WITHDRAWN: On the position of intuitionistic fuzzy set theory in the framework of theories modelling imprecision

2011 IEEE Colloquium on Humanities, Science and Engineering Research (CHUSER 2011), Dec 5-6 2011, Penang

New Computation Approach for Fuzzy Group Decision-Making *Z. Tarmudi, **A.O. Md Tap, ***L. Abdullah *UiTM Sabah, M'sia **International Islamic University M'sia (IIUM) ***Universiti Malaysia Terengganu *[email protected], ** [email protected], *** [email protected] Since Zadeh [4] introduced the concept of a linguistic variable, researchers started to develop computational linguistics or processes of computing with words thoroughly using this idea. Initially, the concepts were very abstract to quantify without the theoretical knowledge, but this idea is becoming more significant as a main issue in different research areas in the near future. According to Herrera et al. [5], the linguistic variables recently used to develop inference and reasoning processes in intelligent information systems and knowledge-based systems compare only for preference modeling and evaluate qualitative aspects in decision-making problems and information-retrieval systems. However, existing research seldom explores the equilibrium approach to assist the evaluator group in the multi-purpose evaluation process. Thus, the aim of this paper is to propose a new aggregation method using conflicting linguistic preference relations in the group evaluation process based on the equilibrium concept (i.e., both positive and negative aspects) concurrently. To do so, the rest of this paper is organised as follows: Section II provides the basic idea of so-called conflicting bifuzzy sets theory and related theory behind it; Section III and IV describe how the linguistic labels and modified aggregation method were developed in the group evaluation process; Section V focuses on the applicability of the proposed method through a hypothetical example, and finally, brief conclusions are pointed out.

Abstract - This paper proposes a new method of using the socalled ‘equilibrium linguistic preference relations’ for group evaluation purposes. The theoretical background is briefly discussed in the early stages; followed by the discussion on the modified linguistic geometric averaging operator to facilitate the proposed computational algorithms. A hypothetical example was employed to demonstrate the suitability and practicability of the proposed method. It was found that the proposed method can reduce the uncertainty of the incomplete initial information that is able to generate more precise and comprehensive results in the evaluation process. Keywords - Bifuzzy; equilibrium linguistic preference relations; group evaluation process;

1. INTRODUCTION The evaluation process basically consists of finding the best option from a set of feasible alternatives. It involves a qualified individual or a group of individuals familiar with company policy to ensure that every single decision made is always in line with the main objectives of the organisation. Usually, a group of evaluators is not able to estimate their preference degree using the numeric scale, especially when attempting to qualify phenomena related to human perception. This occurs due to the variety of the evaluator’s background, such as attitude, feeling, personality, environment, culture, qualification etc. In addition, the simplicity and comfort factors play the main role as to why linguistic preferences relation is preferred rather than exact scale. Therefore, an alternative option should be considered and the best option is by using the linguistic assessments to express their preferences [1–2]. Group evaluators’, is defined as a situation in which i) there are two or more individuals who differ in their preferences, but have the same access to information, each of them characterised by his or her own perceptions, attitudes, motivations, and personalities; ii) a person who recognises the existence of a common problem; and iii) a person who attempts to reach a collective decision [3]. As we mentioned earlier, in many cases, experts are unable to give their preference degree directly by use of the exact values. They feel more comfortable and fairer with the natural languages such as “very good”, “good”, slightly good”, etc., instead of numeric values like 1, 3, 5, etc.

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II. THE IDEA OF BIFUZZY SET THEORY Since Zadeh [6] proposed his idea of fuzzy sets theory, the application of this concept has advanced in various areas. However, fuzzy sets only give a membership degree to each element of the universe, and the non-membership degree is always equivalent to the complimentary basis [7]. Several definitions are put forward that eventually led to the proposed method. Definition 1 A fuzzy set A in a universe of discourse X is characterised by a membership function μA(x) that takes the values in the interval of [0,1]. It can be denoted as follows: A = {(μA(x)/x); x ∈ X}

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2011 IEEE Colloquium on Humanities, Science and Engineering Research (CHUSER 2011), Dec 5-6 2011, Penang

This idea of Zadeh was extended by Atanassov [8] some 20 years later using the concepts of dual membership degrees in each of the sets discourse by giving both a degree of membership and a degree of non-membership which are moreor-less independent from one other with the sum of these two grades being not greater than 1.

Definition 3 Let a set X be fixed. A conflicting bifuzzy set (CBFS) A of X x X is an object having the following form: A = {〈x, μA (x), γA(x)〉 I x ∈ X; 0 < μA(x) + γA(x) ≤ 1.5} where the functions μA : X → [0,1] represent the positive degree of x with respect to A and x ∈ X such that μA (x) ∈ [0,1], and the functions γA : X → [0,1] represent the negative degree of x with respect to A and x ∈ X such that γA(x) ∈ [0,1], and the 0 < μA(x) + γA(x) ≤ 1.5

Definition 2 Let a set A be fixed. An intuitionistic fuzzy set or IFS A of U is an object having the form A = {〈x, μA(x), γA(x)〉 ⎜x ∈U}

III. LINGUISTIC LABELS Suppose that L = {lα / α = 0.1, 0.2,…,t} is a finite and totally-ordered discrete term set, where lα represents a possible value for a linguistic variable and satisfies the following properties as: i. The set is ordered: lα > lβ if and only if α > β ii. There is the negation operator: Neg (lα) = lβ , such that β = t + 0.1 − α

where the function μA(x) : U → [0, 1] and γA(x) : U → [0, 1] define, respectively, the degree of membership and degree of non-membership of the element x∈ U to the set A, which is a subset of U, and every x ∈ U, 0 ≤ μA(x) + γA(x) ≤ 1. However, IFS seems to have the potential of being extended by ignoring or releasing the 0 ≤ μA(x) + γA(x) ≤ 1 condition. Therefore, a new concept of evaluation approach is proposed by utilising both the positive and negative aspects concurrently [9 – 11]. If μ : X → I and γ : X → I are two fuzzy sets, we can define bifuzzy set as (μ,γ): X x X → I as in the following structure shown in Figure 1. X x X

μ

γ I

Table 1: The mean of fuzzy numbers and its label Fuzzy Linguistic Extremely Poor Very Poor Poor Slightly Poor Fair Slightly Good Good Very Good Extremely Good

(μ,γ) I

The mean of fuzzy numbers 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Label l0.1 l0.2 l0.3 l0.4 l0.5 l0.6 l0.7 l0.8 l0.9

In the evaluation process, the continuous linguistic terms of a label system was employed. To preserve this situation, we extend the discrete term set L to a continuous linguistic terms

x I

Figure 1: The bifuzzy set structure[11]

_

set, L = {lα | l1 < lα < lt ,α ∈ [0,1]} , where if lα ∈ L, then we call lα the original linguistic term, otherwise, we call lα the virtual linguistic term [12]. Hence, the transformation is needed from the linguistic scales into the mean of fuzzy numbers [13] (see Table 1). The set on corresponding transformation of nine linguistic labels is given by

Hence, we can see that 0 ≤ μ(x) + γ(x) ≤ 2, where 0 ≤ μ(x) ≤ 1 and 0 ≤ γ(x) ≤ 1. Since the value of fuzzy set is in [0,1], we define 0.5 as the threshold value when the value is greater than 0.5 is said to be dominant. If the value is less than 0.5, it is then said to be under dominant. Now, if we have two fuzzy sets which are conflicting on the same X, we see that the values of μ(x) and γ(x) cannot be both dominant and/or both under dominant, concurrently. Hence, for two conflicting bifuzzy sets (CBFS) μ and γ on the same X, if μ is dominant, then γ must be under dominant and its true conversely. For example, we observe the two fuzzy sets “good’ and “bad” for performance of a candidate. The fuzzy set “bad” need not necessarily be the compliment of the fuzzy set “good”. If the “good” performance with μz(x) = 0.7, the value for “bad” performance need not be 0.3, but may be γ(x) = 0.4 and μ(x) + γ(x) = 0.7 + 0.4 > 1. Thus, we can define the CBFS (μ,γ) as X x X → I such that 0 < μ(x) + γ(x) < 1.5. Thus, we can consider two conflicting bifuzzy set μ : X → [0,1] and γ : X → [0,1] defined on the same premises of X to be given as the following definition [11]:

L = {l0.1 = extremely poor = 0.1, l0.2 = very poor = 0.2, l0.3 = poor = 0.3, l0.4 = slightly poor = 0.4, l0.5 = fair = 0.5, l0.6 = slightly good = 0.6, l0.7 = good = 0.7, l0.8 = very good = 0.8, (1) l0.9 = extremely good = 0.9}. For the purpose of reference, three aggregation operator definitions from Xu [12] were reviewed, namely the Linguistic Ordered Weighted Geometric Averaging (LOWGA), Linguistic Geometric Averaging (LGA), and Linguistic Hybrid Geometric Averaging (LHGA). All these definitions are related to the linguistic approach that has been utilised for aggregating the group decision maker’s evaluation throughout this paper.

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2011 IEEE Colloquium on Humanities, Science and Engineering Research (CHUSER 2011), Dec 5-6 2011, Penang

corresponding to the alternative xi, and then get the equilibrium preference degree ( Ep) i of the ith alternative over all the other alternatives. Step 3: Rank the equilibrium preference degree of ( Ep) i (i = 1, 2, 3,…,n) by descending order. For example, for n alternatives: ( Ep) n+1 = lα1 ⊃ (Ep) 2 = lα 4 ⊃ ( Ep) n−3 = lα 2 ⊃ …….

Definition 4 A LOWGA operator of the dimension n is a −n

−

mapping LOWGA: L → L , which has associated with it an exponential weighting vector w = (w1, w2, … , wn)T, with wj∈ n

[0,1] and ∑ w j = 1 , such that j =1

(l

LOWGAw

α 1 , lα 2 ,..., lα n

) = (l ) β1

w1

( )

⊗ lβ 2

w2

⎛ ⎞ ⎛ ⎞ = ⎜ l w1 ⎟ ⊗ ⎜ l w2 ⎟ ⊗ ... ⊗ ⎛⎜ l wn ⎝ βn ⎝ β1 ⎠ ⎝ β2 ⎠

( )

⊗ ... ⊗ lβ n

wn

⊃….. (Ep)1 = lα

⎞=l ⎟ β ⎠

w

where β = ∏nj=1 β j j , l β j is the jth largest of the lαi .

Definition 5 If the exponential weighting vector w = (1/n, 1/n, …, 1/n)T, then the LOWGA operator is reduced to the LGA operator, i.e., LOWGAw lα 1 , lα 2 ,..., lα = lα (3)

(

where α = ∏nj=1 α j

(

)

1/ n

n

V. A HYPOTHETICAL EXAMPLE Suppose that the federal government decided to build a new Universiti Teknologi MARA (UiTM) campus in the state of Sabah, East of Peninsular Malaysia. Four possible locations may be considered, given as i) x1, Labuan Federal Territory; ii) x2, Sandakan; iii) x3, Lahad Datu; and iv) x4, Beaufort. One main criterion used is the environmental impact assessment (EIA). Three stakeholders were involved and they gave their assessment based on experience and expertise. The stakeholders are i) UiTM panel (D1); ii) environmental experts (D2); and iii) local government authority (D3), whose weight vector τ = (0.3, 0.4, 0.3). The stakeholders compare these four location with respect to the EIA criterion by using the linguistic terms in the set L = {l0.1 = extremely poor = 0.1, l0.2 = very poor = 0.2, l0.3 = poor = 0.3, l0.4 = slightly poor = 0.4, l0.5 = fair = 0.5, l0.6 = slightly good = 0.6, l0.7 = good = 0.7, l0.8 = very good = 0.8, l0.9 = extremely good = 0.9}. The conflicting linguistic preference relations sk (k = 1, 2, 3) denoted by (l1(1j )+ , l1(1j )− ) , are shown in Tables 2, 4 and 6, respectively. Meanwhile, the equilibrium linguistic preference relations denoted by (l1(1j )+ , l1(1j)* ) , are shown in Tables 3, 5, and 7, respectively.

)

. −n

−

Definition 6 A LHGA operator is a mapping LHGA: L → L , which has associated with it an exponential weighting vector n

w = (w1, w2, …, wn)T, with wj ∈[0,1], ∑ w j = 1 , such that

(

) ( )

LHGAω,w lα 1 , lα 2 ,..., lα = lβ n 1

j =1

w1

( )

⊗ lβ 2

w2

( )

⊗ ... ⊗ lβ n

wn

7

which can be written as xn+1 ⊃ x2 ⊃ x n−3 ⊃ ….. ⊃ x7, where the symbol ‘⊃’ means ‘is preferred or superior to’. Step 4: Identify the best alternative from Step 3. Thus, (xn+1) is the best option, followed by (x2) and (x n−3) and so on, and (x7) is the last option among n alternatives.

(2)

(4)

where l β j is the jth largest of the linguistic weighted argument

( )

− ⎛− ⎞ nω l αi ⎜ l αi = lαi i , i = 1,2,3,..., n ⎟ , ω = ( ω1, ω2,…, ωn)T is the ⎝ ⎠ exponential weighting vector of the lαi , with ωj ∈ [0,1],

n

∑ ω j = 1 , and n is the balancing coefficient. j =1

IV. THE ALGORITHM OF THE PROPOSED AGGREGATION METHOD From Section III, the algorithm of the proposed aggregation method was developed in four steps as follows: Step 1: Employ the modified LGA operator from definition 3 – 6 to aggregate the combining positive (lα) with the nonnegative ( l β* ) aspects (henceforth called equilibrium linguistic

Table 2: Conflicting linguistic preference relation D1

preference relations) of preference information to get the equilibrium preference degree ( Ep)i( k ) of the ith alternative over all the other alternatives as follows: (Ep)1(1) = LGA {(l1(1j)+ ⊗ l1(1j)* )1/ 2 ⊗ (l1(1j )+ ⊗ l1(1j )* )1/ 2 ⊗ .... ⊗ (l1(n1)+ ⊗ l1(n1)* )1/ 2 }1/( n−1) (j = 1,2,3, …,n) where (l1(1j )+ , l1(1j)* ) means (‘linguistic positive labels’,

x1 x2 x3 x4

x1 (l0.8, l0.1) (l0.6, l0.4) (l0.7, l0.2)

x2 (l0.2, l0.9) (l0.5, l0.4) (l0.6, l0.5)

x3 (l0.4, l0.6) (l0.5, l0.6) (l0.8, l0.1)

x4 (l0.3, l0.8) (l0.4, l0.5) (l0.2, l0.9) -

Table 3: Equilibrium linguistic preference relation D1 x1 x2 x3 x4

‘linguistic non-negative labels) Step 2: Give exponential weighting vector w = (α1, α2, α3)T to aggregate ( Ep)i( k ) (k = 1, 2, 3,…m; i = 1, 2, 3, …, n)

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x1 (l0.8, l0.9) (l0.6, l0.6) (l0.7, l0.8)

x2 (l0.2, l0.1) (l0.5, l0.6) (l0.6, l0.5)

x3 (l0.4, l0.4) (l0.5, l0.4) (l0.8, l0.9)

x4 (l0.3, l0.2) (l0.4, l0.5) (l0.2, l0.1) -

2011 IEEE Colloquium on Humanities, Science and Engineering Research (CHUSER 2011), Dec 5-6 2011, Penang

Table 4: Conflicting linguistic preference relation D2 x1 (l0.3, l0.6) (l0.6, l0.4) (l0.5, l0.4)

x1 x2 x3 x4

x2 (l0.7, l0.4) (l0.6, l0.3) (l0.4, l0.4)

x3 (l0.4, l0.6) (l0.4, l0.7) (l0.6, l0.2)

D3

x4 (l0.5, l0.6) (l0.6, l0.6) (l0.4, l0.8) -

x1 (l0.3, l0.4) (l0.6, l0.6) (l0.5, l0.6)

x2 (l0.7, l0.6) (l0.6, l0.7) (l0.4, l0.6)

x3 (l0.4, l0.4) (l0.4, l0.3) (l0.6, l0.8)

x4 (l0.5, l0.4) (l0.6, l0.4) (l0.4, l0.2) -

⇒ (l0.240 )

x1 x2 x3 x4

x2 (l0.2, l0.8) (l0.6, l0.5) (l0.7, l0.2)

x3 (l0.6, l0.3) (l0.4, l0.5) (l0.5, l0.5)

x1 (l0.8, l0.8) (l0.4, l0.3) (l0.6, l0.7)

x2 (l0.2, l0.2) (l0.6, l0.5) (l0.7, l0.8)

x4 (l0.4, l0.3) (l0.3, l0.2) (l0.5, l0.5) -

{(l11(1)+ ⊗ l11(1)* )1/ 2 ⊗ (l12(1)+ ⊗ l12(1)* )1/ 2 ⊗ (l13(1)+ ⊗ l13(1)* )1/ 2 ⊗ (l14(1)+ ⊗ l14(1)* )1/ 2}1/(4−1) = { ' _' ⊗ (l0.2 ⊗ l0.1 )1 / 2 ⊗ (l0.4 ⊗ l0.4 )1 / 2 ⊗ (l0.3 ⊗ l0.2 )1 / 2 }1/(4 −1)

⊗l

0.16(1 / 6 )

⊗l

0.06(1 / 6 )

)

= l 0.521 ⊗ l 0.737 ⊗ l 0.626 = l 0.521 ⊗ 0.737⊗ 0.626 = l 0.240 Similarly, the rest of the equilibrium linguistic preference degrees ( Ep)i( k ) (k = 1, 2, 3; i = 1, 2, 3, 4) can be obtained as shown in Table 8. Table 8: The aggregate of (Ep) for all three evaluators (Di) 1

x1

x2

x3

x4

l0.240

l0.554

l0.359

l0.575

2

l0.488

l0.388

l0.478

l0.570

D D

(Ep) 3 = l0.432 ,

(Ep) 4 = l0.600

(Ep)1 = l0.375 which can be written as x4 ⊃ x2 ⊃ x3 ⊃ x1 where the symbol ‘⊃’ means ‘is preferred or superior to’. Step 4: Identify the best location from Step 3. Thus, the Beaufort (x4) is the best options for federal government build a new UiTM campus, followed by Sandakan (x2) and Lahad Datu (x3), and Labuan Federal Territory (x1) is the last option among four locations.

(Ep)1(1) =LGA

0.02(1 / 6 )

= l0.277 ; (l0.488 )3 × 0.4 = l0.423 ; (l0.355 )3 × 0.3 = l0.394

Step 3: Rank the preference degree of ( Ep) i (i = 1, 2, 3, 4) by descending order as follows, (Ep) 4 = l0.600 ⊃ (Ep) 2 = l0.496 ⊃ (Ep) 3 = l0.432 ⊃

Thus, from Section IV, the following algorithm was employed to obtain the best location: Step 1: Employ the modified LGA operator to aggregate the combining non-negative equilibrium linguistic preference information to get the equilibrium preference degree ( Ep)i( k ) of the ith location over all the other locations. For example, the (Ep)1(1) can be obtained as follows:

= (l

3×0.3

(Ep) 2 = l0.496 ,

x4 (l0.4, l0.7) (l0.3, l0.8) (l0.5, l0.5) -

x3 (l0.6, l0.7) (l0.4, l0.5) (l0.5, l0.5)

l0.624

Similarly, the ( Ep) i (i = 2, 3, 4) can be obtained as follows:

Table 7: Equilibrium linguistic preference relation D3 x1 x2 x3 x4

l0.456

and we have (Ep)1 = (l0.423 ) 0.3 ⊗ (l0.394 ) 0.5 ⊗ (l0.277 ) 0.2 = l0.375

Table 6: Conflicting linguistic preference relation D3 x1 (l0.8, l0.2) (l0.4, l0.7) (l0.6, l0.3)

l0.498

Step 2: Give exponential weighting vector w = (0.3, 0.5, 0.2)T to aggregate ( Ep)i( k ) (k = 1, 2, 3; i = 1, 2, 3, 4) corresponding to the locations xi, and then get the equilibrium preference degree ( Ep) i of the ith location over all the other locations. For purpose of calculation, given τ = (0.3, 0.4, 0.3)T, thus the (Ep)1 can be obtained as follows:

Table 5: Equilibrium linguistic preference relation D2 x1 x2 x3 x4

l0.355

VI. CONCLUSION In this paper, we have proposed the modified aggregation method based on the equilibrium linguistic preference relations in the group evaluation environment. Since group evaluation problems generally involve uncertainty, it is important to incorporate an equilibrium approach to derive comprehensively in any proposed aggregation method. It can be seen that the consideration for both positive and negative aspects is more comprehensible in concept and very promising in an evaluation perspective. Moreover, it also demonstrates a highly beneficial method, especially when the original problem lacks precise information. In addition, it gives a new dimension in the group evaluation processes and the hypothetical example in this paper can be applied to other situations as well. Finally, possible future research proposal may include on the effort to develop the specific software which complies with the proposed method to assist the evaluators to reach judgment in a simple manner. Once this issue is well resolved, our new proposed method would fit to be one of the comprehensive evaluation methods available. ACKNOWLEDGMENT The authors would like to acknowledge the editor and two anonymous reviewers for their valuable comments and suggestions.

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2011 IEEE Colloquium on Humanities, Science and Engineering Research (CHUSER 2011), Dec 5-6 2011, Penang

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