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Does “Fuzzifying” AHP Improve the Quality of Multi-Attribute Decision Making?

Li Lun and Poh Kim Leng Department of Industrial & Systems Engineering

National University of Singapore

{u0706517 | isepohkl}@nus.edu.sg

Abstract

The Analytic Hierarchy Process (AHP) model was designed by Thomas L.Saaty (Saaty, 1980) in 1970s as a decision making aid. It is especially suitable for complex decisions which involve the comparison of decision elements which are difficult to quantify. It is based on the assumption that when faced with a complex decision the natural human reaction is to cluster the decision elements according to their common characteristics. Along the way of its development, people found that the uncertainty in assigning priorities for the selected decision elements is not being addressed. This naturally led to the incorporation of the traditional AHP with fuzzy logic to suggest the relative strength of factors in the corresponding criteria, thereby enabling the construction of a fuzzy judgment matrix to facilitate decision making. Fuzzy Analytic Hierarchy Process (FAHP) model became the solution to the problem. No doubt, FAHP has a better performance in dealing with vague input data, but the trade-off is a more complicated calculation it adds to the traditional AHP model. This paper aims to analyse the worthiness of using FAHP instead of traditional AHP to improve the quality of multi-attribute decision making in consideration of its complexity based on outranking method/ELECTRE 1.

Key words: Analytic Hierarchy Process (AHP), Fuzzy Analytic Hierarchy Process (FAHP), ELECTRE 1, Multiple Attribute Decision Making (MADM)

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1. Introduction

The Analytic Hierarchy Process (AHP) model was designed by Thomas L.Saaty in 1970s as a decision making aid. It is especially suitable for complex decisions which involve the comparison of decision elements which are difficult to quantify. It is based on the assumption that when faced with a complex decision the natural human reaction is to cluster the decision elements according to their common characteristics. Along the way of its development, people found that the uncertainty in assigning priorities for the selected decision elements is not being addressed. This naturally led to the incorporation of the traditional AHP with fuzzy logic to suggest the relative strength of factors in the corresponding criteria, thereby enabling the construction of a fuzzy judgment matrix to facilitate decision making. Fuzzy Analytic Hierarchy Process (FAHP) model became the solution to the problem. No doubt, FAHP has a better performance in dealing with vague input data, but the trade-off is a more complicated calculation it adds to the traditional AHP model. One important thing that we have to take note of is that the application of different techniques to a problem may lead to different rankings of alternatives and hence different decisions. Therefore, it is crucial for the decision makers to adopt the most appropriate method.

This paper aims to present a comparative study on AHP and FAHP based on the outranking process of ELECTRE 1 (Roy, 1968). Different people have different ability to deal with the problems. Here, only two types of people will be considered, namely, the real-world businessmen and the mathematically-inclined people.

This paper is organized as follows: Section 2 the review of AHP & FAHP; Section 3 problem solving using both AHP & FAHP; Section 4 the ELECTRE 1 method; Section 5 the comparative study of AHP & FAHP. Finally, Section 6 concludes the paper.

2. Review of AHP & FAHP

The Analytic Hierarchy Process (AHP) was developed by Satty (1980). It is especially suitable for complex decisions which involve the comparison of decision elements which are difficult to quantify. It is based on the assumption that when faced with a complex decision the natural human reaction is to cluster the decision elements according to their common characteristics.

AHP involves building a hierarchy (ranking) of decision elements and then making comparisons between each possible pair in each cluster (as a matrix). This gives a weighting for each element within a cluster (or level of the hierarchy) and also a consistency ratio (useful for checking the consistency of the data).

Users of the AHP first decompose their decision problem into a hierarchy of more easily comprehended sub-problems, each of which can be analyzed independently. The elements of the hierarchy can relate to any aspect of the decision problem—tangible or intangible, carefully measured or roughly estimated, well- or poorly-understood—anything at all that applies to the decision at hand.

Once the hierarchy is built, the decision makers systematically evaluate its various elements, comparing them to one another in pairs. In making the comparisons, the decision makers can use

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concrete data about the elements, or they can use their judgments about the elements' relative meaning and importance. It is the essence of the AHP that human judgments, and not just the underlying information, can be used in performing the evaluations.

The AHP converts these evaluations to numerical values that can be processed and compared over the entire range of the problem. A numerical weight or priority is derived for each element of the hierarchy, allowing diverse and often incommensurable elements to be compared to one another in a rational and consistent way. This capability distinguishes the AHP from other decision making techniques.

In the final step of the process, numerical priorities are derived for each of the decision alternatives. Since these numbers represent the alternatives' relative ability to achieve the decision goal, they allow a straightforward consideration of the various courses of action.

As we know, the AHP assumes that the multi-criteria problem can be completely expressed in a hierarchical structure. The data acquired from the decision makers are pair-wise comparisons concerning the relative importance of each criterion. Since it is difficult to map qualitative preferences to quantitative ones, a degree of uncertainty is associated with some or all pair-wise comparison values in an AHP problem. Fuzzy AHP is there to solve this problem. The earliest study of FAHP was done by Laarhoven (1983), where the fuzzy comparing judgment is represented by triangular membership functions. According to the method of logarithmic least squares, the priority vectors can be obtained. Chang (1996) introduced another approach for handling FAHP. He used triangular fuzzy numbers for pair-wise comparison and extent analysis for the synthetic extent values of the pair-wise comparisons. Besides these two methods, there are also other approaches for FAHP, but the details are not presented in this paper.

3. Problem solving using both AHP and FAHP

In this section, two examples will be solved using both AHP and FAHP to demonstrate their respective features. The experience learned will be used later for the comparative study. The examples are small uncertainties in pair-wise comparison and large uncertainties in pair-wise comparison.

3.1 Small uncertainties involved in pair-wise comparison

In this case, the decision makers are quite sure about their decisions.

Consider a matrix ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

175413581

A

Here the Row Geometric Mean Approximation method is adopted due to its better performance compared to Column Normalization method. So the weight

4

∑ ∏

∏

− =

=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

n

i

nn

iij

nn

iij

i

a

aw

1 1

1 (1)

According to Equation (1), the normalized weights of each criterion are obtained,

Table 1 Normalized weights obtained from matrix A

Criterion X1 X2 X3 Symbols for normalized

weight w₁ w₂ w₃

Normalized weights 0.381 0.255 0.364

Now we use triangular fuzzy numbers to represent matrix A and assume the uncertainties of each element is rather small.

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

1,1,18,7,66,5,45,4,31,1,14,3,26,5,49,8,71,1,1

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

186514691

1UA ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

164312471

1LA

Table 2 Normalized weights obtained from matrix A1

Criterion X1 X2 X3 Symbols for normalized

weight w₁ w₂ w₃

Normalized weights (0.392,0.381,0.373) (0.235,0.255,0.268) (0.377,0.364,0.359)

3.2 Large uncertainties involved in pair-wise comparison

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

1,1,110,7,48,5,27,4,11,1,16,3,08,5,211,8,51,1,1

2A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

142110251

2UA ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

11087168111

2LA

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Table 3 Normalized weights obtained from matrix A2

Criterion X1 X2 X3 Symbols for normalized

weight w₁ w₂ w₃

Normalized weights (0.401,0.381,0.519) (0.284,0.255,0) (0.352,0.364,0.481)

4. The ELECTRE 1 Method

ELECTRE 1 is one of the outranking methods which originated from Europe. Alternatives are compared on the basis of two matrices: concordance and discordance matrices. Concordance matrix reflects those cases where the first alternative in a pair of alternatives is superior to the second. Discordance matrix reflects those cases where the first alternative is inferior to the second. Outranking relationships are developed based on weighted concordance and discordance indices. It only gives us partial ranking of the alternatives (Nijkamp, 1975).

Outranking method uses various mathematical functions to indicate the degree of dominance of one alternative or group of alternatives. The model permits any two alternatives to remain incomparable with each other. For instance, in the context of a choice problem, if alternative A is better than both the alternative B and C, leaving alternative B and C incomparable will not affect the final decision made. However, there are only two candidates for comparison in this paper, so incomparability is not a concern here. Outranking method can be branched in to the ELECTRE (Elimination et Choice Translating Reality) family (Roy 1996) & TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) family, but in this paper, ELECTRE 1 is adopted.

5. Comparative study of AHP & FAHP

5.1 Scenario 1: For real-world businessmen

In previous studies (Liu & Poh, 2003) criteria such as ability, simplicity, compatibility, nature of data and software support were used to evaluate the strengths and weaknesses of various MADM. However, in consideration of the characteristics of AHP and FAHP, robustness, simplicity and software support are chosen as the criteria for comparison in this paper. The definitions of the three criteria are as follows:

Ability to handle uncertainty (C1): This criterion refers to the method’s ability to deal with the uncertainties that may be involved in assigning weights during pair-wise comparisons.

Simplicity (C2): This refers to how easy it is to understand the operation and to interpret the results. Apparently, the simpler the method is more desirable.

Software support (C3): The decision making process will be faster and more accurate with the help of the advanced tools. The effect of using software is evident when dealing with larger sets of data. Hence, the method with existing commercially available and affordable software is preferred.

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Complexity of input data (C4): This refers to how detailed the information of input data is needed for the method.

Table 4 Weight of each criterion

Criterion Description Weight C1 Ability 2 C2 Simplicity 4 C3 Software support 3 C4 Complexity of data 1

In the case of real business world, the decision makers are usually not very familiar with the complicated mathematics, and this makes a decision making method that is easy to understand and use a must. That is why the criterion “simplicity” has the highest weight, followed by the criterion “software support”. This is because we have to consider the time value of money, and we have to complete the work accurately and at the same time, efficiently. A method with existing commercially available and affordable software is necessary to speed up the decision making process.

However, for those decision makers who are mathematics inclined, the weights carried by various criteria will be different. And this will be covered in the sensitivity analysis in the later section of the paper.

Figure 1 shows the hierarchy for this comparative analysis

Method Effectiveness

C1

Ability

C2

Simplicity

C3

Software support

C4

Complexity of data

AHP FAHP

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Table 5 shows the qualitative scale and the corresponding quantitative ratings using the scoring system introduced firstly by Guigou (1975).

Table 5 Rating scale

Qualitative scale Quantitative scale Excellent 20

Above average 15 Average 10

Below average 5 Unsatisfactory 0

The criterion scores are set by the author based on her understanding of these two methods. For C1 (Ability), FAHP is no doubt better than AHP in terms of dealing with uncertainties. For C2 (Simplicity), AHP beats FAHP due to its more straight-forward concept as well as calculations. Since there is no commercial software available for FAHP at all, FAHP scores “0” for C3 (Software support). For C4 (Complexity of data), AHP gets a higher score because it requires less information about the input data than FAHP.

Table 6 Criterion scores

C1 C2 C3 C4 AHP 10 20 20 20

FAHP 20 5 0 5 Weight 2 4 3 1

Table 7 is the standardized criterion scores derived from Table 3 where the highest score of each column is set as “1” and the lowest as “0”.

Table 7 Standard criterion scores

C1 C2 C3 C4 AHP 0 1 1 1

FAHP 1 0 0 0 Weight 2 4 3 1

With these standardized criterion scores, it is now the time for the final step, i.e. to obtain the concordance & discordance matrices. Concordance and discordance matrices are computed using Table 4. The formulae for the computation are as follows:

Concordance index C (A, B) = ∑ (w⁺+0.5w⁼) / (w⁼+w⁺+w⁻)

Discordance index D (A, B) = MAX [ BKZ – AKZ ] / (1-0) for all k where B≥A

Using the above formulae, the detailed calculations are shown below,

C (AHP, FAHP) = [(4+3+1) +0] / [2+4+3+1] = 0.8

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C (FAHP, AHP) = [2] / [2+4+3+1] = 0.2

D (AHP, FAHP) = {MAX [(1-0)]} / (1-0) = 1

D (FAHP, AHP) = {MAX [(1-0), (1-0), (1-0)]} / (1-0) = 1

The concordance matrix C = ⎟⎟⎠

⎞⎜⎜⎝

⎛x

x2.0

8.0

The discordance matrix D = ⎟⎟⎠

⎞⎜⎜⎝

⎛x

x1

1

Then a conclusion can be reached by exploiting the significance/meaning of the concordance and discordance matrices. In order to do this, the concordance threshold (p) and discordance threshold (q) are chosen as the references for the classification of the kernel and not-kernel set. Only the alternative with a concordance index larger than q and at the same time a discordance index smaller than q is dominating and placed in the set of kernel. In this case, p is first set as 0.8, and q is first set as 1. Since both the discordance indices are 1, any change in q will not affect the outranking result and the result only depends on the value of p. With the help of Excel, we found out that AHP outranks FAHP as long as p changes within the range of 0.2< p≤0.8.

5.2 Scenario 2: For mathematically-inclined people

Mathematically-inclined people are more familiar with the mathematics involved in the methods, and thus, they do not value Simplicity as much as real-world businessmen do and value more for the Ability of the method. Due to these differences, each step of the comparative study is performed again to obtain a more comprehensive result for mathematically-inclined people.

Table 8 Weight of each criterion

Criterion Description Weight C1 Ability 4 C2 Simplicity 2 C3 Software support 2 C4 Complexity of data 1

The rating scale remains unchanged.

Table 9 Rating scale

Qualitative scale Quantitative scale Excellent 20

Above average 15 Average 10

Below average 5 Unsatisfactory 0

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The criterion score and standard criterion score are also the same as that in scenario 1 because these are the inherent features of the two methods. They do not vary with scenarios.

Table 10 Criterion score

C1 C2 C3 C4 AHP 10 20 20 20

FAHP 20 5 0 5 Weight 4 2 2 1

Table 11 Standard criterion score

C1 C2 C3 C4 AHP 0 1 1 1

FAHP 1 0 0 0 Weight 4 2 2 1

Then, the concordance and discordance matrices are calculated.

C (AHP, FAHP) = [(4+3+1) +0] / [2+4+3+1] = 0.556

C (FAHP, AHP) = [2] / [2+4+3+1] = 0.44

D (AHP, FAHP) = {MAX [(1-0)]} / (1-0) = 1

D (FAHP, AHP) = {MAX [(1-0), (1-0), (1-0)]} / (1-0) = 1

The concordance matrix C = ⎟⎟⎠

⎞⎜⎜⎝

⎛x

x44.0

556.0

The discordance matrix D = ⎟⎟⎠

⎞⎜⎜⎝

⎛x

x1

1

Then we found out that AHP outranks FAHP when p varies from 0.44 to 0.556 exclusive.

6. Conclusion and Future Work

In this paper, we have studied and compared the AHP and FAHP method using ELECTRE 1 method. Considering the above two scenarios, we can conclude that in general, AHP outranks FAHP in solving decision making problems. This means that it’s usually not worth it to carry out so much calculation for FAHP just because it has better performance in dealing with uncertainties. We should just adopt AHP instead to avoid the troubles.

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According to the results obtained from the Excel, only if FAHP scores higher in C3 (Software support), i.e. there is commercially available software for FAHP, can it outrank AHP. This undoubtedly is an encouragement as well as a motivation for us to develop more user friendly software for FAHP to be widely accepted by users in the future.

References

A. Ozdagoglu & G. Ozdagoglu. (2007). Comparisons of AHP and Fuzzy AHP for the multi-criteria decision making processes with linguistic evaluations. Istanbul Ticret Universitesi Fen Bilimleri Dergisi 6 (11):65-85.

B. Roy. (1968). Classment et choix en presence de points de vue multiples (la method ELECTRE). Revue Francaise d’ Automatique Information et Recherche Operationelle (RIRO) 8:57-75.

D. Chang. (1996). Applications of the extent analysis method on fuzzy AHP. European Journal of Operational Research 95:649-655.

J.L. Guigou. (1971). On French Location Models for Production Units. Regional and Urban Economics 1(2):107-138.

P. Nijkamp. (1975). A Multi-attribute Analysis for Project Evaluation: Economic-Ecological Evaluation of a Land Reclamation Project. Regional science association 35:87-111.

R.J. Liu, K.L. Poh and C.U.Lee (2003). An outranking analysis of MCDM methods. In Z. Jiang, X. Qian and Z Jiang (editors), Proceedings of IE&EM'2003, The 10th International Conference on Industrial Engineering & Engineering Management, Shanghai, pp 273, China Machine Press.

T. Saaty. (1980). A Scaling for Priorities in Hierarchical Structures. Journal of Mathematical Psychology 15: 234-281.

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Appendix

Detailed calculations for the AHP and FAHP example in section 3 are as follows,

Consider a matrix A = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

175413581

Here the Row Geometric Mean Approximation method is adopted due to its better performance compared to Column Normalization method. So the weight

iw =

∑ ∏

∏

− =

=

⎟⎟⎠

⎞⎜⎜⎝

⎛n

i

nn

iij

nn

iij

a

a

1 1

1

A₁ = ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

1,1,18,7,66,5,45,4,31,1,14,3,26,5,49,8,71,1,1

A U1 = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

186514691

A L1 = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

164312471

A M1 = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

175413581

Consider the upper bound matrix, we have

1w = =++ 333

3

8*65*46*96*9 0.373

2w = =++ 333

3

8*65*46*95*4 0.268

3w = =++ 333

3

8*65*46*98*6 0.359

Consider the lower bound matrix, we have

1w = =++ 333

3

6*43*24*74*7 0.392

12

2w = =++ 333

3

6*43*24*73*2 0.235

3w = =++ 333

3

6*43*24*76*4 0.377

A 2 = ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

1,1,110,7,48,5,27,4,11,1,16,3,08,5,211,8,51,1,1

A U2 = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

142110251

A L2 = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

11087168111

A M2 = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

175413581

Consider the upper bound matrix, we have

1w = =++ 333

3

4*21*02*52*5 0.519

2w = =++ 333

3

4*21*02*51*0 0

3w = =++ 333

3

4*21*02*54*2 0.481

Consider the lower bound matrix, we have

1w = =++ 333

3

7*54*35*85*8 0.401

2w = =++ 333

3

7*54*35*84*3 0.284

3w = =++ 333

3

7*54*35*87*5 0.352

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The detailed Excel worksheets for both scenario 1 and scenario 2 are presented below:

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