A Novel Arithmetic Technique for Generalized Interval-Valued Triangular Intuitionistic Fuzzy Numbers and Its Application in Decision Making

Uncertainty is an integral part of decision-making process which arises due to the lack of knowledge, data or information. Initially fuzzy set theory (FST) was used to handle this type of uncertainty. Later, intuitionistic fuzzy set (IFS) was developed to encounter uncertainty in a more specific manner. However, it is observed that due to the existence of different types of uncertainties, the membership function (MF) of IFS itself is uncertain and consequently, the concept of interval-valued intuitionistic fuzzy sets (IVIFS) came into the picture. But IVIFS is also not capable of handling uncertainty. To overcome the limitations of the existing IVIFS, generalized interval valued intuitionistic fuzzy sets (GIVIFS) have been defined and it has been observed that it has utmost applicability in real world situations as the parameter height characterises the degree of buoyancy of judgment of decision maker in a very specific compartment.

For the arithmetic operations on GIVTIFNs, the largest membership function is truncated at the minimum height first and the nonmembership function is truncated at the maximum height.Accordingly, arithmetic operations on GIVTIFNs are defined.For this purpose, Decomposition theorems for GIVTrIFNs are discussed first.

Result:
The outputs are obtained as generalized interval-valued trapezoidal intuitionistic fuzzy numbers (GIVTrIFNs).The interesting part of the proposed approach is that it produces GIVTrIFNs.To check the validity and novelty of the approach, a multi criteria decision making was performed which obtained desirable results.

Conclusion:
The arithmetic GIVTIFNs conventional approach produces invariant output in the form of GIVTIFNs for GIVTIFNs of different height.But for the same input GIVTIFNs, the present approach provided different GIVTrIFNs.It was observed that the proposed approach is efficient, simple, logical, technically sound and general enough for implementation.Researchers may apply this approach in any field where GIVTIFNs are involved.

INTRODUCTION
In the presence of different constraints in real life situation and due to highly complex environment, decision makers may provide their opinion under uncertain and imprecise nature.Due to the involvement of uncertainty crisp data are not always adequate to model in many real-life situations whereas FST introduced by L.A. Zadeh [1], is more suitable On the other hand, an important generalization of fuzzy set theory is the theory of Intuitionistic Fuzzy Set (IFS), introduced by Atanassov [3] describing a membership degree and a non-membership degree separately in such a way that sum of the two degrees must not exceed to one.It is observed that fuzzy sets are IFSs but the converse is not necessarily correct.Further, Atanassov and Gargov [4] developed the notion of Interval-Valued Intuitionistic Fuzzy Sets (IVIFSs) in relation with interval-valued fuzzy sets and IFS.The IVIFSs are characterised by a membership function and a nonmembership function that take interval values rather than the exact number.In human cognitive and decision making processes, it is not absolutely justifiable or technically sound to represent the membership and nonmembership in terms of a single numeric value.Thus, IVIFSs have got more attention due to its ability to handle imprecise and unorganised information in terms of intervals instead of taking a single numeric value [5].IFS and IVIFS have been successfully applied [6 -12] in different areas like decision making, pattern recognition, medical diagnosis.Yager, Yuan and Li [13 -15] studied the cut set characteristic of IVIFS.Following the work of Szmidt and Kacprzyk [16], Xu and Qiansheng [17 -19] applied IVIFS to pattern recognition.Further Yingjie and Qiansheng [20,21] studied the interval-valued intuitionistic fuzzy reasoning.Xu and Li [22 -24], successfully carried IVIFS to decision making problems.The Generalized Intuitionistic Fuzzy Sets (GIFSs) were proposed by Mondal and Samanta [25] under the constraint that the minimum of the two degrees does not exceed half.Shu et al. [26], first introduced the concept of Generalized Intuitionistic Fuzzy Numbers (GIFNs) and defined arithmetic operations between them.But later, it was found that there are some errors and misprints in the definition of the four arithmetic operations and those errors were conducted by Li [27].Zhenhua et al. [28] introduced the construction method of the Generalized Interval-Valued Intuitionistic Fuzzy Sets with Parameters (GIVIFSP), and defined complement operation, intersection operation and union operation on GIVIFS.Furthermore, they proved that like IFS and IVIFS, GIVIFS is a closed algebraic system for all these operations.Bhowmik et al. [29], Zhi et al. [30], and Adak et al. [31] also studied different concepts of GIVIFSs.Baloui and Nadarajah [32] extended the IFSs to the concept of GIFSs and introduced some operators on GIFSs.Based on GIFSs, Shabani and Baloui [33] introduced GIFNs.Baloui [34] considered a new GIVIFSs and introduced some operators on GIVIFSs.He studied different basic operations like union, intersection, subset complement etc. and also transformed the operations on IVIFSs for the GIVIFSs.
Considering two Generalized Interval-Valued Triangular Intuitionistic Fuzzy Numbers (GIVTIFNs) and , the arithmetic operations are as follows: But this approach has some drawbacks and gives illogical results as during the operation, first it reduces the height of their respective (LMF and UMF) higher MFs to the height of the lower ones (i.e., make it as GIVTFN by reducing the height based on Cheng's [26] function principle) and similarly for the respective (LNMF and UNMF) NMFs, it increases the minimum height to maximum one to make it a generalized interval-valued triangular non-membership function.Therefore, this approach produces GIVTIFN with MF at the minimum height of the given respective (LMF and UMF) MFs of the GIVTIFNs and height of the NMF is maximum of the given respective (LNMF and UNMF) NMFs of GIVTIFNs to perform the arithmetic operations.The major drawback of this approach is that when performing arithmetic operations between a fixed GIVTIFN with different GIVTIFNs with the same support but different heights, the height of MFs of the fixed GIVTIFN is lesser and the height of NMFs is higher than other GIVTIFNs; then it is seen that each time, the resultant GIVTIFN remains invariant, which is illogical.For example consider be a fixed GIVTIFN and & be two different GIVTIFNs with different heights.Then performing A*B i (i=1,2), where * is the basic arithmetic operation using the existing approach to provide the same GIVTIFN.That is, the conventional approach gives , which is clearly illogical as and are two different GIVTIFNs and the sum of these two GIVTIFNs with the fixed GIVTIFN A is identical.The following Figs.( )) ))

Motivation
GIVTIFNs play an important role while dealing with uncertainty modeling problems in real life situations as they have the capability to represent imprecision, uncertainty in a proper manner, and are desirable to address such problems.
GIVTIFNs are used as mathematical assessment for different linguistic variables, ratings, and weights in various problems like decision making, medical diagnosis, pattern recognition etc. Proper arithmetic operations on GIVTIFNs are very important for the correct output in different problems.The existing approach produces some illogical results while performing arithmetic operation on GIVTIFNs with different heights.To overcome the shortcomings of the existing approach and for proper evaluation, it is always useful to define novel techniques for arithmetic on GIVTIFNs.

PRELIMINARIES
In this section, some basic definitions of FS, IFS and IVIFS have been discussed.

Definition (Fuzzy Set)
Let X be a universe of discourse; then the fuzzy subset A of X is defined by its membership function which assigns a real number µ A (x) in the interval [0, 1], to each element , where the value of µ A (x) at x shows the grade of membership of x in A.

Definition (Support)
The support of a fuzzy set A defined on X is a crisp set defined as:

Definition (Height)
[1] The height of a fuzzy set A, denoted by h(A), is the largest membership grade obtained by any element in the set and it is denoted as .

Definition (Generalized Fuzzy Numbers (GFN))
The Compared to normal fuzzy number, the GFN can deal with uncertain information in a more flexible manner because of the parameter w that represents the degree of confidence of opinions of decision maker's.
x a

Definition (Intuitionistic Fuzzy Set (IFS))
An Intuitionistic fuzzy set A on a universe of discourse X is of the form: Where is called the "degree of membership of x in A", is called the "degree of nonmembership of x in A", and where µ A (x) and v A (x) satisfy the following condition: ) is called hesitancy of x which is a reflection of lack of commitment or uncertainty associated with the membership or non-membership or both in A.

Definition (Generalized trapezoidal intuitionistic fuzzy number (GTrIFN))
The membership function of trapezoidal GTrIFN , where is defined as: and the non-membership function of the GTrIFN A is defined as: For example, consider the GTrIFS .The MF and NMF of A are shown in Fig. (5).

Definition (Positive and Negative GIFS)
A GIFS is said to be positive GIFS a 2 > 0 if and negative GIFS if a 2 < 0.

Let
be a GIFS then height for MF is defined as: and height for NMF is defined as .

Definition (Level Set of GFIS)
Let be a GIFS then level set for MF is defined as and level set for NMF is defined as

Definition (The α -cut of MF and NMF of the GIFS)
The α -cut of the MF of the GIFS is defined as: The α -cut of the NMF of the GIFS is defined as:

Definition ( The α -cut GIFS)
Let be a GIFS and --cut of MF (µ A ) be and NMF (v A ) be respectively.Then --cut of GIFS A can be evaluated by the following formula: where , A + & A -mF and NMF such that is the height of MF, is the height of NMF and Ø is an empty set.

Definition (Special IFS)
Let be a GIFS defined on the universe of discourse X, then a special IFS can be defined as αA = α.α A.

Definition (Generalized Interval-Valued Triangular Intuitionistic Fuzzy Number (GIVTIFN))
The membership functions (lower MF (LMF) and upper MF (UMF) of GIVTIFN , where is defined as: and the non-membership function of the GIVTIFN A is defined as: For example, consider the GIVTIFN .The MFs and NMFs of A are shown in Fig. (6).

Definition (Generalized Interval-Valued Trapezoidal Intuitionistic Fuzzy Number (GIVTrIFN))
The membership functions (LMF and UMF) of GIVTrIFN , where are defined as: and the non-membership functions (lower NMF (LNMF) and upper NMF (UNMF)) of the GIVTrIFN A are defined as:

Definition (Support of a GIVTIFS)
Let be a, GIVTIFS then support of A is defined as: .

Definition (Height of a GIVTIFS)
Let be a GIVTIFS, then height for , 2 2 1, o th e rw ise ( )  , where respectively.Then α,β-cut of GIVTIFS A can be evaluated by the following formula: . 1 1 . 1 1 , where MFs and NMFs such that are the height of MFs, are the height of NMFs and is an empty set.

DECOMPOSITION THEOREM FOR GIVTrIFN
In this section, decomposition theorems for GIVTrIFN have been discussed.

Theorem (First Decomposition Theorem)
Let X be a universe of discourse.For any GIVTrIFN in X, where are standard fuzzy union and intersection, respectively.

Proof
For MF, let for each which indicates the degree of belonging in A. Then, , ( , ), , & , .
Hence from (3.1), we have Similarly, For NMF, let for each which indicates the degree of non-belonging in A.

Theorem (Second Decomposition Theorem)
Let X be a universe of discourse.
For any GIVTrIFN where are standard fuzzy union and intersection, respectively.

Proof
For MF, let for each which indicates the degree of belonging in A. Then, Hence from (3.3), we have For NMF, let for each which indicates the degree of non-belonging in A. (3.4) Hence from (3.4), we have

Theorem (Third Decomposition Theorem)
Let X be a universe of discourse.

For any GIVTrIFN
where are standard fuzzy union and intersection, respectively, and λ(A) is the level set of A.

PROPOSED ARITHMETIC TECHNIQUE FOR GIVTIFNS
GIVTIFN is the extended version of GTIFN.Arithmetic on GIVTIFNs is a crucial issue.Let us consider that and and 

Generalized Interval-Valued Triangular and Its Application
The Open Cybernetics & Systemics Journal, 2018, Volume 12 87 are two GIVTIFNs with different heights.Here a novel approach will be discoursed to perform the arithmetic operation between GIVTIFNs A and B. In this approach, the MFs are truncated at the smallest height of their respective (LMF and UMF) MFs.Similarly, the NMFs are truncated at the maximum heights of their respective (LNMF and UNMF) NMFs.
The interesting part of this approach is that it produces GIVTrIFNs.

Proof
To determine the addition of GIVTIFNs A and B, we first add the α, βcuts of GITVIFNs A and B using interval arithmetic.
then, the , 1 1 1 1 On the other hand, the , and Hence the LMF of is 3), we get the domain of x, In a similar manner we also have the UMF

of A + B as
To obtain NMFs, we proceed as: Let's equate each component with x, we have Now, expressing in terms of x, we obtain (4.4) 1 1      That is, ) , 11 (1 , ( ) (

Theorem (Subtraction of Two GIVTIFNs with Different Heights Produces a GIVTrIFNs) Proof
To perform subtraction operation of GIVTIFNs A and B, we subtract the α, β-cuts of A and B using interval arithmetic.
For MF,   In a similar way,we can have the UMF as

Theorem (Multiplication of Two GIVTIFNs with Different Heights Produces a GIVTrIFN) Proof
To calculate multiplication of GIVTIFNs A and B, we first multiply the α, β-cuts of generalized fuzzy numbers A and B using interval arithmetic.
To find the LMF , we equate both the first and second component of (4.11) to x which gives which is a quadratic equation and by solving it we obtain  (1  Also with a similar manner, we have the UMF as For NMF a w a a w a a w a a w w w a a a a a a a a a a a a w w w It can be expressed in terms of x by solving this quadratic equation, .
Now, equating both the terms with x, we obtain ) Thus, the LNMF is In a similar fashion, we can have UNMF as follows: Thus, we have where ,  Hence proved the theorem.

Theorem (Division of Two GIVTIFNs with Different Heights Produces a GIVTrIFNs) Proof
To divide two GIVTIFNs A and B, we first divide the α, β-cuts of A and B using interval arithmetic.
For MF, where To find the LMF we equate both the first and second component of In a similar way, we can have the UMF as: Where .
For NMF Equating each component with x, we have Now, expressing β in terms of x, we obtain (4.17) Again, expressing in terms of α we have 1 /, l l l l l u u u u u l l l l l u u u u u f f f f w f f f f w g g g g g g g g AB , , ,  Hence proved the theorem .

Graphical Representation of the Proposed Approach
Let us consider the same example discussed in section-1 where, is a fixed GIVTIFN while and be two different GIVTIFNs with different heights.In section 1, we have seen that the existing approach [26] produces identical GIVTIFN while carrying out the addition operation between A and B 1 & A and B 2 which are depicted in Figs.(8 and 9) respectively.On the other hand, while performing the addition operation for the same pairs of GIVTIFNs by using the proposed approach, it produces two distinct GIVTrIFNs.Thus, the GIVTrIFNs are given as:     Similarly, different outputs will be obtained other arithmetic operations which is logical and correct while the existing approach leads to illogical output.

NUMERICAL EXAMPLES
whose MFs and NMFs are

RANKING OF GIVTRIFNS BASED ON VALUE INDEX
Deng Feng Li [35] introduced the concept of value and ambiguity of GTIFN and the same concept has been put forward for GTrIFN by De and Das [36].In this section, we will extend the concept of the value of GTrIFN to GIVTrIFNs.

Definition: Let be a
GIVTrIFN and be α, β-cuts of the MFs and NMFs of A, respectively.Then the value of MFs and NMFs of A is defined as: Where f(α) is a non-negative and non-decreasing function on with and on .The function f(β) is a non-negative and non-increasing function on also f(β) is a non-negative and non-increasing function on Like [35] and [36], we also choose, ,     Zeng et al. [37], devised value-index to rank trapezoidal IFS and we extend it for GIVTrIFNs.
That is, for a GIVTrIFS the value-index of A is defined as: , where [0,1] is a weight which represents the decision maker's preference information.

MULTI-CRITERIA GROUP DECISION-MAKING USING ARITHMETIC OPERATIONS ON GIVTIFNs
In general, multi-criteria group decision-making problems include uncertain imprecise data and information.For validity and justification of the approach and to show the application in real-world problem of the proposed approach, a multi-criteria group decision-making model has been carried out to rank the best alternative among the available alternatives based on GIVTIFNs.

Methodology
Let us suppose that a committee of K expert decision makers D 1 , D 2 ...D K will choose the best alternative among n alternatives A 1 , A 2 ,..., A n based on m criteria where C 1 ,C 2 ,...,C m are for each alternative respectively.
The procedure for the decision process is given below: Step-I: Decision makers choose linguistic weighting variables with respect to the importance weight of criteria and the linguistic ratings variable to evaluate the ratings of alternatives with respect to each criterion which are expressed in terms of positive GIVTIFNs.
Step-II: Decision makers evaluate the importance weight of each criterion using linguistic weighting variables.
Step-III: The weights of criteria are aggregated using.
to get the aggregated fuzzy weight of the criterion C j .
The new weight vector can be written as: , , , ;  Step-IV: Decision makers give their opinion to get the aggregated fuzzy ratings of alternative under criterion C j .That is, where each is GIVTIFNs.
Step-V: If all the weights and ratings are in the interval [0, 1] (i.e., W and R are normalized) the next step is followed and if not, they can be normalized by: Where is the ceiling function, Step-VI: Construct the weighted normalized fuzzy decision matrix Where using our proposed arithmetic operations which are normalized positive GIVTIFNs.
Step-VII: Decision makers evaluate using our proposed arithmetic operations.
Step-VIII: Based on maximum value-index of , decision-makers will choose the suitable alternative A i .

Hypothetical Case Study
Let us suppose a committee of three expert decision makers, D1, D2 and D3 which has been formed to conduct the interview for the post of the professor to select the most suitable candidate among the three eligible candidates, namely A1, A2 and A3.Five benefit criteria are considered:

Computational Procedure is Discussed in Detail Below:
Step-I: Decision makers choose the linguistic weighting variable (Table 1) for the importance weight of criteria and the linguistic ratings variable Table 2 to evaluate the ratings of alternatives with respect to each criterion.Step-II: To assess the importance of the criteria (Table 3) linguistic weighting variables are used Table 1  Step-III: The weights of criteria are aggregated using equation (1) to get the aggregated fuzzy weight of the criterion C j and decision makers give their opinion (Table 4) to get the aggregated fuzzy ratings of alternative A i under criterion C j .0, 0.0, 0.08;0.60, 0.0, 0.1;0.7 0, 0.0, 0. [ ][ ], 0, 0, 0.08;0.80.0, 0.0, 0.1;0.90, 0, 0.12; 0.1 0.0, 0.0, 0.15;0.
Since all the weights and ratings are in the interval [0, 1], so the matrix R is the normalized fuzzy decision matrix.
The weighted normalized fuzzy decision matrix is now constructed by using equation (2). 1.
To evaluate using our proposed arithmetic operations, we have
A in X and any real number α [0, 1].Then, (a) (α-cut) the α -cut fuzzy set A, denoted by α A is the crisp set: (b) (Strong a-cut) the strong a -cut, denoted by α+ A is the crisp set: membership function of GFN A= [a,b,c,d;w] where a ≤ b ≤ c ≤ d, 0 < w ≤ 1 is defined as: If w = 1, then GFN A is a normal trapezoidal fuzzy number A = [a, b, c, d].If a = b and c = d, then A is a crisp interval.If b = c then A is a generalized triangular fuzzy number.If a = b = c = d and w = 1 then A is a real number.

and 1 
 in (4.5), we have 92 The Open Cybernetics & Systemics Journal, 2018 is clearly a GIVTrIFN with height of the MFs are and NMFs are .

( 4 8 )
To find the membership function we equate both the first and second component of (4.6) to x which gives Now, expressing in terms of xSetting α ≥ 0 & α ≤ w L in (4.7) and α ≤ w L & α ≥ 0 in (4.8) we get the domain of xThe required LMF is where and .
equate each component with x, we have Now, expressing β in terms of x, we obtain -B is also a GIVTrIFN, where height of the MFs are and NMFs are .
Now, setting α ≥ 0 & α ≤ w L and α ≤ w L & α ≥ 0 in (4.12) and (4.13), we get the LMF of the resulting GIVTrIFN after multiplication of A and B together with the domain of x AB is clearly a type of GIVTrIFN, where height of the MFs are and NMFs are .
(4.16) to x, which gives Now, expressing α in terms of x and setting α ≥ 0 & α ≤ w and α ≤ w & α ≥ 0 in the above expressions, we get α together with the domain of x /B is also a GIVTrIFN, where height of the MFs are and NMFs are .