Total ordering defined on the set of all intuitionistic fuzzy numbers

1.1 Motivation

The capacity to handle dubious and uncertain data is more effectively done by stretching out intuitionistic fuzzy values to TrIFNs because the membership and non-membership degrees are better expressed as trapezoidal values rather than exact values. TrIFNs are generalisation of intuitionistic fuzzy values and IVIFNs. As a generalisation, the set of TrIFNs should contain the set of all intuitionistic fuzzy values and IVIFNs. But the existing defintion for TrIFNs [4, 19] does not contain the set of intuitionistic fuzzy values which means the existing definition for TrIFN is not the real generalisation of intuitionistic fuzzy values. Hence the study about new structure for intuitionistic fuzzy number [15] is essential. In the application point of view our proposed method is more applicable and more natural when it is compared with the existing methodology. More precisely, the existing definition for Trapezoidal intuitionistic fuzzy numbers (TrIFN) present in the literature [4, 19] is defined as A = 〈 (a, b, c, d) (e, f, g, h) 〉 with e ≤ a ≤ f ≤ b ≤ c ≤ g ≤ d ≤ h does not generalise even the intuitionistic fuzzy value of the kind A = (a, c)  with a + c ≤ 1 and a < c. That is, if we write A = (a, c) in trapezoidal intuitionistic form, we get A = 〈 (a, a, a, a) (c, c, c, c) 〉 with a < c which contradicts the above definition for TrIFNs. So till today the real generalization of intuitionistic fuzzy values and interval valued intuitionistic fuzzy numbers have not been studied in detail. This problem motivate us to study this type of intuitionistic fuzzy numbers [15] and its ordering principles for ranking.

3.1 Upper dense sequence in (0, 1]

The major difficulty in defining total ordering on the set of all fuzzy numbers is that there is no effective tool to identify an arbitrarily given fuzzy number by only finitely many real-valued parameters. To over come this difficulty, W. Wang, Z. Wang [21] has introduced the concept of upper dense sequence in the interval (0, 1], and defined the total ordering on the set of all fuzzy numbers using this sequence. This upper dense sequence gives values for α in the α-cut of FNs. But for IFNs, two sequences are needed to give values for α  &  β in the (α, β)-cut where α ∈ (0, 1]   &  β ∈ [0, 1). For convenience upper dense sequence of α is denoted by S₁ ={ α _i |i = 1, 2, . . . } and upper dense sequence of β is denoted by S₂ ={ β _i |i = 1, 2, . . . }. In this section, the upper dense sequence in (0, 1] is briefly reviewed.

Definition 3.1.1. (W. Wang, Z. Wang, [21]) A sequence S = { α _i |i = 1, 2, . . . } ⊂ (0, 1] is said to be upper dense in (0, 1] if, for every point x ∈ (0, 1] and ε > 0, there exists some α _i  ∈ S such that α _i  ∈ [x, x + ε) for some i. A sequence S ⊂ [0, 1) is said to be lower dense in [0, 1) if, for every point x ∈ [0, 1) and any ε > 0, there exists α _i  ∈ S such that α _i  ∈ (x - ε, x] for some i.

From definition 3.1.1 we note that, any upper dense sequence in (0, 1] is nothing but a dense sequence with real number 1 and it is also a lower dense sequence.

Definition 3.1.2. Let S₁ ={ α _i |i = 1, 2, . . . } and S₂ ={ β _i |i = 1, 2, . . . } be two upper dense sequences in (0, 1], then the double upper dense sequence is defined as S = (S₁, S₂) ={ (α _i , β _i ) |i = 1, 2, . . . }.

Example for an upper dense sequence and double upper dense sequence in (0, 1] is given as follows.

Example 3.1.1. Let S₁ ={ α _i |i = 1, 2 . . . } be the set of all rational numbers in (0, 1], where α₁ = 1, α 2 = 1 2 , α 3 = 1 3 , α 4 = 2 3 , α 5 = 1 4 , α 6 = 3 4 , α 7 = 1 5 ,α 8 = 2 5 , α 9 = 3 5 , α 10 = 4 5 , . . ..

Then sequence S₁ is an upper dense sequence in (0, 1]. If we allow a number to have multiple occurences in the sequence, the general members in upper dense sequence S _β  ={ s _{β i} |i = 1, 2, . . . } can be expressed by s β i = ( i k - k - 1 2 ) , i = 1 , 2 , . . . where k = ⌈ 2 i + 1 4 - 1 2 ⌉. That is, s β 1 = 1 , s β 2 = 1 2 , s β 3 = 1 ,s β 4 = 1 3 , s β 5 = 2 3 , s β 6 = 1 , s β 7 = 1 4 , s β 8 = 2 4 , s β 9 = 3 4 , . . .. In sequence S _β , for instance, s _{β 3} is the same real number as s _{β 1} .

Example 3.1.2. Consider S₁ as in example 1. Let S = (S₁, S₂) = { (α _i , β _i ) |α _i  = β _i , α _i  ∈ S₁ } = {(1, 1) , (1/2, 1/2) , (1/3, 1/3) , (2/3, 2/3) , (1/4, 1/4) , (3/4, 3/4) , (1/5, 1/5) , . . .}. Clearly S is a double upper dense sequence in (0, 1].

In the forthcoming sections this double upper dense sequence will be very useful for defining total orderings on the set of IFNs.

3.2 Decomposition theorem for intuitionistic fuzzy number using upper dense sequence

In this section, a new decomposition theorem for IFSs is established using the double upper dense sequence defined in (0, 1]. Before establishing a new decomposition theorem for intuitionistic fuzzy sets, it is needed for us to define decompostion theorems for intuitionistic fuzzy sets using special α - β cuts.

Definition 3.2.1. Let X be a nonempty universal set and A = (μ _A , ν _A ) be an intuitionistic fuzzy set of X with membership function μ _A and with a nonmembership function ν _A . Let α, β ∈ (0, 1]

The α - β cut, denoted by ^α - βA is defined by  ^(α - β) A = ^αμ _A  × ^βν _A where ^α μ _A  = {x ∈ X|μ _A  (x) ≥ α} and ^β ν _A  = {x ∈ X|ν _A  (x) ≥ β}. Equivalently ( α - β ) A = μ A - 1 ( [ α , 1 ] ) × ν A - 1 ( [ β , 1 ] ) is a subset of ℘ (X) × ℘ (X).

Example 3.2.1. Let A = 〈(0.17, 0.3, 0.47, 0.56) , (0.05, 0.13, 0.16, 0.23) 〉 be a trapezezoidal intuitionistic fuzzy number. Then  ^(α - β) A = ([0.17 + (0.3 - 0.17) α, 0.56 - (0.56 - 0.47) α] , [0.05 + (0.13-0.05) β, 0.23 - (0.23 - 0.16) β])  =  ([0.17 + 0.13α, 0.56 - 0.09α] , [0.05 + 0.08β, 0.23 - 0.07β]) , ∀ α, β∈[0, 1]. (0.05 + 0.08β, 0.23 - 0.07β)) , ∀ α, β ∈ [0, 1]. L _{μ A}  =({0.17 + (0.3 - 0.17) α, 0.56 - (0.56 - 0.47α} , {0.05+(0.13 - 0.05) β, 0.23 - (0.23 - 0.16) β)} = ({ 0.17 + 0.13α, 0.56 - 0.09α } , {0.05 + 0.08β, 0.23 - 0.07β}) , ∀α, β ∈ [0, 1].

One way of representing a fuzzy set is by special fuzzy sets on α-cuts and another way of representing a fuzzy set is by special fuzzy sets on Strong α-cuts. As a generalisaton of fuzzy sets, any intuitionistic fuzzy set can also be represented by the use of special intuitionistic fuzzy sets on α - β cuts and special intuitionistic fuzzy sets on strong α - β cuts.

Definition 3.2.2. The special intuitionistic fuzzy set  _(α,β) A = ( _α μ _A , _β ν _A ) is defined by its membership ( _α μ _A ) and non-membership function ( _β ν _A ) as follows,

α μ A ( x ) = { α , x ∈ α μ A 0 , otherwise

β ν A ( x ) = { β , x ∈ β ν A 0 , otherwise.

The following decomposition theorems will show the representation of an arbitrary IFS in terms of the special IFSs  _(α,β) A.

Theorem 3.1. First Decomposition theorem of an IFS: LetXbe a non-empty set. For an intuitionistic fuzzy subsetA = (μ _A , ν _A ) inX, A = (⋃ _{α∈[0,1]  α}μ _A , ⋃ _{β∈[0,1] β} ν _A ) , where ⋃ is standard union.

Proof. Let x be an arbitrary element in X and let μ _A  (x) = a  &  ν _A  (x) = b.

Then ( ( ∪ α ∈ [ 0 , 1 ] α μ A ) ( x ) , ( ∪ β ∈ [ 0 , 1 ] β ν A ) ( x ) ) = ( S u p α ∈ [ 0 , 1 ] α μ A ( x ) , S u p β ∈ [ 0 , 1 ] β ν A ( x ) ) = ( m a x S u p α ∈ [ 0 , a ] α μ A ( x ) , S u p α ∈ ( a , 1 ] α μ A ( x ) , m a x S u p β ∈ [ 0 , b ] β ν A ( x ) , S u p β ∈ ( b , 1 ] β ν A ( x )

For each α ∈ [0, a], we have μ _A  (x) = a ≥ α, therefore _α μ _A  (x) = α. On the other hand, for each α ∈ (a, 1], we have μ _A  (x) = a < α and _α μ _A  (x) =0.

Similarly, for each β ∈ [0, b], we have ν _A  (x) = b ≥ β, therefore _β ν _A  (x) = β. On the other hand, for each β ∈ (b, 1], we have ν _A  (x) = b < β and _β ν _A  (x) =0. Therefore ((⋃ _{α∈[0,1] α}μ _A ) (x) , (⋃ _{β∈[0,1] β}ν _A ) (x))=(Sup_α∈[0,a]α, Sup_β∈[0,b]β) = (a, b) = (μ _A  (x) , ν _A  (x)). Hence the theorem.

To illustrate the above theorem, let us consider a trapezoidal intuitionistic fuzzy number A as in Fig. 3.

For each α, β ∈ [0, 1], the α - β cut of A = {(a, b₁, b₂, c) , (e, f₁, f₂, g)} is given by  ^(α - β) A =([a + (b₁ - a) α, c - (c - b₂) α] , [e + (f₁ - e) β, g - (g - f₂) β]) and the special intuitionistic fuzzy set  _(α,β) A employed in definition 3.2.2 is defined by its membership ( _α μ _A ) and non-membership function ( _β ν _A ) as follows

α μ A ( x ) = { α , x ∈ [ a + ( b 1 - a ) α , c - ( c - b 2 ) α ] 0 , otherwise

β ν A ( x ) = { β , x ∈ [ e + ( f 1 - e ) β , g - ( g - f 2 ) β ] 0 , otherwise

Examples of sets ^α μ _A , ^β ν _A , _α μ _A and _β ν _A for three values of α (namely α₁, α₂, α₃) and β (namely β₁, β₂, β₃) are shown in Fig. 3.

According to theorem 3.1, A is obtained by taking the standard fuzzy union of sets ( _α μ _A , _β ν _A ) for all α, β ∈ [0, 1].

Theorem 3.2. Second Decomposition theorem of an IFS: LetXbe a non-empty set. For an intuitionistic fuzzy subsetAinX,

A = (⋃ _{α∈[0,1]  α+}μ _A , ⋃ _{β∈[0,1]  β+}ν _A ), where ⋃ is standard union.

Proof. Let x be an arbitrary element in X and let μ _A  (x) = a  &  ν _A  (x) = b.

Then ((⋃ _{α∈[0,1]α+}μ _A ) (x) , (⋃ _{β∈[0,1]β+}ν _A ) (x)) = (Sup_{α∈[0,1]  α+μ A  (x) , Supβ∈[0,1] β+ν A  (x)) = (max[Supα∈[0,a) α+μ A  (x) , Supα∈[a,1] α+μ A  (x)] , max[Supβ∈[0,b)  β+ν A  (x) , Supβ∈[b,1]  β+ν A  (x)]).}

For each α ∈ [0, a), we have μ _A  (x) = a > α, therefore _α +μ _A  (x) = α. On the other hand, for each α ∈ [a, 1], we have μ _A  (x) = a ≤ α and _α +μ _A  (x) =0. Similarly, for each β ∈ [0, b), we have ν _A  (x) = b > β, therefore _β +ν _A  (x) = β. On the other hand, for each β ∈ [b, 1], we have ν _A  (x) = b ≤ β and _β +ν _A  (x) =0. Therefore

((⋃ _{α∈[0,1]  α+}μ _A ) (x) , (⋃ _{β∈[0,1]  β+}ν _A ) (x))=

(Sup_α∈[0,a)α, Sup_β∈[0,b)β)= (a, b) = (μ _A  (x) , ν _A  (x)). Hence the theorem.□

Theorem 3.3. Third Decomposition theorem of an IFS: LetXbe a non-empty set. For an intuitionistic fuzzy subsetAinX,

A = (⋃ _{α∈L μ A   α}μ _A , ⋃ _{β∈L ν A   β}ν _A ), where ⋃ is standard union.

Proof. The proof of this theorem is similar to the Theorem 3.1.

Regarding IFN as special intuitionistic fuzzy subset (IFS) of R, these decomposition theorems are also available for IFNs. Since they identify an intuitionistic fuzzy number by uncountably infinite real valued parameters generally and therefore lexicography can not be used anymore, unfortunately none of them can be used to define a total ordering on the set of IFNs. Thus, establising a new decomposition theorem, which identifies any IFN by only countably many real valued parameters is essential.

Theorem 3.4. Fourth Decomposition theorem of an IFS:LetA = (μ _A , ν _A ) be an intuitionistic fuzzy subset ofX, andS = (S₁, S₂) be a given double upper dense sequence in [0, 1].

ThenA = (⋃ _{α∈S1  α}μ _A , ⋃ _{β∈S2  β}ν _A ).

Proof. Let x be an arbitrary element in X and let μ _A  (x) = a  &  ν _A  (x) = b. Since S₁ ⊆ [0, 1]   &  S₂ ⊆ [0, 1], Thus we have ⋃_{α∈S1  αμ A  ⊆ ⋃ α∈[0,1]  αμ A  = μ A and ⋃β∈S2  βν A  ⊆ ⋃ β∈[0,1]  βν A  = ν A .}

So (⋃ _{α∈S1  α}μ _A , ⋃ _{β∈S2  β}ν _A ) ⊆ A            ........(1)

Now we have to show that μ _A  (x) ≤ ⋃ _{α∈S1  α}μ _A  (x) and ν _A  (x) ≤ ⋃ _{β∈S2  β}ν _A  (x) to prove the theorem.

From second decomposition theorem we know that, for each x ∈ X, (μ _A  (x) , ν _A  (x)) = (Sup_α∈[0,1] (x) , Sup_{β∈[0,b)  β+ν A  (x)).}

For each α ∈ [0, a), since S₁ is upper dense in (0, 1]. We may find a real number K₁ ∈ S₁ such that K₁ ≥ α, which implies that K₁ ∈ [α, a) and . Thus taking the supremum with respect to α ∈ (0, μ _A  (x)), we obtain μ _A  (x) = Sup_{α∈(0,a) α+μ A  (x) ≤ SupK1∈S1 K1μ A  (x) = Supα∈S1 αμ A  (x). i.e., μ A  (x) ≤ ⋃ α∈S1  αμ A  (x).}

Similarly for each β ∈ [0, b), since S₂ is upper dense in [0, 1]. We may find a real number K₂ ∈ S₂ such that K₂ ≥ β, which implies that K₂ ∈ [α, b) and . Thus taking the supremum with respect to β ∈ (0, ν _A  (x)), we obtain ν _A  (x) = Sup_{β∈(0,b)  β+ν A  (x) ≤ SupK2∈S2  K2ν A  (x) = Supβ∈S2 βν A  (x).i.e., ν A  (x) ≤ ⋃ β∈S2  βν A  (x).}

Hence A (x) = (μ _A  (x) , ν _A  (x)) ⊆ (⋃ _α∈S1 αμ _A  (x) , ⋃ _β∈S2 βν _A  (x)).                   ......(2) (1) and (2) concludes the proof.□

5.1 Comparision between our proposed method with the score function defined in Lakshmana et al. [15]

In this subsection, our proposed method is compared with the total score function defined in Lakshmana et al. [15], which is explained here with an illustrative example.

Definition 5.1.1. The membership score of the triangular intuitionistic fuzzy number (TIFN) M = { (a, b, c) (e, f, g)} is defined by T ( M ) = [ 1 + R ( M ) - L ( M ) ] 2, where L ( M ) = 1 - a 1 + b - a and R ( M ) = c 1 + c - b.

Definition 5.1.2. (Lakshmana et al. [15]) The new membership score of the triangular intuitionistic fuzzy number M ={ (a, b, c) (e, f, g) } is defined by NT ( M ) = [ 1 + NL ( M ) - NR ( M ) ] 2, where NL ( M ) = e 1 + e - f and NR ( M ) = 1 - g 1 + f - g.

Definition 5.1.3. (Lakshmana et al. [15]) The nonmembership score of the triangular intuitionistic fuzzy number M ={ (a, b, c) (e, f, g) } is defined by NT _c  (M) =1 - NT (M).

Definition 5.1.4. (Lakshmana et al. [15]) Let M =〈 (a, b, c) , (e, f, g) 〉 be an triangular intuitionistic fuzzy number. If e ≥ b and f ≥ c, then the score of the intuitionistic fuzzy number M is defined by (T, NT _c ), where T is the membership score of M which is obtained from (a, b, c) and NT _c is the nonmembership score of M obtained from (e, f, g).

Definition 5.1.5 (Lakshmana et al. [15]) Ranking of Intuitionistic Fuzzy Numbers: Let M₁ =〈 (a₁, b₁, c₁) , (e₁, f₁, g₁) 〉 and M₂ = 〈 (a₂, b₂, c₂) , (e₂, f₂, g₂)〉 be two triangular intuitionistic fuzzy numbers with e _i  ≥ b _i   &  f _i  ≥ c _i . Then M₁ ≤ M₂ if the membership score of M₁≤ the membership score of M₂ and the nonmembership score of M₁≥ the nonmembership score of M₂.

In this example definition 5.1.1 to 5.1.5 are demonstrated and also the illogicality of Lakshmana et al.’s [15] method is shown.

Example 5.1.1. Let M =〈 (0, 0.2, 0.4) , (0.4, 0.45, . 5) 〉 with (0, 0.2, 0.4)  ^c  ≤ (0.4, 0.45, 0.5) and 0.4 ≥ 0.2  &  0.45 ≥ 0.4 and N = 〈(0.25, 0.25, 0.25) , (0.4, 0.45, 0.5) 〉 with (0.25, 0.25, 0.25)  ^c  ≥ (0.4, 0.45, 0.5) and 0.4 ≥ 0.25  &  0.45 ≥ 0.25 be the two triangular intuitionistic fuzzy numbers. Using definition 5.1.1 we get L (M) =0.8333 and R (M) =0.3333 which implies T (M) =0.25. Applying definition 5.1.2, 5.1.3, 5.1.4 to M we get NL (M) =0.421053, NR (M) =0.526316, NT (M) =0.447368 and NT _c  (M) =0.553. Therefore the total score of membership and nonmembership functions of M are T = 0.25 and NT _c  = 0.553 respectively. Then the intuitionistic fuzzy score is represented by (T, NT _c ) = (0.25, 0.553). Similarly using definitions 5.1.1 to 5.1.3, we get the total score of membership and nonmembership functions of N are T = 0.25,  and NT _c  = 0.553 respectively.

Therefore from definition 5.1.5, this method ranks M and N are equal but M  &  N are different triangular intuitionistic fuzzy numbers.

The total ordering < defined by using double upper dense sequence D_(α,β) given in example 3.1.2 and the way shown in definition 4.0.3 are now adapted. For i = 1 and  (α₁, β₁) = (1, 1), we have C₁ (M) = -0.16 and C₁ (N) = -0.0875.

i.e., C₁ (M) < C₁ (N). Hence our proposed method ranks N as better one.

5.2 Comparision of the proposed method with some existing methods

In this sub section significance of our proposed method over some existing methods are explained with examples. The table 1 shows that our proposed method is significant over the methods presented in [3, 26], and it is supported by Geetha et al. [9].

For example, let A = ([0.2, 0.25] , [0.4, 0.45]) and B = ([0.15, 0.3] , [0.35, 0.5]) be two IVIFNs. Then by applying Lakshmana and Geetha [13] approach we get LG ( A ) = LG ( B ) = 0.45 + δ ( 0.70 ) 2 this implies that A and B are equal which is illogical. By applying the total ordering < defined by using double upper dense sequence D_(α,β) given in example 3.1.2 and the way shown in definition 4.0.3 are now adapted. For i = 1 and  (α₁, β₁) = (1, 1), we have C₁ (A) = -0.10375 and C₁ (B) = -0.09875, i.e., C₁ (A) < C₁ (B). Hence A < B which is supported by Geetha et al.’s approach. The double upper dense sequence S in example 3.1.2 is used as a necessary reference system for our proposed method in table 1 (i.e., for i = 1, (α₁, β₁) = (1, 1)).

5.3 Trapezoidal Intuitionistic Fuzzy Information System(TrIFIS)

Information system (IS) is a decision model used to select the best alternative from the all the alternatives in hand under various attributes. The data collected from the experts may be incomplete or imprecise numerical quantities. To deal with such data the thoery of IFS provided by Atanassov [1] aids better. In information system, dominance relation rely on ranking of data, ranking of intuitionistic fuzzy numbers is inevitable.

Definition 5.3.1. An Information System

S = (U, AT, V, f) with V = ∪ _a∈ATV _a where V _a is a domain of attribute a is called trapezoidal intuitionistic fuzzy information system if V is a set of TrIFN. We denote f (x, a) ∈ V _a by f (x, a) =〈 (a₁, a₂, a₃, a₄) , (a₁^′, a₂^′, a₃^′, a₄^′) 〉 where a _i , a _i ^′ ∈ [0, 1]. The numerical illustration is given in example 5.5.1.

Definition 5.3.2. A TrIFIS, S = (U, AT, V, f) together with weights W = {w _a /a ∈ AT} is called Weighted Trapezoidal Intuitionistic Fuzzy Information System (WTrIFIS) and is denoted by S = (U, AT, V, f, W).

Definition 5.3.3. Let a ∈ AT be a criterion. Let x, y ∈ U. If f (x, a) > f (y, a) (as per definition 4.0.3) then x >  _a y which indicates that x is better than (outranks) y with respect to the criterion a. Also x =  _a y means that x is equally good as y with respect to the criterion a, if f (x, a) = f (y, a).

Definition 5.3.4. Let S = (U, AT, V, f, W) be a WTrIFIS and A ⊆ AT.

Let B _A  (x, y) ={ a ∈ A|x >  _a y } and let C _A  (x, y) ={ a ∈ A|x =  _a y }. The weighted fuzzy dominance relation WR _A  (x, y) : U × U → [0, 1] is defined by WR A ( x , y ) = ∑ a ∈ B A ( x , y ) w a + ∑ a ∈ C A ( x , y ) w a 2.

Definition 5.3.5. Let S = (U, AT, V, f, W) be a WTrIFIS and A ⊆ AT. The entire dominance degree of each object is defined as

WR A ( x i ) = 1 | U | ∑ j = 1 | U | WR A ( x i , y j )

5.4 Algorithm for ranking of objects in WTrIFIS

Let S = (U, AT, V, f, W) be a WTrIFIS. The objects in U are ranked using following algorithm.

Algorithm:5.3.

1. Using Definition 4.0.3 find C j ′ s accordingly, to decide whether x _i   >  _a  x _i  or x _j   >  _a  x _i  or x _i   =  _a  x _j for all a ∈ A (A ⊆ AT) and for all x _i , x _j  ∈ U.

2. Enumerate B _A  (x _i , x _j ) using

B _A  (x _i , x _j ) ={ a ∈ A|x _i  >  _a x _j  } and C _A  (x _i , x _j ) using C _A  (x _i , x _j ) ={ a ∈ A|x _i  =  _a x _j  }.

3.Calculate the weighted fuzzy dominance relation using WR _A  (x, y) : U × U → [0, 1] defined by WR A ( x i , x j ) = ∑ a ∈ B A ( x i , x j ) w a + ∑ a ∈ C A ( x i , x j ) w a 2.

4. Calculate the entire dominance degree of each object using WR A ( x i ) = 1 | U | ∑ j = 1 | U | WR A ( x i , x j ).

5. The objects are ranked using entire dominance degree. The larger the value of WR _A  (x _i ), the better is the object.

5.5 Numerical illustration

In this subsection, Algorithm 5.4 is illustrated by an example 5.5.1.

Example 5.5.1. In this example we consider a selection problem of the best supplier for an automobile company from the available alternatives {x _i |i = 1 to 10} of pre evaluated 10 suppliers, based on WTrIFIS with attributes {a _j |j = 1 to 5} as product quality, relation- ship closeness, delivery performance, social responsib- ility and legal issue.

An TrIFIS with U ={ x₁, x₂, . . . , x₁₀ }, AT = {a₁, a₂, . . . , a₅} is given in Table 2, and weights for the each attribute W _a is given by W ={ w _a |a ∈ AT } = { 0.3, 0.2, 0.15, 0.17, 0.18 }.

In the table 2, f (x _i , a _i ) = (a₁, a₂, a₃, a₄) , (c₁, c₂, c₃, c₄) denotes the trapezoidal intuitionistic fuzzy numbers which include intuitionistic fuzzy values and interval valued intuitionstic fuzzy numbers and they stand for the evaluation of alternative x _i under the criteria a _i with acceptance of (a₁, a₂, a₃, a₄) and nonacceptance of (c₁, c₂, c₃, c₄). For example, f (x₁, a₁) denotes supplier x₁ is evaluated under the criteria ‘product quality’ (a₁) with “20% of acceptance and 40% of non acceptance” and f (x₅, a₃) denotes the supplier x₅ is evaluated under the criteria ‘delivery performance (a₃)’ with “around 30% to 35% of accepatance and around 45% to 50% of non acceptance”.

Step:1 For i = 1, (α, β) = (1, 1): By step 1, C₁ (f (x _i , a _j )) using defintion 4.0.3 and note 4.0.1, for all a _i  ∈ AT and for all x _i  ∈ U is found and tabulated in table 3. If C₁ (f (x _i , a _j )) = C₁ (f (x _j , a _j )) for any alternatives x _i , x _j then C₂ and other necessary score functions (C₃ and C₄) are found wherever required. The bold letters are used in table 3 to represent the equality of scores. From table 3 we observe that in many places C j ′ s are not distiguishable for different IFNs.

Hence from table 3 we do not get the best alternative. Therefore the same procedure which is explained in step 1 is repeated for i = 2, (α, β) = (1/2, 1/2) and it is shown in table 4.

The weighted fuzzy dominance relation using WR A ( x , y ) = ∑ a ∈ B A ( x , y ) w a + ∑ a ∈ C A ( x , y ) w a 2 is calculated and is tabulated in table 5. For example, B _A  (x₁, x₂) ={ a₁, a₂, a₃ } and C _A  (x₁, x₂) ={  } and hence WR _A  (x₁, x₂) =0.3 + 0.2 + 0.15 = 0.65. Now the entire dominance degree of each object using WR A ( x i ) = 1 | U | ∑ j = 1 | U | WR A ( x i , y j ) is found by definiton 5.3.4.

For example, WR A ( x 1 ) = 1 10 ∑ j = 1 10 WR A ( x 1 , x j ) = 0.469. So by step 5, x₈ is selected as the best object from the weighted trapezoidal intuitionistic fuzzy information system is seen from table 6.