Comparison Of Moderate Analytic Hierarchy Process And Moderate Fuzzy Analytic Hierarchy Process

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Published On: 18th April, 2016

International Journal of Informative & Futuristic Research ISSN: 2347-1697

Volume 3 Issue 8 April 2016 Original Paper

Abstract

In the conventional AHP, the pair wise comparisons for each level with respect to the goal of the best alternative selection are conducted using a nine-point scale. So, the applications of Saaty’s AHP has some shortcomings, in addition, a decision-maker’s requirements on evaluating alternatives always contain ambiguity and multiplicity of meaning. Furthermore, it is also recognized that human assessment on qualitative attributes is always subjective and thus imprecise. Therefore, conventional AHP seems inability to capture decision maker’s requirements explicitly (Kabir & Hasin, 2011). A fuzzy AHP comes into implementation in order to overcome the compensatory approach and inability of the conventional AHP in handling linguistic variables. So AHP and fuzzy AHP comparable is required. In this paper, AHP and Fuzzy AHP, Ideal AHP and Ideal Fuzzy AHP and Moderate AHP and Moderate Fuzzy AHP modes have been compared and presented.

1. INRODUCTION

To achieve the evaluation for the problems in this paper, the (Analytic Hierarchy Process)

AHP has been preferred among the other methods of MADM. The AHP is a structured

technique for dealing with complex decision. Rather than prescribing a ‘correct’ decision, the AHP and Fuzzy AHP help the decision makers find the one that best suits their needs

and their understanding of the problem.

Comparison Of Moderate Analytic

Hierarchy Process And Moderate Fuzzy

Analytic Hierarchy Process Paper ID IJIFR/V3/ E8/ 009 Page No. 2874-2883 Subject Area Mathematics

Keywords Multi Criteria Decision Making, Pairwise Comparison Model, AHP, Ideal

AHP, Moderate AHP, Fuzzy AHP, Ideal Fuzzy AHP, Moderate Fuzzy AHP

1st G. Marimuthu

Ph.D. Scholar & Associate Professor

Department of Mathematics,

A.V.V.M. Sri Pushpam College (Autonomous),

Poondi -Thanajvur

2nd Dr. G. Ramesh

Associate Professor

Department of Mathematics,

Government Arts College (Autonomous), Kumbakonam

http://www.ijifr.com/

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ISSN: 2347-1697

International Journal of Informative & Futuristic Research (IJIFR)

Volume - 3, Issue -8, April 2016

Continuous 32nd Edition, Page No.:2874-2883

G. Marimuthu, Dr. G. Ramesh:: Comparison Of Moderate Analytic Hierarchy Process And Moderate Fuzzy Analytic Hierarchy Process

1.1 Analytic Hierarchy Process (AHP)

The main essence of the AHP method is analyzing complex problems into a hierarchy

with aim at the top of the hierarchy, criterions at levels of the hierarchy and decision

alternative at the bottom of the hierarchy. Elements are given in hierarchy level are

compared in pair wise to calculate their relative importance with respect to each of the

elements at the next higher level. The AHP method calculates and totalizes their

eigenvectors until the composite last vector of weight comparisons for alternatives is

achieved. The entries of last weight comparisons vector reflect the importance value of

each alternative with respect to the aim stated at the top of hierarchy [22].

Ideal AHP

Also we can get ideal AHP matrix (model), by dividing their entries in the column

of the original AHP matrix for the corresponding criterion with the largest entry in that

selected column. Multiply these values of the alternatives with corresponding the

resulting criterion weights (Global priorities). Sum these values to get the final priority

vector for the respective alternative. In such away we find the final priority vectors for the

remaining alternatives. After normalization of the final priority vector to have the values

with ranking.[22]

Moderate AHP

It can be extended to find the final alternative priority vectors for all alternatives

from the original AHP decision matrix. It can be obtained from the following formula.

MAj =

n

j

ijjj AWW1

1 )(

Where Wj is the weight vector for corresponding resulting criteria and A1

ij is the

weight vector (scores) of the ith

alternative and jth

resulting criteria of the original AHP

decision matrix we get moderate AHP decision matrix [22].

1.2 Fuzzy Analytic Hierarchy Process

The Analytic Hierarchy Process is a powerful and flexible decision making process

to help manager set priorities and make the best decision when both qualitative and

quantitative aspect of a decision need to be considered. By reducing complex decisions to

a series of one-on-one comparisons, then synthesizing the results, many researchers have

concluded that AHP is a useful. Using AHP to perform problem analysis can reduce risk

of mistakes in decision making. However, AHP use cannot overcome the subjectivity,

inaccuracy and fuzziness produced when making decision. Then introducing fuzzy set

theory and fuzzy operation to AHP can ameliorate these failures [19].

In fuzzy AHP, the pair wise comparisons of both criteria and alternatives are

performed through the linguistic variables, which are represented by triangular fuzzy

numbers [2][23]

Ideal fuzzy AHP

Also we can get ideal fuzzy AHP matrix (model), by dividing their entries in the

column of the original fuzzy AHP decision matrix for the corresponding criterion with the

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largest entry in that selected column. Multiply these values of the alternatives with

corresponding the resulting criterion weights (Global priorities). Sum these values to get

the final priority vector for the respective alternative. In such away we find the final

priority vectors for the remaining alternatives. After normalization of the final priority

vector have the values with ranking [23].

Moderate fuzzy AHP

It can be extended to find the final alternative priority vectors for all alternatives

from the original fuzzy AHP decision matrix. It can be obtained from the following

formula.

MAj =

n

j

ijjj AWW1

1 )(

Where Wj is the weight vector for corresponding resulting criteria weight and A1

ij

is the weight vector (scores) of the ith

alternative and jth

resulting criteria of the original

fuzzy decision matrix we get moderate fuzzy decision matrix [23].

2. COMPARISON OF MODERATE AHP PAIR WISE COMPARISON MODEL (

NUMERICAL EXAMPLE)

Suppose the expert has to select a movie with a theatre for the award. Two main

criteria as theatres have been chosen for evaluation of alternative with better service,

namely Joy theatre and Modern theatre. Each theatre is screening two different movies as

sub criteria. In joy theatre, Beldom is screened for the days until the collection of amount

has minimum average. In Modern theatre, Zenith is screened for the days while the

collection has minimum average, similarly for the movie Zillah. Three decision makers

are looking each movie until last one and give the mark for award. Ultimately, select a

movie with highest score of the sum of the marks given by the decision makers in each

movie. Determine the best feature film for award chosen by the decision markers. The

diagrammatic representation is as follows.

Goal

Joy Modr

Beldo Bell Zenit Zilla

D1 D1 D1

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Table –1: Scale of Relative Importance (According to Saaty 1980)

Intensity of

Importance Definition

1 Equal importance

3 Weak importance of one over another

5 Essential or strong importance

7 Demonstrated importance

9 Absolute importance

2, 4, 6, 8 Intermediate values between the two adjacent judgments

Reciprocals of

above

If activity i has one of the above nonzero numbers assigned to it when

compared with activity j, then j has the reciprocal value when compared

with i.

Table 2: Pairwise Comparison Matrix of Main Criteria with respect to goal

= 2 , CI = 0

Table 2: Pairwise Comparison Matrix with respect to Joy

= 2 , CI = 4.4409e-16

Table 3: Pairwise Comparison Matrix with respect to Modern

= 2 , CI = 0

Table 4: Pairwise Comparison Matrix with respect to Beldom

Beldom D1 D2 D3 Priority

Vector

D1 1 3 7 0.6491

D2 1

3 1 5 0.2789

D3 1

7

1

5

1 0.0719

CJ = 0.0324438, = 3.06489

Goal Joy Modern Priority

Vector

Joy 1 7 0.875

Modern 1/7

1 0.125

Goal Beldom Belle Priority

Vector

Beldom 1 5 0.8333

Belle 1/5

1 0.1667

Modern Zenith Zillah Priority

Vector

Zenith 1 3 0.75

Zillah 1/3

1 0.25

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Table 5: Pairwise Comparison Matrix with respect to Belle

CJ = 0.081673, = 3.1632

Table 6: Pairwise Comparison Matrix with respect to Zenith

CJ = 0.01925, = 3.03851

Table 7: Pairwise Comparison Matrix with respect to Zillah

CJ = 0.0145319, = 3.02906

Table 8: Original AHP decision matrix

Main Criterion Joy Modern

0.8750 0.1250

Sub Criterion Beldom Belle Zenith Zillah

Alternative 0.8333 0.1667 0.7500 0.2500

D1 0.6491 0.0791 0.1047 0.4053

D2 0.2789 0.2118 0.6398 0.4806

D3 0.0719 0.7090 0.2582 0.1139

Table 9: Modified Original AHP decision matrix

Alternative

/criterion Beldom Belle Zenith Zillah Final

Priority

Vector Criterion

Weight 0.7291 0.1459 0.0938 0.0313

D1 0.6491 0.0791 0.1047 0.4053 0.5071

Belle D1 D2 D3 Priority

Vector

D1 1 1

4 1/6 0.0791

D2 4

1 1/5 0.2118

D3 6

5

1 0.7090

Zenith D1 D2 D3 Priority

Vector

D1 1 1/5

1/3 0.1047

D2 5

1 3 0.6398

D3 3

1/3

1 0.2582

Zillah D1 D2 D3 Priority

Vector

D1 1 1

3 0.4053

D2 1

1 5 0.4806

D3 1/3

1/5

1 0.1139

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D2 0.2789 0.2118 0.6398 0.4806 0.3092

D3 0.0719 0.7090 0.2582 0.1139 0.1835

Table 10: Ideal AHP decision matrix

Alternative

/criterion Beldom Belle Zenith Zillah

Final

Priority

Vector

After Normalization

Criterion Weight 0.7291 0.1459 0.0938 0.0313

D1 1.0000 0.1115 0.1636 0.4833 0.7869 0.5108

D2 0.4296 0.2987 1.0000 1.0000 0.4818 0.3127

D3 0.1107 1.0000 0.4035 0.2369 0.2718 0.1764

Table 11: Moderate AHP decision matrix

Alternative

/criterion Beldom Belle Zenith Zillah

Criterion

Weight 0.7291 0.1459 0.0938 0.0313

D1 1.0048 0.0328 0.0186 0.0136

D2 0.7349 0.0521 0.0688 0.0160

D3 0.5840 0.1247 0.0330 0.0046

If any one priority value exceeds 1, normalization for corresponding column for the

priority value should be taken for further evaluation. We get the modified moderate the

AHP.

Table 12: Modified moderate AHP decision matrix

Alternative


Priority

Vector

After Normalization

Criterion

Weight 0.7291 0.1459 0.0938 0.0313

D1 0.4324 0.0328 0.0186 0.0136 0.8848 0.3597

D2 0.3162 0.0521 0.0688 0.0160 0.8078 0.3284

D3 0.2513 0.1247 0.0330 0.0046 0.7673 0.3119

Using the above Numerical example, we shall find the rank by using Moderate F-AHP

pair wise comparison model below.

Table 13: Moderate Fuzzy AHP pair wise comparison model (Numerical Example)

Sl.

No. Definition

Triangular Fuzzy

Number

Reciprocal Fuzzy

Number

1.

2

3.

4.

5.

6.

7.

Equally Importance

Weakly Importance

Moderately Importance

Moderately Plus Importance

Strongly Importance

Very Strongly Importance

Extremely Importance

(1, 1, 1)

( 12

, 1, 2)

(2, 3, 4),

(3, 4, 5)

(4, 5, 6)

(5, 6, 7)

(6, 7, 8)

(1, 1, 1)

( 12

, 1, 2)

( 14

, 13

, 12

)

( 15

, 14

, 13

)

( 16

, 15

, 14

)

( 17

, 16

, 15

)

(18

, 17

, 16

)

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Table 14: Pairwise Comparison Matrix of Main Criteria with respect to goal

Goal Joy Modern Priority

Vector

Joy (1,1,1) (6,7,8) 0.875

Modern 1 1 1

, ,8 7 6

(1, 1, 1) 0.125

CI = 0.0, = 2

Table 15: Pairwise Comparison Matrix with respect to Joy

CI = 4.4409 C-16, = 2 Table 16: Pairwise Comparison Matrix with respect to Modern

Modern Zenth Zillah Priority

Vector

Zenith (1,1,1) (2, 3, 4) 0.75

Zillah 1 1 1

, ,4 3 2

(1, 1, 1) 0.25

CI = 0.0 = 2,

Table 17: Pairwise Comparison Matrix with respect to Beldom

Table 18: Pairwise Comparison Matrix with respect to Belle

= 0

Joy Beldom Belle Priority

Vector

Beldom (1,1,1) (4,5,6) 0.8333

Belle 1 1 1

, ,6 5 4

(1, 1, 1) 0.1667

Belle D1 D2 D3 Priority Vector

D1 (1,1,1) 1 1 1

, ,5 4 3

1 1 1, ,

7 6 5

0.085

D2 (3,4,5) (1,1,1) 1 1 1

, ,6 5 4

0.209

D3 (5,6,7) (4,5,6) (1,1,1) 0.706

Beldom D1 D2 D3 Priority Vector

D1 (1,1,1) (2,3,4)

(6,7,8) 0.615

D2 1 1 1

, ,4 3 2

(1,1,1) (4,5,6) 0.308

D3 1 1 1

, ,8 7 6

1 1 1, ,

6 5 4

(1,1,1) 0.037

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Table 19: Pairwise Comparison Matrix with respect to Zenith

Table 20: Pairwise comparison Matrix with respect to Zillah

Table 21: Original Fuzzy AHP Decision Matrix

Main Criterion J M

0.8750 0.1250

Sub Criterion Beldom Belle Zenith Zillah

Alternative 0.8333 0.1667 0.7500 0.2500

D1 0.615 0.085 0.113 0.385

D2 0.308 0.209 0.626 0.503

D3 0.077 0.706 0.261 0.111

Table 22: Modified Original Fuzzy AHP Decision Matrix

Alternative


Priority

Vector Criterion

Weight 0.729 0.145 0.093 0.031

D1 0.615 0.085 0.113 0.385 0.4830

D2 0.308 0.209 0.626 0.503 0.3285

D3 0.077 0.706 0.261 0.111 0.1860

Table 23: Ideal Fuzzy AHP Decision Matrix

Alternative


Priority

Vector

After

Normalization

Criterion

Weight 0.729 0.145 0.093 0.031

D1 1.000 0.120 0.180 0.765 0.7868 0.4919

D2 0.501 0.296 1.000 1.000 0.5321 0.3326

D3 0.125 1.000 0.416 0.220 0.2815 0.1760

Zenith D1 D2 D3 Priority Vector

D1 (1,1,1) 1 1 1

, ,6 5 4

1 1 1, ,

4 3 2

0.113

D2 (4,5, 6) (1,1,1) (2,3,4) 0.626

D3 (2,3, 4) 1 1 1

, ,4 3 2

(1,1,1) 0.261

Zillah D1 D2 D3 Priority Vector

D1 (1,1,1) 1

,1,22

(2,3,4) 0.385

D2 1,1,2

2

(1,1,1) (4,5, 6) 0.503

D3 1 1 1, ,

4 3 2

1 1 1, ,

6 5 4

(1,1,1) 0.111

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Table 24: Moderate Fuzzy AHP Decision Matrix

Alternative


Priority

Vector

After Normalization

Criterion

Weight 0.729 0.145 0.093 0.031

D1 0.979 0.033 0.019 0.012 1.043 0.3896

D2 0.755 0.051 0.066 0.016 0.888 0.3317

D3 0.587 0.123 0.032 0.004 0.746 0.2786

3. COMPARISON RESULT

Selection of alternative is D1, which is followed by D2 and is followed by D3 for both

numerical examples above. The priority vectors after normalization of the alternatives D1,

D2 and D3 of moderate AHP decision matrix are 0.3597, 0.3284 and 0.3119 respectively.

And the priority vectors after normalization of the alternatives D1, D2, and D3 of moderate

Fuzzy AHP decision matrix are 0.3896, 0.3317 and 0.2786 respectively. There is a slight

difference of priority vectors between moderate AHP and moderate Fuzzy AHP matrices

with same rankings of the alternatives.

4. CONCLUSION

In this chapter comparison of AHP and F-AHP, ideal AHP and ideal Fuzzy AHP and

moderate AHP and Moderate Fuzzy AHP have been presented. AHP is one of the most

commonly used techniques that were evaluated by linguistic variable. Single numbers

were introduced for comparison Multicriterion decision making techniques based on the

linguistic evaluations in which AHP make a best selection decision by using a weighting

process with in the current alternatives via pairwise comparison. The ranking of the

criteria have been made according to their final scores on the basis of weight.

But for the uncertain or fuzzy environment, a fuzzy numbers that are triangular

fuzzy numbers were introduced into the conventional AHP in order to improve the

judgments of decision makers and experts. The ranking of the criteria have been made

according to their final scores on the basis of weight. For single Criterion and

multicriteria, ranking of the alternatives of original AHP, Ideal AHP and moderate AHP

decision matrices are the same under the criteria weights in the corresponding problem.

Similarly for single criteria and multicriteria, ranking of the alternative of the original

Fuzzy AHP, Ideal fuzzy AHP and moderate Fuzzy AHP are the same under the criteria

weights in the corresponding problems.

In the moderate AHP and Moderate Fuzzy AHP, priority vectors of the alternatives

are closer different values compared with original and ideal models when alternatives are

ranked. Therefore the priority values of the moderate model are sensitive.

5. REFERENCES

[1] Amy H.I. Lee (2009) “A Fuzzy supplier selection model with the consideration of benefits,

opportunities, costs and risks” 36, 2879 – 2893.

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[2] Bellman. R.E., Zadeh. L.A., Decision making in fuzzy environment, Management

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[3] Boender C.G.E., deGrann J.E., F.A. 1989. Lootsma, Multi-criteria decision analysis with

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[4] Buckley.J.J, (1985) “Fuzzy hierarchical analysis”, Fuzzy Sets and Systems, 17(3), 233-

247.

[5] Cengiz Kahraman: Fuzzy Multi Criteria Decision Making: Theory and Application with

Recent Development, Istanbul Technical University, Turkey: Springer: May 2008.

[6] Chang. D.Y. (1996) “Applications of the extent analysis method of fuzzy AHP”. European

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[7] Chang. D.Y., (1992), “Extent analysis and synthetic decision”, Optimization Techniques

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[8] Marimuthu G, Ramesh. G: On moderate analytic Hierarchy process pariwise comparison:

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[10] Mikhailov L., 2003. Deriving priorities from fuzzy pairwise comparison judgements,

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[11] Saaty. T.L., (1980) “The analytic hierarchy process”, New York, NY: McGraw-Hill.

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[14] Van Laarhoven. P.J.M, Pedrycz. W., “A fuzzy extension of Saaty’s priority theory”, Fuzzy

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Comparison Of Moderate Analytic Hierarchy Process And Moderate Fuzzy Analytic Hierarchy Process