Fusion of Fuzzy Information

Introduction

I: DC J: Depot

7.4 Fusion of Fuzzy Information

Fusion approach of fuzzy information is proposed by Herrera et al.

(2000) to carry out the aggregation step of a decision process in a group decision-making problem defined using nonhomogeneous information.

This approach consists of obtaining a collective performance profile on the alternatives according to the individual performance profiles. It is performed in two phases (Herrera et al., 2000):

1. Making the information uniform

2. Aggregating individual preference values 7.4.1 Making the Information Uniform

The nonhomogeneous information will be unified into a specific lin- guistic domain, called basic linguistic term set (BLTS) denoted as ST, chosen so as not to impose useless precision to the original evaluations and to allow an appropriate discrimination of the initial performance values. The process of unifying the information involves the compari- son between fuzzy sets. These comparisons are usually carried out by means of a measure of comparison.

The transformation function is defined as follows (Herrera et al., 2000).

Let Ω = {l₀,l1,…,lH} and ST = {s0,s1,…,sG} be two linguistic term sets, such that G ≥ H. Then, the transformation function, τAS_T, is defined as

τ

τ γ

γ

AS T

AS h

g gh

h

g T

T

F S

l s

g G l

:

, , , , ,

Ω

Ω

→

( )

( )

^=⎧

( )

_∈

_{{ }}

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

0 1… ∀ ∈

hh

y lh y sg y

=^{max min}

{

^μ

( )

^,μ

( ) }

(7.2)

where

F(ST) is the set of fuzzy sets defined in ST

y ( )

lh

µ and µsg( )y are the membership functions of the fuzzy sets associated with the terms lh and sg, respectively

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The transformation function is also appropriate to convert the stan- dardized fuzzy assessments into a BLTS (Chuu, 2009). The max–min operation has been chosen in the definition of the transformation function since it is a classical tool to set the matching degree between fuzzy sets (Herrera et al., 2000).

7.4.2 Aggregating Individual Preference Values

The input information, which was denoted by means of fuzzy sets, is expressed on a BLTS by the earlier-mentioned transformation func- tion. Then, in order to obtain a collective preference value for each alternative, an aggregation function is used. This collective perfor- mance value is a new fuzzy set defined on a BLTS.

This chapter employs ordered weighted averaging (OWA) operator, initially proposed by Yager (1988), as the aggregation operator. This operator provides aggregations that lie between two extreme cases of MCDM problems that lead to the use of and and or operators to com- bine the criteria function. OWA operator encompasses several opera- tors since it can implement different aggregation rules by changing the order weights.

The OWA operator provides a unified framework for decision making under uncertainty, in which different decision criteria such as maximax, maximin, equally likely (Laplace), and Hurwicz’s criteria are characterized by different OWA operator weights. To apply the OWA operator for decision making, a crucial issue is to determine its weights, which can be accomplished as follows.

Let A = {a1,a2,…,an} be a set of values to be aggregated, then OWA operator F is defined as

F a a an T w b

i i i

n

1 2

1

, , ,…

( )

^{= =}

∑

=

wb (7.3)

where w = {w1,w2,…,wn} is a weighting vector, such that wi∈ [0,1] and wi

∑

i ^=1, and b is the associated ordered value vector where bi∈ b is the ith largest value in A.

15 2 mehtaP dursun e t al .

The weights of the OWA operator are calculated using fuzzy lin- guistic quantifiers, which for a nondecreasing relative quantifier Q are given by

w Q i

n Q i

n i n

i = ⎛

⎝⎜ ⎞

⎠⎟ − ⎛ −

⎝⎜ ⎞

⎠⎟ =

1 , 1, ,… (7.4)

The nondecreasing relative quantifier, Q, is defined as (Herrera et al., 2000)

Q y

y a y a

b a a y b y b

( )

⁼

<

−

− ≤ ≤

>

⎧

⎨⎪⎪

⎩

⎪⎪ 0

1 ,

, ,

(7.5)

with a,b,y ∈ [0,1] and Q(y) indicating the degree to which the propor- tion y is compatible with the meaning of the quantifier it represents.

Some nondecreasing relative quantifiers identified by terms most, at least half, and as many as possible, with parameters (a,b) are (0.3,0.8), (0,0.5), and (0.5,1), respectively.

7.5 2-Tuple Fuzzy Linguistic Representation Model

The 2-tuple linguistic model, composed of a linguistic term and a real number, presented by Herrera and Martínez (2000a) is based on the concept of symbolic translation. It can be denoted as (sg,α), where sg represents the linguistic label of the predefined linguistic term set ST, and α is a numerical value representing the symbolic translation.

Since the 2-tuple linguistic model can express any counting of infor- mation in the universe of discourse and avoid the loss of information, it has been widely employed in decision making. This model is well suited to deal with uniformly and symmetrically distributed linguistic term sets. Moreover, the results of the Herrera and Martínez model can match the elements in the initial linguistic term set.

The process of comparison between linguistic 2-tuples is carried out according to an ordinary lexicographic order as follows (Herrera and Martínez, 2001).

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integr ated deCision model

Let r1 = (sc,α1) and r2 = (sd,α2) be two linguistic variables represented by 2-tuples:

• If c < d, then r₁ is smaller than r₂.

• If c = d, then

• If α₁ = α₂, then r1 and r2 represent the same information.

• If α₁ < α₂, then r1 is smaller than r2.

• If α₁ > α₂, then r₁ is bigger than r₂.

In the following, we define a computational technique to operate with the 2-tuples without loss of information.

Definition 7.1 (Herrera and Martínez, 2000b) Let L = (γ₀,γ₁,…,γ_G) be a fuzzy set defined in ST. A transformation function χ that trans- forms L into a numerical value in the interval of granularity of ST,[0,G]

is defined as χ

χ χ γ

γ γ

: ,

, , , , ,

F S G

F S s g G g

T

T g g

g g G

g g

( )

^→

[ ]

( ( ) )

⁼

( { ( )

⁼

} )

^{= =}

=

∑

0

0 1 ⁰

0

… _GG

∑

^{= β (7.6)}

where F(ST) is the set of fuzzy sets defined in ST .

Definition 7.2 (Herrera and Martínez, 2000a) Let S = {s0,s1,…,sG} be a linguistic term set and β∈[0,G] a value supporting the result of a symbolic aggregation operation, then the 2-tuple that expresses the equivalent information to β is obtained from the following function:

Δ Δ

: , . , .

,

, . , .

0 0 5 0 5

0 5 0 5

G S

s g

g

g

[ ]

^{→ × −}

[ )

( )

^{= =}

( )

= − ∈ −

[ )

⎧⎨

β β

α β α

round

⎪⎪

⎩⎪

(7.7)

where

round is the usual round operation sg has the closest index label to β

α is the value of the symbolic translation

15 4 mehtaP dursun e t al .

Proposition 7.1 (Herrera and Martínez, 2000a) Let S = {s0,s1,…,sG} be a linguistic term set and (sg,α) be a 2-tuple. There is a Δ⁻¹ func- tion such that from a 2-tuple, it returns its equivalent numerical value β ∈[ , ]0 G ⊂ ℜ. This function is defined as

Δ Δ

−

−

× −

[ )

^→

[ ]

( )

^{= + =}

1

1

0 5 0 5 0

: . , . ,

,

S G

sg α g α β (7.8)

Definition 7.3 (Herrera-Viedma et al., 2004) Let x = {(s₁,α₁),…, (sG,αG)} be a set of linguistic 2-tuples and W = {w1,…,wG} be their asso- ciated weights. The 2-tuple weighted average x^w is computed as

x s s s w

w

w G G

g g g

g G

g g 1 1 G

1 1

1

, , , , ,

α α α

( ) ( )

⎡⎣ ⎤⎦ = ^⎛

( )

^⋅

⎝

⎜⎜

⎜

⎞

⎠

− ⎟

=

=

∑

… Δ

∑

Δ

⎟⎟⎟

=

⎛ ⋅

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟

=

=

∑

Δ

∑

βg g g

G

g g G

w w

1

1

(7.9)

Definition 7.4 (Herrera-Viedma et al., 2004; Wang, 2010) Let x = {(s1,α₁),…,(sG,α_G)} be a set of linguistic 2-tuples and W ={( ,w1 α^w1), ,(… wG,αGw)} be their linguistic 2-tuple-associated

weights. The 2-tuple linguistic weighted average xlw is calculated by the following function:

xlw s w w s w

G G G Gw

g w

g ^g

1,α1 , 1,α1 , , ,α , ,α

β β

( ) ( )

⎡⎣ ⎤

⎦ ^⎡_⎣

( ) ( )

^⎤_⎦

( )

=

= ⋅

…

Δ ¹¹

1 G

g w G

g

∑

∑

⁼

⎛

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

β ⎟ (7.10)

with β_g = Δ⁻¹(sg,α_g) and βw g αwg

g =Δ⁻¹(w , ).