analyze sensitivity to changes in judgments

- AHP also uses a weighted average approach idea, but it uses a method for assigning ratings and weights that is considered more reliable and consistent

Purposes of AHP

- To structure complexity in gradual steps from the large to the small, or from the general to the particular, so we can relate them with greater accuracy according to our understanding
- To improve our awareness by richer synthesis of our knowledge and intuition; AHP is a learning tool rather than a means to discover the TRUTH

Phases in AHP

- Phase 1: Decompose the problem into a hierarchy
- Start with an identification of the criteria to be used in evaluating different alternatives, organized in a tree-like hierarchy

- Phase 2: Collect input data by pairwise comparisons of criteria at each level of the hierarchy and alternatives
- Phase 3: Estimate the relative importance (weights) of criteria and alternatives and check the consistency in the pairwise comparisons
- Phase 4: Aggregate the relative weights of criteria and alternatives to obtain a global ranking of each alternative with regards to the goal

Job Satisfaction

GOAL

Reputation

Research

Location

CRITERIA

Growth

Benefits

Colleagues

Job C

Job B

ALTERNATIVES

Job A

Hierarchy for a Job Selection Decision

Hierarchy (Cont.)

- How to Structure a Hierarchy
- Identify the overall objective or goal
- Identify criteria to satisfy the goal
- Identify, where appropriate, sub-criteria under each criterion
- Identify alternatives to be evaluated in terms of the sub-criteria at the lowest level
- If the relative importance of the sub-criteria can be assessed and the alternatives can be evaluated in terms of the sub-criteria, the hierarchy is finished
- Otherwise, continue inserting levels until it is possible to link levels and set priorities (relative weights) on the elements at each level in terms of the elements at the level above it

Goal

C3

C1

C2

C21

C22

C31

C32

C33

C11

C13

C12

Sub-criteria at the lowest level

More Specific

Alternatives

Structure of a Hierarchy

Overall Benefit

Political

Factors

Health, Safety

Environment

National

Economy

Centra-

lization

Political

Cooperative-

ness

Independ-

ence

Foreign

Trade

Capital

Resources

Cheap

Electricity

Natural

Resources

Accidents

Long-Term

Risks

Unavoidable

Pollution

Coal-fired

power plant

No big

power plants

Nuclear

power plant

Energy Decision in the Parliament of Finland

Hierarchy (Cont.)

- How Large Should a Hierarchy Be?
- Large enough to capture decision maker’s major concerns
- Small enough to remain sensitive to change in what is important

Judgment and Preference

- In AHP, we use subjective judgment to express preference and its intensity
- e.g. Which of two apples is more red and how strongly more red we perceive it to be

- From this preference we derive a scale of relative strength of preference

Pairwise Comparison

- Define the relative importance of criteria at each level of the hierarchy and relative importance of alternatives by means of pairwise comparisons

Pairwise Comparison Matrix

(Wi is the relative weight of ith criterion (alternative))

Pairwise Comparison (Cont.)

- Scale for Pairwise Comparisons
- 1. Equally preferred
- 3. One is moderately preferred over the other
- 5. One is strongly preferred over the other
- 7. One is very strongly preferred over the other
- 9. One is extremely preferred over the other
- 2,4,6,8 intermediate values
- Reciprocals for inverse comparison

Relative Weights

Denote the pairwise comparison matrix and the weight matrix as

Then

Or

λis the eigenvalue of A and W is its corresponding right eigenvector

There are n eigenvalues and n corresponding eigenvectors for Anxn

Relative Weights (Cont.)

- Consistency
- Transitivity
- a1>a2 and a2>a3, then a1>a3

- Measurement consistency
- aij∙ ajk = aik (aij – the cell at the ith row and jth column of the comparison matrix)

- Transitivity
- Consistency Index (CI)
- A measure of deviation of consistency

λmax is the maximum eigenvalue of the pairwise comparison matrix

CI = 0 or λmax = n implies perfect consistency

CI = 0.1 is the generally accepted threshold value

Find Eigenvalues and Eigenvectors

, where

The solution to the equation is given by

You can solve eigenvalue and eigenvector problems in Matlab using the command

“[V,E]=eig(A)”, where E is the eigenvalue and V is the corresponding eigenvector

Find Eigenvalues and Eigenvectors Using Matlab

Suppose the comparison matrix of three criteria is

There is only one non-zero real eigenvalue for the pairwise comparison matrix in AHP

eigenvectors

λmax =3.0385

Its corresponding eigenvector is

(0.9161, 0.3715, 0.1506)T; the three numbers in the eigenvector are proportional to the relative weights of the three criteria

eigenvalue

Relative Weights (Cont.)

Because relative weights must sum up to 1, we have to normalize the eigenvector by dividing each number in it by the sum of all numbers

e.g. In the previous example, the eigenvector is (0.9161, 0.3715, 0.1506)T

The normalized eigenvector is

=(0.64, 0.26, 0.10)T

Relative weights for the three criteria

CI=(λmax-n)/(n-1) = (3.0385-3) /(3-1) = 0.0193

GOAL

Friends

Vocational

Training

Music

Classes

Learning

School

Life

College

Preparation

CRITERIA

School A

School C

ALTERNATIVES

School B

School Selection ExamplePairwise Comparison Matrix of Six Criteria

(L,F,SL,VT,CP, and MC denote Learning, Friends, School Life, Vocational Training, College Preparation, and Music Classes, respectively)

Using Matlab, we can get λmax =6.2397, and its corresponding eigenvector is ( -0.6773, -0.2198, -0.1908, -0.5780, -0.3212, 0.1389)T

The normalized eigenvector is (0.32, 0.10, 0.09, 0.27, 0.15, 0.07)T

CI =( 6.2397 – 6) / (6 – 1) = 0.048 < 0.1

0.16

0.59

0.25

Pairwise Comparison Matrix of Three Alternative Schools With Respect to Learning

λmax=3.0536, , and the normalized eigenvector is (0.16, 0.59, 0.25)T

CI =( 3.0536 – 3) / (3 – 1) = 0.027

0.33

0.33

0.33

Pairwise Comparison Matrix of Three Alternative Schools With Respect to Friends

λmax=3, and the normalized eigenvector is (1/3, 1/3, 1/3)T

CI =( 3 – 3) / (3 – 1) = 0

0.46

0.09

0.46

Pairwise Comparison Matrix of Three Alternative Schools With Respect to School Life

λmax=3, and the normalized eigenvector is (0.46, 0.09, 0.46)T

CI =( 3 – 3) / (3 – 1) = 0

0.77

0.05

0.17

Pairwise Comparison Matrix of Three Alternative Schools With Respect to Vocational Training

λmax=3.2085, and the normalized eigenvector is (0.77, 0.05, 0.17)T

CI =( 3.2085 – 3) / (3 – 1) = 0.104

0.25

0.50

0.25

Pairwise Comparison Matrix of Three Alternative Schools With Respect to College Preparation

λmax=3, and the normalized eigenvector is (0.25, 0.50, 0.25)T

CI =( 3 – 3) / (3 – 1) = 0

0.69

0.09

0.22

Pairwise Comparison Matrix of Three Alternative Schools With Respect to Music Classes

λmax=3.053, and the normalized eigenvector is (0.69, 0.09, 0.22)T

CI =( 3.0536 – 3) / (3 – 1) = 0.027

Relative Weights (Cont.)

- Other than computing the eigenvector of a pairwise comparison matrix to find the weights of compared criteria or alternatives, we can also approximate the weight by:
- First, normalizing each column in the comparison matrix
- Then, calculating the average of each row in the normalized matrix as the estimate of the relative weight for its corresponding criterion or alternative

T2

Tn

CM2=

CM1=

CMn=

Tn

T2

Tn

T2

…

Total

Tn=

T2=

T1=

CI ≈

Consistency

Measure

Approximated Weight

T1

T1

…

…

T1

School Selection Example (Cont.)

Sum

3.17

10.50

12.50

3.87

7.00

14.00

Pairwise Comparison Matrix of Six Criterion

Approximated Weight

0.32

0.32

0.10

0.10

0.09

0.09

0.27

0.27

0.15

0.15

0.07

0.07

Normalized Pairwise Comparison Matrix of Six Criteria

CM1=

Consistency

Measure

Approximated Weight

6.22

0.32

6.5

0.10

6.29

0.09

6.30

0.27

6.31

0.15

5.86

0.07

CM2=

Likewise, we can calculate CM3, CM4, CM5, and CM6

CI =0.048 from eigenvalue

CI ≈

0.16

0.59

Sum

6

1.67

4.5

0.25

Pairwise Comparison Matrix of Three Alternative Schools With Respect to Learning

Normalized Pairwise Comparison Matrix of Three Alternative Schools With Respect to Learning

Composition and Synthesis

- Combine the relative importance of criteria and alternatives to obtain a global ranking of each alternative with regards to the goal

Criteria Cj (j=1,2,…,n) and their corresponding weights

Composite impact

OA1 = wC1wA1C1+wC2wA1C2+…+wCnwA1Cn

OA2 = wC1wA2C1+wC2wA2C2+…+wCnwA2Cn

…

OAm = wC1wAmC1+wC2wAmC2+…+wCnwCnAm

Weights of alternatives Ai (i=1,2,…,m) w.r.t. criteria Cj (j=1,2,…,n)

OA=0.32*0.16+0.10*0.33+0.09*0.45+0.27*0.77+0.15*0.25+0.07*0.69 = 0.42

OB=0.32*0.59+0.10*0.33+0.09*0.09+0.27*0.05+0.15*0.5+0.07*0.09 = 0.33

OC=0.32*0.25+0.10*0.33+0.09*0.46+0.27*0.17+0.15*0.25+0.07*0.22 = 0.25

In conclusion, school A seems to be the best, and C seems to be the worst

Summary of AHP

- Applications of AHP
- e.g. resource allocation , conflict resolution, prediction, planning, etc.

- Advantages
- Decision hierarchy and pairwise comparisons make the AHP process easy to comprehend
- The use of a subjective scale, such as “strongly preferred”, rather than a quantitative scale is particularly useful when it is difficult to formalize some criteria (attributes) quantitatively
- It is usually much easier to compare two items at a time than to compare many items all at once

- Disadvantages
- The decision hierarchy in AHP assumes independence among criteria, which is not always appropriate
- The subjective scale is subject to human errors and biases
- The number of pairwise comparisons becomes quite extensive when the number of attributes and alternatives is large

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