Possibility Intuitionistic Fuzzy Soft Set

Possibility intuitionistic fuzzy soft set and its operations are introduced, and a few of their properties are studied. An application of possibility intuitionistic fuzzy soft sets in decision making is investigated. A similarity measure of two possibility intuitionistic fuzzy soft sets has been discussed. An application of this similarity measure in medical diagnosis has been shown.


Introduction
In most real-life problems in social sciences, engineering, medical sciences, and economics the data involved are imprecise in nature. The solutions of such problems involve the use of mathematical principles based on uncertainty and imprecision. A number of theories have been proposed for dealing with uncertainties in an efficient way. Fuzzy set was introduced by Zadeh in 1 as a mathematical way to represent and deal with vagueness in everyday life. Then Atanassov 2 defined the concept of intuitionistic fuzzy set which is more general than fuzzy set. Molodtsov 3 initiated the theory of soft sets as a new mathematical tool for dealing with uncertainties which traditional mathematical tools cannot handle. Maji et al. 4,5 have further studied the theory of soft sets and used this theory to solve some decision-making problems. Alkhazaleh et al. 6 introduced soft multiset as a generalisation of Molodtsov's soft set. Alkhazaleh and Salleh 7 defined the concept of soft expert set and they gave an application of this concept to decision making. Also Maji et al. 8 introduced the concept of fuzzy soft set and studied its properties and also Roy and Maji used this theory to solve some decision-making problems 9 . Majumdar and Samanta 10 defined and studied the generalised fuzzy soft sets, where the degree is attached with the parameterizat ion of fuzzy sets while defining a fuzzy soft set. In 2010 Baesho 11 introduced the concept of generalised intuitionistic fuzzy soft sets, where the degree is attached with the 2 Advances in Decision Sciences parameterization of fuzzy sets while defining an intuitionistic fuzzy soft set see also Baesho et al. 12 .D i n d ae ta l . 13 introduced the same concept independently, and Agarwal et al. 14 introduced the same concept in 2011. Alkhazaleh et al. defined in 15 the concept of fuzzy parameterized interval-valued fuzzy soft set and gave its applications in decision making and medical diagnosis. Alkhazaleh et al. 16 defined the concept of possibility fuzzy soft set and gave its applications in decision making and medical diagnosis. Maji 17 defined the concept of intuitionistic fuzzy soft set, and Liang and Shi in 18 defined some new operations on intuitionistic fuzzy soft sets and studied some results relating to the properties of these operations. Salleh 19 gave a brief survey from soft set to intuitionistic fuzzy soft set. In this paper, we generalise the concept of possibility fuzzy soft set to the possibility intuitionistic fuzzy soft set. In our generalisation, a possibility of each element in the universe is attached with the parameterization of fuzzy sets while defining an intuitionistic fuzzy soft set. We also give some applications of the possibility intuitionistic fuzzy soft set in decisionmaking problem and medical diagnosis.

Preliminaries
In this section we recall some definitions and properties regarding intuitionistic fuzzy soft set and a possibility fuzzy soft set required in this paper. Let U be a universe set and E be a set of parameters. Let P U denote the power set of U and A ⊆ E. Definition 2.1 see 12 . Consider U and E as a universe set and a set of parameters, respectively. Let P U denote the set of all intuitionistic fuzzy sets of U.L e t A ⊆ E. Ap a i r F, E is an intuitionistic fuzzy soft set over U, where F is mapping given by F : A → P U . Definition 2.2 see 2 . An intuitionistic fuzzy set IFS A in a nonempty set U a universe of discourse is an object having the form A { x, µ A x ,v A x : x ∈ U}, where the functions µ A x : U → 0, 1 , v A x : U → 0, 1 , denotes the degree of membership and degree of nonmembership of each element x ∈ U to the set A, respectively, and 0 ≤ µ A x v A x ≤ 1 for all x ∈ U.
The following definitions and propositions are due to Alkhazaleh et al. 16 .
.,x n } be the universal set of elements and E {e 1 ,e 2 ,...,e m } be the universal set of parameters. The pair U, E will be called a soft universe. Let F : E → I U and µ be a fuzzy subset of E,t h a ti sµ : E → I U , where I U is the collection of all fuzzy subsets of U.LetF µ : E → I U × I U be a function defined as follows: Proposition 2.11. Let F µ ,G δ , and H ν be any three PFSSs over U, E , then the following results hold:

Possibility Intuitionistic Fuzzy Soft Sets
In this section we generalise the concept of possibility fuzzy soft sets as introduced by Alkhazaleh et al. 16 . In our generalisation of a possibility fuzzy soft set, a possibility of each element in the universe is attached with the parameterization of fuzzy sets while defining an intuitionistic fuzzy soft set.

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.,x n } be the universal set of elements and E {e 1 ,e 2 ,...,e m } be the universal set of parameters. The pair U, E will be called a soft universe. Let F : E → I × I U × I U where I × I U is the collection of all intuitionistic fuzzy subsets of U and I U is the collection of all fuzzy subsets of U.Letp be a fuzzy subset of E,thatis,p : E → I U and let F p : E → I × I U × I U be a function defined as follows: Then F p is called a possibility intuitionistic fuzzy soft set P I F S Si ns h o r t over the soft universe U, E . For each parameter e i , F p e i F e i x ,p e i x indicates not only the degree of belongingness of the elements of U in F e i , but also the degree of possibility of belongingness of the elements of U in F e i , which is represented by p e i . So we can write F p e i as follows: ,p e i x 2 ,..., x n F e i x n ,p e i x n .

3.2
Sometime we write F p as F p ,E .IfA ⊆ E we can also have a PIFSS F p ,A . 3 } be a set of parameters and let p : E → I U . We define a function F p : E → I × I U × I U as follows:

3.3
Then F p is a PIFSS over U, E . In matrix notation we write Definition 3.3. Let F p and G q be two PIFSSs over U, E . F p is said to be a possibility intuitionistic fuzzy soft subset (PIFS subset of G q and one writes F p ⊆ G q if i p e is a fuzzy subset of q e , for all e ∈ E, ii F e is an intuitionistic fuzzy subset of G e , for all e ∈ E. 3 } be a set of parameters and let p : E → I U . We define a function F p : E → I × I U × I U as follows:

3.5
Let G q : E → I × I U × I U be another PIFSS over U, E defined as follows:

3.6
It is clear that F p is a PIFS subset of G q .
Definition 3.5. Let F p and G q be two PIFSSs over U, E . Then F p and G q are said to be equal and one writes F p G q if F p is a PIFS subset of G q and G q is a PIFS subset of F p . In other words, F p G q if the following conditions are satisfied: i p e is equal to q e , for all e ∈ E, ii F e is equal to G e , for all e ∈ E.
Definition 3.6. A PIFSS is said to be a possibility null intuitionistic fuzzy soft set, denoted by φ 0 , where F e 0,v e ,andp e 0, for all e ∈ E.
Definition 3.7. A PIFSS is said to be a possibility absolute intuitionistic fuzzy soft set, denoted by where F e 1, 0 and P e 1, for all e ∈ E. 3 } be a set of parameters and let p : E → I U . We define a function F p : E → I × I U × I U as follows:

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Then F p is a possibility null intuitionistic fuzzy soft set. Let p : E → I U , and we define the function F p : E → I × I U × I U which is a PIFSS over U, E as follows:

3.10
Then F p is a possibility absolute intuitionistic fuzzy soft set.

Union and Intersection of PIFSS
In this section we introduce the definitions of union and intersection of PIFSS, derive some properties and give some examples.

4.2
Let G q be another PIFSS over U, E defined as follows:

4.3
By using the Atanassov union which is the basic intuitionistic fuzzy union we have

4.7
By using the Atanassov intersection which is the basic intuitionistic fuzzy union we have

4.8
Similarly we get

4.9
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11
Remark 4.6. The Atanassov intersection can be replaced by any T-norm which is a general intuitionistic fuzzy intersection see Fathi 20 .
Proposition 4.7. Let F p , G q , and H r be any three PIFSSs over U, E , then the following results hold: Proof. The proof is straightforward by using the fact that the union and intersection of fuzzy sets and intuitionistic fuzzy sets are commutative and associative.

Proposition 4.8. Let F p be a PFSSs over U, E . Then the following results hold:
Proof. The proof is straightforward by using the definitions of union and intersection. Proposition 4.9. Let F p ,G q , and H r be any three PIFSS over U, E . Then the following results hold: Advances in Decision Sciences

4.11
We can use the same method in i .

AND and OR Operations on PIFSS with Applications in Decision Making
In this section we introduce the definitions of AND and OR operations on possibility intuitionistic fuzzy soft sets. Applications of possibility fuzzy soft sets in decision-making problem are given.  H λ e 1 ,e 1 x 1 0, 0.6 , 0.1 ,

5.3
Similarly we get

5.4
In matrix notation we have

5.5
Now to determine the best machine we first calculate the difference between the membership and non-membership values and then we mark the highest numerical grade indicated in parenthesis in each row. Now the score of each of such machine is calculated by taking the sum of the products of these different numerical grades with the corresponding value of λ.
The machine with the highest score is the desired machine. We do not consider the different numerical grades of the machine against the pairs e i ,e i ,i 1, 2, 3, as both parameters are the same. Matrix of different numerical grades is shown below:

5.7
Score Then the firm will select the machine with the highest score. Hence, they will buy machine x 3 .

Advances in Decision Sciences
In matrix notation we have

5.11
Now to determine the best machine we first calculate the difference between the membership and non-membership values and then we mark the highest numerical grade indicated in parenthesis in each row. Now the score of each of such machine is calculated by taking the sum of the products of these different numerical grades with the corresponding value of λ.
The machine with the highest score is the desired machine. We do not consider the different numerical grades of the machine against the pairs e i ,e i ,i 1, 2, 3, as both parameters are the same. Matrix of different numerical grades is shown below:

5.13
Score Then the firm will select the machine with the highest score. Hence, they will buy machine x 3 .

An Application of PIFSS in Decision Making
In this section we present an application of PIFSS in decision making problem. We shall use the algorithm introduced by Dinda et al. 13 . Suppose that there are three schools in universe U {x 1 ,x 2 ,x 3 }and the parameter set E {e 1 ,e 2 ,e 3 ,e 4 ,e 5 ,e 6 }, each, e i ,1≤ i ≤ 6 indicates a specific criterion for the schools e 1 stands for "international". e 2 stands for "English". e 3 stands for "high efficiency". e 4 stands for " modern". e 5 stands for "full day". e 6 stands for "half day". Suppose Madam X wants to pick a good school for her son on the basis of her wishing parameters among those listed above. Our aim is to find out the most appropriate school for her son.
Suppose the wishing parameters of Madam X is A ⊆ E where A {e 1 ,e 3 ,e 6 }. Let p : E → I U be a fuzzy subset of E, defined by Madam X.

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Consider the PIFSS defined as follows: , 0.1 .

6.1
Now we introduce the following operations: i for membership function: α e i µ i γ i − µ i γ i , ii for non-membership function β υ i γ i ,fori 1, 2, 3.
Actually we have taken these two operations to ascend the membership value and descend the non-membership value of F e i on the basis of the degree of preference of Madam X. Then the PIFSS F µ e i reduced to an intuitionistic fuzzy soft set Ψ e i given as follows: i Input the set A ⊆ E of choice of parameters of Madam X.
ii Consider the reduced intuitionistic fuzzy soft set.
iii Consider the tabular representation of membership function andnon-membership function see Table 1 and Table 4 respectively .
iv Compute the comparison table membership function andnon-membership function see Table 2 and Table 5 respectively .
v Compute the membership score and non-membership score see Table 3 and Table 6 respectively .
vi Compute the final score by subtracting non-membership score frommembership score see Table 7 .
vii Find the maximum score, if it occurs in ith row then Madam X will choose school x i .

Decision:
Madam X will choose the school x 2 . In case, if she does not want to choose x 2 due to certain reasons, her second choice will be x 1 .

Similarity between Two Possibility Intuitionistic Fuzzy Soft Sets
Similarity measures have extensive application in several areas such as pattern recognition, image processing, region extraction, and coding theory and so forth. We are often interested to know whether two patterns or images are identical or approximately identical or at least to what degree they are identical. Several researchers have studied the problem of similarity measurement between fuzzy sets, intuitionistic fuzzy sets and Liang and Shi 18 have studied the similarity measures on intuitionistic fuzzy sets. Shawkat et al. 16 have studied the similarity between two possibility fuzzy soft sets.
In this section we introduce a measure of similarity between two PIFSSs. The set theoretic approach has been taken in this regard because it is easier for calculation and is a very popular method too.

Application of Similarity Measure in Medical Diagnosis
In the following example we will try to estimate the possibility that a sick person having certain visible symptoms is suffering from swamp fever. For this we first construct a model possibility intuitionistic fuzzy soft set for swamp fever and the possibility intuitionistic fuzzy soft set of symptoms for the sick person. Next we find the similarity measure of these two sets. If they are significantly similar then we conclude that the person is possibly suffering from swamp fever. Let our universal set contain only two elements "yes" and "no," that is, U {y, n}. Here the set of parameters E is the set of certain visible symptoms. Let E e 1 ,e 2 ,e 3 ,e 4 ,e 5 ,e 6 ,e 7 ,e 8 ,e 9 ,e 10 , where e 1 trembling, e 2 cough with chest congestion, e 3 muscles pain, e 4 nausea, e 5 headache, e 6 low heart rate bradycardia , e 7 pain upon moving the eyes, e 8 fever, e 9 a flushing or pale pink rash comes over the face, e 10 vomiting.
Our model possibility intuitionistic fuzzy soft set for swamp fever F µ is given in Table 8 and this can be prepared with the help of a physician.
After talking to the sick person we can construct his PIFSS G δ as in Table 9.Nowwe find the similarity measure of these two sets as in Example 7.4 , here S F µ ,G δ ∼ 0.38 < 1/2. Hence the two PIFSSs are not significantly similar. Therefore we conclude that the person is not suffering from swamp fever.

Conclusion
In this paper we have introduced the concept of possibility intuitionistic fuzzy soft set and studied some of its properties. Applications of this theory have been given to solve a decision-making problem. Similarity measure of two possibility intuitionistic fuzzy soft sets is discussed and an application of this to medical diagnosis has been shown.