quantitative analysis for management multifactor evaluation process and analytic hierarchy process...
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Quantitative Analysis for Management
Multifactor Evaluation Processand
Analytic Hierarchy Process
Dr. Mohammad T. Isaai
Graduate School of Management & EconomicsSharif University of Technology
Quantitative Analysis for Management 2
Multifactor Evaluation Process (MFEP)
Assume you have to select among some alternatives
To evaluate each alternative, some factors are considered.
For example, someone intends to buy a house; then, the factors can be price, location, distance to their workplace, neighborhood etc.
Two types of evaluation is required:• Factor Weight• Alternative Evaluation
The Decision-Making Process:
Develop DecisionCriteria
Allocate Weightsto Criteria
DevelopAlternatives
AnalyzeAlternatives
Select Alternative
ImplementAlternative
Evaluate Results
Identify Problem
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MFEP Example
someone decides to buy one of the following cars: A B C D
There are three factors to consider: Style Reliability Fuel Economy
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Factor Weights
From decision maker’s point of view, importance weight of the factors are:
Style 0.5 Reliability 0.3 Fuel Economy 0.2
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Alternative EvaluationCar
Factor A B C D
Style 0.1 0.3 0.1 0.5
Reliability 0.4 0.3 0.1 0.2
Fuel Economy 0.2 0.2 0.5 0.1
The Results
A: 0.1*0.5+0.4*0.3 +0.2*0.2= 0.21
B: 0.3*0.5+0.3*0.3 +0.2*0.2= 0.28
C: 0.1*0.5+0.1*0.3 +0.5*0.2= 0.18
D: 0.5*0.5+0.2*0.3 +0.1*0.2= 0.33
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Analytic Hierarchy Process (AHP)
The difficulty with traditional MFEP is the evaluation approach. It is usually difficult or even impossible to evaluate all alternatives, especially when there are too many.
The AHP, developed by Tom Saaty in 1980, is a decision-making method for prioritizing alternatives when multi-criteria must be considered.
AHP is based on two concepts:• Hierarchy Design• Pair-wise Comparisons
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Comparisons How does AHP capture human judgments?How does AHP capture human judgments?
AHP AHP nevernever requires you to make an requires you to make an absolute judgmentabsolute judgment or or assessment. You would never be asked to directly estimate the assessment. You would never be asked to directly estimate the weight of a stone in kilograms.weight of a stone in kilograms.
AHP AHP doesdoes require you to make a require you to make a relative assessmentrelative assessment between between twotwo items at a time. AHP uses a ratio scale of measurement. items at a time. AHP uses a ratio scale of measurement.
For example, if there are four cars and you want to evaluate For example, if there are four cars and you want to evaluate their their Appearance . Appearance . Clearly it is difficult to evaluate them Clearly it is difficult to evaluate them absolutely.absolutely.
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Pair-wise Comparisons
However, it is easy to compare any two of them. The question is how do you prefer car A to car B, as far
as the appearance is concerned. The reply may be “I prefer it strongly” or “I prefer it
moderately”
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AHP Pair-wise Comparison ScaleNumerical Ratings Verbal Description of
Judgment 1 Equally preferred3 Moderately preferred5 Strongly preferred7 Very strongly preferred9 Extremely strongly preferred
We may also use numerical ratings 2, 4, 6 and 8. They describe something between two ratings.
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Complete AHP Scale
Numerical Verbal Description of Ratings Judgement 1 Equally preferred2 Equally to moderate preferred3 Moderately preferred4 Moderately to strongly preferred5 Strongly preferred6 Strongly to very strongly preferred7 Very strongly preferred8 Very strongly to extremely strongly pr9 Extremely strongly preferred
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AHP problems are structured in at least AHP problems are structured in at least three levels:three levels:
1.1. The goalThe goal,, such as selecting the best car to such as selecting the best car to purchase,purchase,
2.2. The criteriaThe criteria,, such as style, Reliability, and Fuel such as style, Reliability, and Fuel Economy,Economy,
3.3. The alternativesThe alternatives,, namely the cars themselves. namely the cars themselves.
Hierarchy Structure
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Example: Car Selection
Objective Selecting a car
Criteria Style, Reliability, Fuel-economy Cost?
Alternatives Civic Coupe, Saturn Coupe, Ford Escort,
Mazda Miata
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Hierarchical tree
S tyle R e lia b ility F u e l E con o m y
S e lec tinga N e w C ar
- Civic- Saturn- Escort- Miata
- Civic- Saturn- Escort- Miata
- Civic- Saturn- Escort- Miata
Level 1: Goals
Level 2: Criteria
Level 3: Alternatives
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First, we deal with
FACTOR WEIGHTS
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Pair-wise Comparisons
Q: How important is Style with respect to Fuel Economy?
A: Moderately Important. Then numerical rating is 3
As a result the importance of Fuel Economy with respect to Style is 1/3.
This type of comparison continues for every pair of factors
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Pair-wise Comparisons
Similarly Reliability with respect to style is equally to moderately important (Rating=2).
Reliability with respect to fuel economy is moderately to strongly important (Rating=4).
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Pair-wise Comparisons Matrix
Clearly, each factor is as important as itself. Then, all diagonal entries are equal to 1.
The criteria matrix is as follows.
Style Reliability Fuel Economy
Style
Reliability
Fuel Economy
1 1/2 3
2 1 4
1/3 1/4 1
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Ranking the Factors-1
Step 1. Normalize each column, i.e. divide the elements of each column by its total.
Factors Style Rel. F.E
Style 1 0.5 3
Rel. 2 1 4
F.E. 0.33 0.25 1
Total 3.33 1.75 8
Factors Style Rel. F.E
Style 0.3 0.285 0.375
Rel. 0.6 0.57 0.5
F.E. 0.1 0.145 0.125
Total 1 1 1
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Ranking the Factors-2
Step 2. Calculate Factor Weights.
Factors Style Rel. F.E Average
Style 0.3 0.285 0.375 (0.3+0.285+0.375)/3= 0.3196
Rel. 0.6 0.57 0.5 (0.6+0.57+0.5)/3= 0.5584
F.E. 0.1 0.145 0.125 (0.1+0.145+0.125)/3= 0.122
Total 1 1 1 1
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Preference Style 0.3196 Reliability 0.5584 Fuel Economy 0.1220
S tyle.3 196
R e lia b ility.5 584
F u e l E con o m y.1 220
S e lec tinga N e w C ar
1 .0
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Second, we deal with
Alternative Evaluations
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Ranking alternatives
Style
Civic
Saturn
Escort
1 1/4 4 1/6
4 1 4 1/4
1/4 1/4 1 1/5
Miata 6 4 5 1
Civic Saturn Escort Miata
Reliability
Civic
Saturn
Escort
1/1 2/1 5/1 1/1
1/2 1/1 3/1 2/1
1/5 1/3 1/1 1/4
Miata 1/1 1/2 4/1 1/1
Civic Saturn Escort Miata
.1160
.2470
.0600
.5770
Average
.3790
.2900
.0740
.2570
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Fuel Economy(quantitative information)
Civic
Saturn
Escort
MiataMiata
34
27
24
28 113
Miles/gallon Normalized
.3010
.2390
.2120
.2480 1.0
Since we have quantitative information, it is not required to compare alternatives pair-wise.
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S tyle.3 196
R e lia b ility.5 584
F u e l E con o m y.1 220
S e lec tinga N e w C ar
1 .0
- Civic .1160- Saturn .2470- Escort .0600- Miata .5770
- Civic .3790 - Saturn .2900- Escort .0740- Miata .2570
- Civic .3010- Saturn .2390- Escort .2120- Miata .2480
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Ranking of alternatives
Style Reliability Fuel Economy
Civic
Escort
MiataMiata
Saturn
.1160 .3790 .3010
.2470 .2900 .2390
.0600 .0740 .2120
.5770 .2570 .2480
* .3196
.5584
.1220
=
.3060
.2720
.0940
.3280
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Complex decisions
•Many levels of criteria and sub-criteria
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Example
Level 1: Evaluation of Representatives Level 2:
Commercial Evaluation Service Evaluation Management Evaluation Technical Evaluation
Level 3 Attributes of Level 2 Level 4 Representatives
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Level 3
Commercial Attributes Amount of Spare parts Purchasing Credit Ratio of Returns to Purchasing Ratio of Debits to Purchasing Ratio of Purchasing to Forcast Ratio of Purchasing to Guarantee
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Level 3
Service Attributes Initial Service Guarantee Repair
Management Attributes Manpower Systems Communications Management
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Level 3
Technical Attributes Building Equipment Tools Security system
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Consistency
In pair-wise comparison, if alternative A is preferred over B and B is preferred over C, then clearly A must be preferred over C. Now if in pair-wise comparison C is preferred over A, it is said that the system is not consistent.
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Consistency Test
Step 1. Multiply the pair-wise comparison matrix by the average rating. The result is called weighted sum vector.
Step 2. Divide the vector of Step 1 by the average rating. The result is called consistency vector.
Step 3. is the sum of consistency vector elements. Step 4. Calculate ,where n is the number of
items. Step 5. Determine RI, from table on page 526, for
example for n=3, RI=0.58. Step 6. Calculate CR= CI/RI. If CR <0.1, then it is
consistent.
1
nnCI
AHP and Related Software Expert Choice (Forman)
Criterium DecisionPlus (Hearne Scientific Software)
HIPRE 3+ (Systems Analysis Laboratory, Helsinki)
Web-HIPRE
Super Decisions (Saaty)
EC Resource Aligner combines optimization with AHP to select the optimal combination of alternatives or projects subject to a budgetary
constraint
The first web-based multiattribute decision analysis tool
This software implements the analytic network process (decision making with dependence and feedback)
7
Quantitative Analysis for Management 35
Group Decision Making
Group Problem Solving Techniques
Brainstorming process to generate a quantity of ideas Delphi Technique process to generate ideas from physically
dispersed experts Nominal Group Technique process to generate ideas and
evaluate solutions Computer-Aided Decision Making
Special topics in AHP
Quantitative Analysis for Management 36
Modeling Group Decisions
Suppose there are n decision makers Most common approach
Have each decision maker k fill in a comparison matrix independently to obtain [ ak
ij ]
Combine the individual judgments using the geometric mean to produce entries A = [ aij ] where
EM is applied to A to obtain the priority vector
aij = [ a1ij x a2
ij x … x anij ] 1/n
38
Extending the 1-9 Scale to 1-
•The 1-9 AHP scale does not limit us if we know how to use clustering of similar objects in each group and use the largest element in a group as the smallest one in the next one. It serves as a pivot to connect the two.
•We then compare the elements in each group on the 1-9 scale get the priorities, then divide by the weight of the pivot in that group and multiply by its weight from the previous group. We can then combine all the groups measurements as in the following example comparing a very small cherry tomato with a very large watermelon.
.07 .28
.65
Unripe Cherry Tomato
Small Green Tomato
Lime
.08
.22
.70
Lime
1=.08
.08
.65 .651x
Grapefruit
2.75=.08
.22
.65 1.792.75
Honeydew
8.75=.08
.70
.65 5.698.75x
.10
.30
.60
Honeydew
1=.10
.10
5.69 5.691
Sugar Baby Watermelon
3=.10
.30
5.69 17.073
Oblong Watermelon
6=.10
.60
5.69 34.146
This means that 34.14/.07 = 487.7 unripe cherry tomatoes are equal to the oblong watermelon
40
Clustering & ComparisonColor
How intensely more green is X than Y relative to its size?
Honeydew Unripe Grapefruit Unripe Cherry Tomato
Unripe Cherry Tomato Oblong Watermelon Small Green Tomato
Small Green Tomato Sugar Baby Watermelon Large Lime
Application Areas
Quantitative Analysis for Management 41
Illustrative Problem: Best Site Selection
The ABC Restaurant Corporation is offering franchise opportunities. After completing all the requirements from the applicants, the company
seeks the best site location from a set of alternative locations. There are three DMs to make the judgments: CEO, CFO, and CIO.
Best SiteSelection
AccessibilityVisibility Traffic Convenience
Location1
Level 0: Goal
Level 1: Criteria
Level 2: Alternatives
Location1 Location1 Location1
Location2
Location3
Location2 Location2 Location2
Location3 Location3 Location3
The Analytic Hierarchy Process
Level 2: Criteria Scientific Economic Political
Level 3:Subcriteria
Level 4:Alternatives
Statewide Local
CloseRestricted
AccessOpen Access
Partial Hierarchy: Management of a Fishery
Best FisheryManagement Policy
Level 1: Focus
Illustrative example
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