Chapter 2 ANALYTIC HIERARCHY PROCESS A H P

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Chapter 2 ANALYTIC HIERARCHY PROCESS(A H P)
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Overview

• AN MADM TECHNIQUE
• Developers Prof. T.L. Saaty Associates
  University of pittsburgh, U.S.A
• Software Expert Choice (Developed by Saaty
  Associates)
• AHP a MADM technique
• Approach
• Analysis of a complex problem through
  decomposition and synthesis
• Structured in hierarchy
• Hierarchy
• A system of levels, each consisting of elements
  or factors
• Hierarchical structuring
• How do we structure the functions of a system
  hierarchically?
• How do we measure the impacts of any element in
  the hierarchy?

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• Examples of Hierarchy

Level 1 Focus
Best New Car to Buy
Level 2 Criteria
Price
Running cost
Comfort
Status
Level 3 Alternatives
A
B
C
Hierarchy model To Buy Best New Car
Level 1 Focus Level 2 Criteria Level 3
Sub-Criteria Level 4 Alternatives
Hierarchy Model To Rank Schools
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Advanced Manufacture System
Selection of Alternatives
NPV
Flexibility
Productivity
Existing
Flexible Manufacturing System
Flexible Manufacturing Cell
Performance Evaluation Hierarchy
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Profitability
Dependability
Flexibility
Cost
Quality
Number of products
Capital Cost
Producers view
Production process
Operating Cost
Consumers view
Design Changes
Labor
Volume Changes
Prob. of Failure
Time to Repair
Reject Rate
Rework Rate
Availability
Skill Level
Lead time to manufacturing
Prob. of meet. demand

A
B
Z
C
Hierarchy Model For Capacity Expansive Model
Decision
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Societys Overall Benefit
Health,,Safety Environment
Political Factors
National Economy
Cheap Electricity
Foreign Trade
Capital Resource
National Resource
Unavailable Pollution
Accident Long-term Risks
Independence
Centralization
Political co-operativeness
No big Power plants
Coal Fired Power Plant
Nuclear Power Plant
Hierarchy Model ENERGY DECISION
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Level 1 Focus
Dam Decision
Govt. Authority
Parliament
Local Citizens
Conserva-tionists
Level 2 Actors
Flood Control
Economic Stimulus
Preservation of Species
Protecting Environ.
Level 3 Criteria
Alt-1 Completed the Dam
Alt-2 Halt Construction
Level 4 Alternatives
Alt-3 Replace
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Aim of Decision
Criteria 1
Criteria 1
Criteria 1
Sub-criteria 1
Sub-criteria 1
Sub-criteria 1
Criteria 1 in lowest level
Criteria 1 in lowest level
Criteria 1 in lowest level
Project 1
Project 1
Project 1
Stratified Hierarchic Structure Pattern (Relative
and Absolute ranking)
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Pair-wise ComparisonScale of Relative Importance

• Intensity Definition Explanation
• 1 Equal importance Two activities contribute
  equally to the objective
• 3 Moderate importance Slightly favors one over
  another
• 5 Essential or strong Strongly favors one over
  importance another
• 7 Demonstrated Dominance of one
  importance demonstrated in practice
• 9 Extreme importance Evidence favoring one over
  another of higher possible order of
  affirmative
• (2,4,6,8) Intermediate value When compromise is
  needed

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Comparison Matrix An Example
A B C A 1 3 9 B 1/3 1 3 C 1/9 1/3 1
n(n-1)/2 comp.
Column
9/13 3/13 1/13
0.69 0.23 0.08
1 1/3 1/9
Normalized to
or
Represents the relative importance of A,B,C
corresponding to the judgments in the first
row. For the other columns, when procedure is
repeated, the same results are obtained.
3 1 1/3
0.69 0.23 0.08
9 3 1
0.69 0.23 0.08
This is a situation where call consistent. The
normalized principal eigenvector is identical to
the normalized columns of the comparison matrix.
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ESTIMATION OF PRINCIPAL EIGENVECTOR

• 1. Normalizing the sum of row elements
• 2. Taking the normalized form of the reciprocals
  of the sum of the elements of each column
• 3. Averaging over the normalized columns
• 4. Normalizing the geometric mean of each row
• (Recommended)
• Note
• For a non-consistent matrix, generally the above
  methods give different results
• For a consistent matrix the results are identical

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Example
A B C A 1 4 9 B 1/4 1 2 C 1/9 1/2 1
Normalized relative weights of 3 columns
0.735 0.184 0.084
0.727 0.182 0.091
0.750 0.167 0.083

• The inconsistency shown is not an error but that
  it reflects the process of
• obtaining as much data as possible.
• The principal eigenvector may be obtained by
  averaging across the rows

0.737 0.178 0.085
(Note A matrix is said to be consistent if aij
ajk aik for all i,j,k )
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Consistency

• Consistency should not be forced.
• For example BgtA 2gt1 and CgtB 3gt1
• We do not insist that CgtA 6gt1
• The rationed is independent judgement To add
  new info. and connection and balance for existing
  info. But, too much inconsistency is undesirable.
  
• It can be proved that, for a consistent
  reciprocal matrix ?max n
• where ?max largest eigenvalue of matrix order
  n.
• Saatys measure of consistency (Consistency
  Index)
• C.I. ( ?max n ) / (n-1)

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• How to determine ?max ?
• Add entries in each column of the judgment matrix
  and multiply the first column sum by the
  normalized weight of first row, etc. and add.
• A B C Norm. Wts..
• A 1 3 9 0.69
• B 1/3 1 3 0.23
• C 1/9 1/3 1 0.08
• Col. sum 1.44 4.33 13
• ?max 1.44(0.69) 4.33(0.23) 13(0.08) 3
• C.I. (3-3)/(3-1) 0

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• Example of inconsistency
• 1.
• A B C Norm. Wts.
• A 1 4 9 0.737
• B 1/4 1 2 0.178
• C 1/9 1/2 1 0.085
• S 1.36 5.5 12
• ?max 1.36(0.737) 5.5(0.178) 12(0.085)
  3.002
• C.I. 0.002/2 0.001
• 2.
• A B C Norm.Wts.
• A 1 2 1/2 0.333
• B 1/2 1 2 0.333
• C 2 1/2 1 0.333
• S 3.5 3.5 3.5

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• Randomly Generated C.I. (R.I.)
• n 1 2 3 4 5 6 7 8
• R.I 0 0 0.58 0.90 1.12 1.24 1.32 1.41
• 9 10 11 12 13 14 15
• 1.45 1.49 1.51 1.48 1.56 1.57 1.59
• Consistency Ratio C.R. C.I./R.I.
• Acceptable C.R. is 0.10 or less
• USING PREVIOUS EXAMPLES
• C.R. 0.001/0.58 1.6
• C.R. 0.248/0.58 50

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• ILLUSTRATIVE EXAMPLE
• Problem Hierarchy
• Level 1 Best Car to Buy
• Level 2 Price Running Cost
  Comfort Status
• Level 3 A B C
• Paired Comparison Matrix at Level 1
• Price Running Cost Comfort Status Norm.Wts.
  (Norm geom.mean)
• Price 1 3 7 8 0.582
• Cost 1/3 1 5 5 0.279
• Comfort 1/7 1/5 1 3 0.090
• Status 1/8 1/5 1/3 1 0.050
• S 1.6 4.4 13.3 17
• ?max 1.6(0.582) 4.4(0.279) 13.3(0.90)
  17(0.05) 4.198
• C.I. (4.198 4)/(4-1) 0.066
• C.R. 0.066/0.9 0.073 (Acceptable)

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• Paired Comparison Matrices at Level 2
• Price A B C Norm.Wts.
• A 1 2 3 0.540
• B ½ 1 2 0.297
• C 1/3 ½ 1 0.163
• ?max 3.009
• C.I. 0.005
• C.R. 0.008 (Acceptable)
• Running cost
• A B C Norm.Wts.
• A 1 1/5 ½ 0.106
• B 5 1 7 0.745
• C 2 1/7 1 0.150
• ?max 3.119
• C.I. 0.059
• C.R. 0.10 (Acceptable)
• Do the same procedure for Comfort and Status.
  Since their weights are very small, 0.09 and
  0.05. respectively , we assume that the effect of
  leaving them out from further consideration is
  negligible.
• Adjusted weight of Price 0.582 / (0.582
  0.279) 0.67

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• Overall
• Price Running Cost Composite Overall
  
  (0.67) (0.33)
  Wt. Ranking
• Car A 0.540 0.106 0.395 2
• Car B 0.297 0.745 0.444 1
• Car C 0.163 0.150 0.159 3
• Composite Wts.
• Car A 0.540(0.67) 0.106(0.33) 0.395
• Car B 0.279(0.67) 0.745(0.33) 0.444
• Car C 0.163(0.67) 0.150(0.33) 0.159
• Overall Consistency of Hierarchy
• 0.066(1) 0.055(0.67) 0.059(0.33) /
  0.9(1) 0.58(0.61) 0.58(0.33)
• 0.06

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SCALING OBJECTIVE FACTORS

• ORIGINAL SCALE

AHP SCALE
97531
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• Why tolerate 10 inconsistency ?
• Inconsistency itself is important for without it,
  new knowledge which changes preferences order
  cannot be admitted.
• Assuming all knowledge to be consistent
  contradicts experience which requires continued
  adjustment in understanding
• From experience it has been found that 10 is a
  resonable threshold value
• Why limit seven elements in paired comparison?
• Cognitive psychologists believe that an
  individual cannot simultaneously compare more
  than (7 2) elements (without being confused)

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Example of AHPs Ability to Predict

• ELECTION CHOICE

• PERSONALITY
• Region
• Charisma
• Media
• Appearance

• POLITICS
• Party
• Running mate
• Money
• Religion

• APTITUDE
• Leadership
• Experience

• PHYSICAL
• Age
• Appearance

• ISSUES
• Economic
• Security
• Foreign affairs
• Social order

CANDIDATES A B .