Quantitative Analysis for Management Multifactor Evaluation Process and Analytic Hierarchy Process Dr. Mohammad T. Isaai Graduate School of Management

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


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


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


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


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


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.


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”


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.


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


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


Example: Car Selection

Objective Selecting a car

Criteria Style, Reliability, Fuel-economy Cost?

Alternatives Civic Coupe, Saturn Coupe, Ford Escort,

Mazda Miata


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



Level 1: Goals

Level 2: Criteria

Level 3: Alternatives


First, we deal with

FACTOR WEIGHTS



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



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).


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


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


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


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


1 .0


Second, we deal with

Alternative Evaluations


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


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.


S tyle.3 196

R e lia b ility.5 584

F u e l E con o m y.1 220


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


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


Complex decisions

•Many levels of criteria and sub-criteria


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


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


Level 3

Service Attributes Initial Service Guarantee Repair

Management Attributes Manpower Systems Communications Management


Level 3

Technical Attributes Building Equipment Tools Security system


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.


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


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


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


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



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

Documents

Quantitative Analysis for Management Multifactor Evaluation Process and Analytic Hierarchy Process Dr. Mohammad T. Isaai Graduate School of Management