A Novel Approach to Generalized Intuitionistic Fuzzy Soft Sets and Its Application in Decision Support System

: The basic idea underneath the generalized intuitionistic fuzzy soft set is very constructive in decision making, since it considers how to exploit an extra intuitionistic fuzzy input from the director to make up for any distortion in the information provided by the evaluation experts, which is redeﬁned and clariﬁed by F. Feng. In this paper, we introduced a method to solve decision making problems using an adjustable weighted soft discernibility matrix in a generalized intuitionistic fuzzy soft set. We deﬁne the threshold functions like mid-threshold, top-bottom-threshold, bottom-bottom-threshold, top-top-threshold, med-threshold function and their level soft sets of the generalized intuitionistic fuzzy soft set. After, we proposed two algorithms based on threshold functions, a weighted soft discernibility matrix and a generalized intuitionistic fuzzy soft set and also to show the supremacy of the given methods we illustrate a descriptive example using a weighted soft discernibility matrix in the generalized intuitionistic fuzzy soft set. Results indicate that the proposed method is more effective and generalized over all existing methods of the fuzzy soft set.


Introduction
The real world is full of imprecision, vagueness and uncertainty. In our daily life, we deal mostly with unclear concepts rather than exact ones. Dealing with imprecision is a big problem in many areas The purpose of this paper is to use weighted soft discernibility matrix (W SDM) for GIFSS using an adjustable perspective to solve decision making problems. In the literature, the GIFSS set is defined and applied for decision making problems using intuitionistic fuzzy weighted averaging operators. This technique can not only give the best alternative but also an order relation of all alternatives easily by scanning the W SDM at most one time. In this paper, Sections 1 and 2 consists of an introduction and preliminaries which include the basic definition related to fuzzy sets, intuitionistic fuzzy set and SDM. Section 3 is devoted to the threshold functions and their level soft sets. In Section 4, two algorithms are proposed basis on W SDM to solve decision making problem using GIFSS. Section 5 consists of a case study of scholarship for a doctoral degree. Finally, comparison and conclusion are given in Sections 6 and 7.

Preliminaries
In this section, we present the basic definitions of fuzzy set, intuitionistic fuzzy set, soft set, generalized intuitionistic fuzzy soft set, soft discernibility matrix and weighted soft discernibility matrix.
Throughout this paper, finite setŶ = { 1 , 2 , ..., n } andP = { 1 , 2 , ..., m } represents the set of n alternatives and m attributes (parameters). The abbreviations IFS, IFSS, GIFSS, SDP, SDM and W SDM represents the intuitionistic fuzzy set, intuitionistic fuzzy soft set, generalized intuitionistic fuzzy soft set, soft discernibility parameters, soft discernibility matrix and weighted soft discernibility matrix, respectively. Moreover, the abbreviation "w.r.t." is used for "with respect to." A fuzzy set is defined by Zadeh [1], which handles uncertainty based on the view of gradualness effectively. Like a membership degree on an element in a fuzzy set, human intuition suggests that there is a non-membership degree of an element in a set. In Reference [4], an IFS defined by Atanassov to sketch the imprecision of human beings when needing the judgments over the elements. Definition 2 ([4]). An IFSÂ over the universeŶ is defined aŝ where ξÂ :Ŷ → [0, 1] and ϑÂ :Ŷ → [0, 1] are the degree of positive membership and degree of negative membership, respectively. Furthermore, it is required that 0 ≤ ξÂ + ϑÂ ≤ 1.
A soft set is defined by Molodtsov [6], which provides an effective framework to dealings with imprecision with the parametric point of view, that is, each element is judged by some criteria of attributes. Definition 3 ([6]). LetŶ be a universal set,P a parameter space,Â ⊆P and P(Ŷ ) the power set ofŶ. A pair (F ,Â) is called a soft set overŶ, whereF is a set valued mapping given byF :Â → P(Ŷ ).
The idea of GIFSS is very encouraging in decision-making since it considers how to capitalize an additional intuitionistic fuzzy input from the director to minimize any possible perversion in the data provided by evaluating specialists. First Agarwal [22], defines the GIFSS, but it has some problems, therefore, in Reference [23], Feng redefines the GIFSS as follows.
In [40], Q. Feng defines the SDM for soft sets which provide not only the best alternative but also an order relation among all the alternatives.
The symbol i p (or j p ) represents the elements in N i (or N j ) have the value 1 at the attribute p , that is, The W SDM is defined as follows. Definition 7 ([40]). Let (F ,Â) be a soft set overŶ.F determined the partition U|I ND(F ,Â) = {N i : i ≤ |Ŷ |} ofŶ. The W SDM is defined as M = (M(N i , N j )) i,j≤|Ŷ | , where M(N i , N j ) is called the SDPs among N i and N j and defined as The symbol i * ω i p (or j * ω j p ) represents the elements in N i (or N j ) have the value 1 at the parameter p , that is, In [40], Q. Feng also give some properties of SDM and W SDM. Proposition 1 ([40]). Let (F ,Â) be a soft set overŶ, whereŶ = { 1 , 2 , ..., n } and ϕ(N i , N j ) = |M(N i , N j )|. Then the SDM has the following characteristics: and |Ê i | = |Ê j | then the elements of N i and N j have same rank (order); and there is an order relation between the elements of N i and N j .
L(Γ; tb) is a notation used to represents the corresponding level soft set ofΓ and called the top-bottom-level soft set ofΓ.
Definition 12. LetΓ = (F ,Â,ρ) be a GIFSS overŶ andÂ ⊆P be a set of attributes. Then top-top-level threshold function ttΓ : L(Γ; tt) is a notation used to represents the corresponding level soft set ofΓ and called the top-top-level soft set ofΓ.
L(Γ; bb) is a notation used to represents the corresponding level soft set ofΓ and is called the bottom-bottom-level soft set ofΓ.

Applications of the GIFSS Model Based on Weighted Soft Discernibility Matrix
The idea of GIFSS is very encouraging in decision-making since it considers how to capitalize an additional intuitionistic fuzzy input from the director to minimize any possible perversion in the data provided by evaluating specialists. Also, in our daily life decision making problems, different attributes are not of equal importance. Some are more important than others; therefore, the decision maker assigns different values (weights) to different attributes and imposed different thresholds functions when we need to put restriction on membership function and non-membership function. First, Skowron and Rauszer [43] initiated the concept of discernibility matrix and extensively used in rough sets to solve attribute reduction and the influence of it are significant and easy to understand. In this paper, we use an adjustable perspective to GIFSS and get level soft sets. Then each GIFSS can be seen as the level soft set and composed a crisp soft set, therefore, for solving decision making problems we apply SDM. The soft equivalence relation in soft sets is a bridge among soft sets and rough sets which is introduced by Ali [44].

Definition 15.
The accuracy weighted choice value of an alternative i ∈Ŷ is f i given by f i = ∑ e ij , where e ij = ω ij × ij , whereω ij are the accuracy weights calculated from PIFS using accuracy function defined asω ij = ξ j + ϑ j .

Definition 16.
The expectation score weighted choice value of an alternative i ∈Ŷ is f i given by f i = ∑ e ij , where e ij =ω ij × ij , whereω ij are the expectation score weights calculated from PIFS using expectation score function defined asω ij = (ξ j − ϑ j + 1)/2.
2m + 1, m ∈ N + } from SDM. 7: For every element of M 1 , we compare the weighted choice values of |Ê i | and |Ê j |, if |Ê i | = | p | ×ω p = |Ê j | = | q | ×ω q , where p ∈Ê i and q ∈Ê j , then the elements i ∈ N i and j ∈ N j have same rank, otherwise there is an order relation between elements of N i and N j . 8: If there is a global relation between elements ofŶ in step 7, then choose the superior one as the optimal, otherwise move to next step. 9: Use the elements of M 2 to find the order relation of the elements ofŶ together with the step 8. 10: From the order relation of elements from step 8 and step 9, choose the optimal alternative which is superior.
From Reference [44], we have no concern with the values of decision parameter D = ∑ ij but actually we are interested on investigation its classification ability. Only from the parameters value in the set of SDPs, classification ability of decision parameter is determined. Thus, we only need to compare the values of the decision parameter restricted within every set of SDPs.
Example 3. Given a GIFSS (F ,Â,ρ) as in Example 1, find the order relation among all the alternatives of GIFSS using Algoritm 1. We use the med-level threshold function and compute the level soft set L(Γ; med) as in Table 7. From Table 7, the parameters 2 , 4 , 5 have value 1, therefore, the weighted value of 2 , 4 and 5 are 1 (say) and the parameters 1 , 3 , 6 have value 0, therefore, the weight value of 1 , 3 and 6 are 0.7. Table 7, we can obtained the partition ofŶ is Table 9. Table 9. The W SDM of Example 3.

From
, so the alternatives inÊ 1 andÊ 3 have same rank, in other words 1 and 3 are in same decision class. Similarly, in M(N 1 , N 5 ), |Ê 1 | > |Ê 5 |, the alternative in N 1 is superior to the alternative in N 5 , that is, 1 is superior to 5 . From M(N 1 , N 3 ), we have 1 , 3 are superior to 5 . From M(N 1 , N 2 ), |Ê 1 | > |Ê 2 |, the alternative in N 1 is superior to the alternative in N 2 , that is, 1 is superior to 2 . From M (N 1 , N 3 ), we have 1 and 3 are superior to 2 . From M (N 1 , N 4 ), |Ê 4 | > |Ê 1 |, the alternative in N 4 is superior to the alternative in N 1 , that is, 4 is superior to 1 , hence 4 is superior to 3 . From M(N 2 , N 5 ), |Ê 2 | > |Ê 5 |, the alternative in N 2 is superior to the alternative in N 5 , that is, 2 is superior to 5 .
Combining the above results, we have 4 { 1 , 3 } 2 5 , so an order relation among all the alternatives is obtained. And the best alternative is 4 .
We have seen that when we solve decision making problem using SDM, some attributes will be erased unintentionally. Thus, by constructing SDM for a soft set some attributes which have no impact on the final conclusion will be erased.   Table 1, the (k med , l med )-level soft set have calculated in Table 7, and the partition M 1 and M 2 are calculated in Example 3. The accuracy weights of each parameter is given as follows: ω 1 = 0.6,ω 2 = 1,ω 3 = 0.8,ω 4 = 0.8,ω 5 = 0.8 andω 6 = 0.8. Then we have a accuracy W SDM with the tabular representation shown in Table 10.   Table 10, we know that in M (N 1 , N 3 ), |Ê 1 | = | 1 | × 0.6 + | 5 | × 0.8 = 1.4 and |Ê 3 | = | 2 | × 1 + | 5 | × 0.8 = 1.8, thus the alternative in N 3 is superior to the alternative in N 1 , that is, 3 is superior to 1 . Similarly, we can obtained the order relation among the all alternatives is 4 3 1 2 5 . Hence the optimal alternative is 4 .

Case Study for Selecting Candidates for Ph.D. Scholarships
A mathematics department of university Y has three scholarships for a doctoral degree. Many students apply for a scholarship but due to initial conditions on CGPA (cumulative grade points average), only seven students are short listed for further evaluation. LetŶ = { 1 , 2 , ..., 7 } represent the alternatives (students) andP = { 1 , 2 , ..., 6 } are the attributes (criteria), where each i stands for "CGPA", "no. of research papers", "research quality", "research proposal", "personal statement" and "interview". For selection, the vice chancellor of the university set up a committee of experts to make an evaluation on the basis of given criteria (attributes). The committee evaluates students and given their evaluation in the form of BIFSS and vice chancellor scrutinizes the general quality of evaluation made by an expert group and gives his view in the form of PIFS, which completes the construction of GIFSS,Γ = (F ,Â,ρ). The tabular representation of GIFSS is given in Table 11.  We use med-level threshold function for BIFSS and the med-level soft set with its tabular representation (Table 12)  From Table 12, we can obtained the partition ofŶ is We find the expectation score weights from PIFS, which are ω 1 = 0.7,ω 2 = 0.7,ω 3 = 0.45,ω 4 = 0.55,ω 5 = 0.5 andω 6 = 0.8. From partition, we constructed W SDM which is given in Table 13.

Comparison Analysis
In this section, we compare our proposed algorithm with some related methods to indicate its advantages.
At first, we compare our method with the method proposed in Reference [22], where Agarwal define GIFSS and used it for different problems like medical diagnose and decision making but in Reference [23], Feng pointed out some problems in the proposed method and provided a counter example that shows that in calculating the NAE in the fifth step, the relation is not compatible with the ⊗ operation. Also in Reference [45], Khalil gives a counter example to some propositions of GIFSS. If we compare our method with the method proposed in Reference [23], we have seen that the we get same result as in Reference [23]. But it is not necessary that we get the same results because our method is different and helps in situations when all attributes are not of equal importance and we need to put some restrictions/boundaries on membership and non-membership functions. In our proposed method, we can find not only the best choice alternative, but also an order relation of all alternatives easily by scanning the W SDM at most one time. If we compare our method with the method proposed in Reference [40], we have seen that the previous method deals with the intuitionistic fuzzy soft sets and in his method he did not give any information on how to calculate the weights of parameters but in our proposed method we are working in GIFSS and give different criteria to find the weights of parameters by using parametric intuitionistic fuzzy soft set. We compare our method with the methods proposed in References [23,40,[46][47][48][49] and results summarized in Table 14 taken from [23], we have seen that the order relation among all alternatives and optimal alternative is the same as we have. Only in Reference [46], we have seen that the 3 is superior to 1 , but in remaining all methods 1 is superior to 3 . Actually Lin use the measure S Lin (ξÂ, ϑÂ) = 2ξÂ + ϑÂ − 1, which is increasing w.r.t the ϑÂ. This is counter intuitive and might cause difficulties in some practical applications. Our main concern to compare with the method proposed in Reference [23] because both papers dealing with GIFSS. But when we solved the case study Section 5, the order of the alternatives not remains the same, that is, The reason behind the changing of order is that, in our proposed method we put some threshold function or values initially that minimize the effect of alternatives that are not fulfil the initial criteria.
The proposed method depends on the initial threshold function, like if we use the top-bottom threshold function then we get the order of alternatives It means it is important to select the threshold function or threshold value according to the situation required in decision making. Actually this is an advantage in a sense that our proposed method response when we change the threshold function or values because the threshold values suggested according to the decision problem situation and secondly we strengthen the attributes by assigning the weights to the attributes.
The limitations of our approach are that in Algorithm 1, the attributes categorized in two groups and the weights are more effective only when the difference between positive membership and negative membership function is maximum. In Algorithm 2, we assign different weights to the attributes but to make weights more effective, the difference between positive membership and negative membership function is high.

Conclusions
Keeping in mind the idea of decision making, in this paper, we have strengthened the director/administrator/decision maker point of view because first, he adjusts the initial conditions according to a situation like in the case study in Section 5, he makes an initial condition that the CGPA of a candidate should be greater than a particular value. Secondly, he differentiates the different attributes/criteria by assigning different weights like in the case study in Section 5, he gave the interview more weightage than the personal statement. Thirdly, we provide the criteria for obtaining weights of different attributes and the weights obtained from PIFS provided by director/administrator/decision maker.
The idea of GIFSS is very encouraging in decision-making since it considers how to capitalize an additional intuitionistic fuzzy input from the director to minimize any possible perversion in the data provided by evaluating specialists. First, Skowron and Rauszer [43] initiated the concept of discernibility matrix and extensively used in rough sets to solve attribute reduction, and the influence of it are significant and easy to understand. In this paper, we use an adjustable perspective to GIFSS and get level soft sets. Then each GIFSS can be seen as a level soft set and composed a crisp soft set, therefore, for solving decision making problems we apply W SDM. In literature, GIFSS is defined and applied for decision making problems using intuitionistic fuzzy weighted averaging operators. But in our daily life decision making problems, different attributes are not of equal importance. Some are more important than others; therefore, the decision maker assigns different values (weights) to different attributes and imposed different threshold functions when we need to put a restriction on membership functions and non-membership functions. Our proposed technique can not only give the best alternative but also an order relation of all alternatives easily, by scanning the W SDM at one time. We define the threshold functions like mid-threshold, top-bottom-threshold, bottom-bottom-threshold, top-top-threshold, med-threshold function and their level soft sets. After, we proposed two algorithms based on threshold functions, W SDM, and GIFSSs. In Algorithm 1, the attributes are categorized in two groups while in Algorithm 2, each attribute is weighted differently. To show the supremacy of the given methods we illustrate a descriptive example using Algorithm 2. Results indicate that the proposed method is more effective and generalized over all the existing methods of fuzzy soft sets.
In future directions, we will introduce the different methods to get the weights of attributes by using similarity or entropy measures. We apply this method to the best concept selection and multiattribute classification or sorting problems.