analytic hierarchy process ahp

Analytic Hierarchy Process

Vikas Sao – [email protected]

Vinit Modak – [email protected]

Vinothkumar – [email protected]

Vipul Singh – [email protected]

Introduction

• Developed by T. Saaty

• Best known and widely used Multi Criteria Analysis (MCA) approach

• Relative priority of each criteria

• Real world decision problems

• Spend on Defence or Agriculture?

• Buy an Ecosport or Koleos?

• Integrated manufacturing (Putrus, 1990),

• In the evaluation of technology investment decisions (Boucher and McStravic, 1991)

• In flexible manufacturing systems (Wabalickis, 1988)

• Layout design (Cambron and Evans, 1991)

SIOM | Symbiosis Institute of Operations Management, Nashik

History

• AHP first introduced by Saaty in 1977

• Apparent problems with the way pairwise comparisons were used pointed out by Belton and Gear in 1983

• Belton and Gear introduced Revised-AHP

• Saaty accepted the change and introduced Ideal Mode AHP in 1994


Procedure

• Structure a decision problem and selection of criteria

• Priority setting of the criteria by pairwise comparison (WEIGHING)

• Pairwise comparison of options on each criterion (SCORING)

• Obtaining an overall relative score for each option


Structuring a decision problem and selection of criteria

• Divide the problem into its constituent parts

• Goal at the Topmost Level

• Criteria at the Intermediate Level

• Options at the Lowest Level

• What it does?

• Provides an overall view of the complex relationships

• Access whether the element in each level are of the same magnitude to compare accurately

Selecting a Car

Style Reliability Mileage

Ecosport (E) Koleos (K) Scorpio (S) Duster (D)


Ranking of Criteria and Alternatives

• Pairwise comparisons are made with the grades ranging from 1-9.

• A basic, but very reasonable assumption for comparing alternatives:

If attribute A is absolutely more important than attribute B and is rated at 9, then B must be absolutely less important than A and is graded as 1/9.

• These pairwise comparisons are carried out for all factors to be considered, usually not more than 7, and the matrix is completed.


Priority setting of the criteria by pairwise comparison (WEIGHING)

• How important is criterion A relative to criterion B?

• Assign a weight between 1 and 9

• Reciprocal of the value is assigned to the other criterion in the pair

• How important is criterion B relative to criterion A?

• Normalize and average the weighing to obtain average weight for each criterion

9 Extreme

Importance

1 Equal

Importance

Scale Definition Explanation

1 Equal

importance

Two activities contribute

equally to the objective

3 Weak

importance of

one over

another

Experience and judgment

slightly favour one activity

over another

5 Essential or

strong

importance


strongly favour one

activity over another

7 Demonstrated

importance

An activity is strongly

favoured and its

dominance demonstrated

in practice

9 Absolute

importance

The evidence favoring one

activity over another is of

the highest possible

order of affirmation

2,4,6,8 Intermediate

values

between the

two

adjacent

judgments

When compromise is

needed



Scale Definition Explanation

1 Equal

importance

Two activities contribute

equally to the objective

3 Weak

importance of

one over

another


slightly favour one activity

over another

5 Essential or

strong

importance


strongly favour one

activity over another

7 Demonstrated

importance

An activity is strongly

favoured and its

dominance demonstrated

in practice

9 Absolute

importance

The evidence favoring one

activity over another is of

the highest possible

order of affirmation

2,4,6,8 Intermediate

values

between the

two

adjacent

judgments

When compromise is

needed

STYLE RELIABILITY MILEAGE

STYLE 1 1/2 3

RELIABILITY 2 1 4

MILEAGE 1/3 1/4 1



STYLE RELIABILITY MILEAGE G.M. EIGEN

VECTOR

STYLE 1 1/2 3 1.14 0.3196

RELIABILITY 2 1 4 2.00 0.5584

MILEAGE 1/3 1/4 1 0.44 0.1220

SUM 3.33 1.75 8 3.58


Consistency Ratio

• The next stage is to calculate max so as to lead to the Consistency Index and the Consistency Ratio.

• Consider [Ax = max x] where x is the Eigenvector.

0.3196

0.5584

0.1220

1 0.5 3

2 1 4

0.333 0.25 1.0

0.9648

1.6856

0.3680 = = max

λmax=average{0.9648/0.3196, 1.6856/0.5584, 0.3680/0.1220}=3.0180

0.3196

0.5584

0.1220


Consistency Index

• The final step is to calculate the Consistency Ratio.

• CI=(max –n)/(n-1)

• CR=CI/RI=0.0090/0.58=0.01552

less than 0.1 so the evaluations are consistent

• An inconsistency of 10% or less implies that the adjustment is small compared to the actual values of the eigenvector entries.

• A CR as high as, say, 90% would mean that the pairwise judgment are just about random and are completely untrustworthy


Pairwise comparison of options on each criterion (SCORING)

• Using Pairwise Comparisons, the Relative Importance Of One Criterion Over Another can be expressed.

E K S D

E 1 1/4 4 1/6

K 4 1 4 1/4

S 1/4 1/4 1 1/5

D 6 4 5 1

Km / L

E 34

K 27

S 24

D 28

E K S D

E 1 2 5 1

K 1/2 1 3 2

S 1/5 1/3 1 1/4

D 1 1/2 4 1



3.00 1.75 8.00

5.33 3.00 14.0

1.17 0.67 3.00 A2=

Row sums 12.75

22.33

4.83

39.92

Normalized Row Sums 0.3194

0.5595

0.1211

1.0

Iteration 1:

Initialization:

A2xA2= 27.67 15.83 72.50

48.33 27.67 126.67

10.56 6.04 27.67

A= 1 0.5 3

2 1 4

0.33 0.25 1.0

Row sums

12.75

22.33

4.83

39.92

Normalized Row Sums

0.3196

0.5584

0.1220

0.0002

-0.0011

0.0009

E1-E0 = - 0.3194

0.5595

0.1211

0.3196

0.5584

0.1220 =

Almost zero, so

Eigen Vector, X = E1.

E0

E1


Pairwise comparison of options on each criterion (SCORING)

• Using Pairwise Comparisons, the Relative Importance Of One Criterion Over Another can be expressed.

Km / L E.V

E 34 0.3010

K 27 0.2390

S 24 0.2120

D 28 0.2480


E K S D E.V.

E 1.00 0.25 4.00 0.17 0.1160

K 4.00 1.00 4.00 0.25 0.2470

S 0.25 0.25 1.00 0.20 0.0600

D 6.00 4.00 5.00 1.00 0.5770

E K S D E.V

E 1.00 2.00 5.00 1.00 0.3790

K 0.50 1.00 3.00 2.00 0.2900

S 0.20 0.33 1.00 0.25 0.0740

D 1.00 0.50 4.00 1.00 0.2570


Obtaining an overall relative score for each option

• The option scores are combined with the criterion weights to produce an overall score for each option.

Final Ranking Of Alternatives = Ranking Of Alternative (Category Wise)*Criteria Weights

E

D

S

K

.1160 .3790 .3010

.2470 .2900 .2390

.0600 .0740 .2120

.5770 .2570 .2480

*

.3196

.5584

.1220

=

.2854

.2700

.0864

.3582


In the industry Areas Company Application

Choice Xerox Product selection

Prioritization General Motors Prioritizing the design alternatives and arrive at a

cost effective design

Resource Allocation Korea

Telecommunication

Authority

Allocation of R&D Budget for 10 technologies

Benchmarking IBM Compare IBM CIM with best of breed companies

Quality Management Steel & Magnetic

Division, Italy

Comparison with its competitors and improve the

quality

Public policy Japan Formulate policy to maintain Sea of Japan

Health care Medical Center,

Washington

Type of team to be sent in case of different

disaster

Strategic Planning 3M A computerized AHP for quick evaluation of

strategy.


AHP with feedback

• Priorities of elements in a level is not dependent on the lower level elements. NOT ALWAYS TRUE

• GOAL : Construction of bridge

• CRITERIA : STRENGTH, COST, LOOK

• OPTIONS : A,B

• A – MEETS ALL THE STRENGTH REQUIREMENTS, LOOKS BEAUTIFUL

• B – MOST STRONGEST, EQUAL COST, UGLY


WEB BASED

• http://www.isc.senshu-u.ac.jp/~thc0456/EAHP/AHPweb.htm

• Input

• Size of pairwise comparison Matrix

• Pairwise comparison Matrix

• Output

• Eigen Vector

• Consistency Index

• www.superdecisions.com

• Provide software for AHP and ANP

• Exhaustive tutorials (PDF and Video)


http://www.isc.senshu-u.ac.jp/~thc0456/EAHP/AHPweb.html




http://www.superdecisions.com/

http://www.superdecisions.com/

Thank You


analytic hierarchy process ahp

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