In this paper, we define OR and AND-products between intuitionistic fuzzy parameterized intuitionistic fuzzy soft sets [13] (ifp-intuitionistic fuzzy soft sets) and we then propose a decision making method called ∧-aggr decision making method based on ifp-intuitionistic fuzzy soft sets. Finally, we give an application of this decision making method on a problem including ifp-intuitionistic fuzzy soft sets.
In the real life, we encounter some problems in economy, engineering, environmental science and social science and many fields involving data that contain uncertain. Researchers have performed to cope with such problems for years and they have proposed some theories such as fuzzy set theory [20], intuitionistic fuzzy set theory [2] and rough set theory [19] etc. But these theories have some difficulties individually. To overcome difficulties as determining of membership function or non-membership function, in 1999, Molodtsov [17] introduced soft set theory as a general mathematical tool for dealing with uncertainty and vagueness. Maji et al. [15] combined the soft set theory with the intuitionistic fuzzy set theory and suggested the concept of intuitionistic fuzzy soft set (IFSS). Çağman and Karataş [6] redefined notion of intuitionistic fuzzy soft set and gave an application in decision making. The concepts of fuzzy parameterized soft set and fuzzy parameterized fuzzy soft set and their operations were introduced by Çağman et al. in [4, 5]. In 2016, Deli and Çağman [8] defined intuitionistic fuzzy parameterized soft sets and their operations. In 2013, Karaaslan [13] proposed intuitionistic fuzzy parameterized intuitionistic fuzzy soft sets and defined set theoretical operations of these sets. Also he gave a decision making method using these sets. In 2015, Deli and Karataş [9] defined concepts of interval valued intuitionistic fuzzy parameterized fuzzy soft set and gave an application in decision making. In this study, firstly we present some definitions and operations required for our study, and then we define OR and AND-products of intuitionistic fuzzy parameterized (ifp) intuitionistic fuzzy soft sets and investigate their some properties. Subsequently we construct a new decision making method called ∧-aggr based on AND-product and give an algorithm to solving some problems. We finally present an example to show the method can be successfully applied to some problems that contain uncertainty.
Preliminaries
In this section, we present some basic definitions and operations required in next sections. Throughout this paper U, E and denote set of objects, set of parameters and power set of U, respectively
Definition 2.1. [20] Let X be a nonempty set. A fuzzy set μ on X is a mapping μ : X → [0, 1]. The value μ (x) represents the degree of membership of x ∈ X in the fuzzy set μ. All fuzzy sets over X is denoted by . If μ (x) =0 for all x ∈ X, then μ is denoted by .
Definition 2.2. [1] Let X be a nonempty set. An intuitionistic fuzzy set (or namely if-set) A over X is defined as an object of the following form
where the functions μA : X → [0, 1] and νA : X → [0, 1] define the degree of membership and the degree of non-membership of the element x ∈ X, respectively, and for every x ∈ X,
In addition, for all x ∈ X, and .
Note that the set of all the if-sets over X will be denoted by .
In this paper, we will prefer the notationA = {(μA (x) , νA (x))/x : x ∈ X} instead of Atanassov’s notation [1] for if-sets.
Definition 2.3. [1] Let . Then, fundamental set operations inclusion, union, intersection and complement of if-sets are defined as follow, respectively:
A ⊆ B iff μA (x) ≤ μB (x) and νA (x) ≥ νB (x) for all x ∈ X
A ∪ B = {〈x, max {μA (x) , μB (x)} , min {νA (x) , νB (x)} 〉 : x ∈ X}
A ∩ B = {〈x, min {μA (x) , μB (x)} , max {νA (x) , νB (x)} 〉 : x ∈ X
}
Ac = {〈x, νA (x) , μA (x) 〉 : x ∈ X}.
Definition 2.4. [17] Let consider a nonempty set A, A ⊆ E. A pair (F, A) is called a soft set over X, where F is a mapping given by .
In this study, we will benefit following definition which was defined by Çağman [7] to define intuitionistic fuzzy soft sets, fuzzy parameterized fuzzy soft sets and intuitionistic fuzzy parameterized intuitionistic fuzzy soft sets.
Definition 2.5. [7] A soft set F over U is a set valued function from E to . It can be written a set of ordered pairs
Note that if F (x) =∅, then the element (x, F (x)) is not appeared in F.
Definition 2.6. [6] An intuitionistic fuzzy soft (or namely ifs) set over U is a function from E into . Therefore it can written as
where, the value is an if-set over U, that is,
for all x ∈ X.
Note that, the set of all ifs-set over U is denoted by .
It must be noted that the sets of all fpfs-sets over U will be denoted by .
Definition 2.7. [13] Let A be an if-set over E and ΩA (x) be an if-set over U for all x ∈ E. Then, an intuitionistic fuzzy parameterized ifs-set ΩA (or namely ifpifs-set) is defined by
where
for all x ∈ E. If μA (x) =0 and νA (x) =1 for x ∈ E, then and such elements are not appeared in ΩA.
Also, it must be noted that the sets of all ifpifs-sets over U will be denoted by Ω (U).
Example 2.8. Assume that U = {u1, u2, u3} is a set of objects and E = {x1, x2, x3, x4} is a set of parameters. Moreover an if-set A is defined as A = {(0.5, 0.2)/x1, (0.0, 1.0)/x2, (0.6, 0.3)/x3, (1.0, 0.0)/x4}. Then,
is an ifpifs-set over U.
Definition 2.9. [13] Let ΩA, ΩB ∈ Ω (U). Then, union of ΩA and ΩB, denoted by ΩA ⊔ ΩB, is defined by
Definition 2.10. [13] Let ΩA, ΩB ∈ Ω (U). Then, intersection of ΩA and ΩB, denoted by ΩA ⊓ ΩB, is defined by
OR-product and AND-product of ifpifs-sets
In this section, we define OR-product and AND-product of ifpifs-sets.
Definition 3.1. Let ΩA, ΩB ∈ Ω (U). Then, ∨-product of two ifpifs-setsΩA and ΩB, denoted by ΩA ∨ ΩB, is defined by
where ΩC = ΩA ∨ ΩB and
for all x, y ∈ E.
Definition 3.2. Let ΩA, ΩB ∈ Ω (U). Then, ∧-product of two ifpifs-setsΩA and ΩB, denoted by ΩA ∧ ΩB, is defined by
where ΩC = ΩA ∧ ΩB and
for all x, y ∈ E.
Example 3.3. Let U = {u1, u2, u3, u4, u5} and E = {x1, x2, x3, x4}. Assume that
are two intuitionistic fuzzy sets over E. Let us consider ifpifs-sets given as follow:
Then,
and
For the simplicity, we can denote the ΩA, ΩB, ΩA ∨ ΩB and ΩA ∧ ΩB ifpifs-sets as in Tables 1, 2, 3 and 4, respectively.
Proposition 3.4.If ΩA, ΩB, ΩC ∈ Ω (U), then
ΩA ∨ ΩA = ΩA
ΩA ∧ ΩA = ΩA
(ΩA ∨ ΩB) ∨ ΩC = ΩA ∨ (ΩB ∨ ΩC)
(ΩA ∧ ΩB) ∧ ΩC = ΩA ∧ (ΩB ∧ ΩC)
Proof. The proofs of 1. and 2. are easily obtained from Definition 3.3.
3. Then, we have
4. Likewise, the proof can be made in a similar way in 3.
Proposition 3.5.If ΩA, ΩB, ΩC ∈ Ω (U), then
ΩA ∨ (ΩB ∧ ΩC) = (ΩA ∨ ΩB) ∧ (ΩA ∨ ΩC)
ΩA ∧ (ΩB ∨ ΩC) = (ΩA ∧ ΩB) ∨ (ΩA ∧ ΩC)
Proof. Let ΩA, ΩB, ΩC ∈ Ω (U).
Then, we have that
The proof can be made similar way 1.
Definition 3.6. Let ΩAi ∈ Ω (U) such that i = 1, 2, …, n. Then,
The OR-product of ifpifs-setsΩAi is defined by
The AND-product of ifpifs-setsΩAi is defined by
Definition 3.7. Let ΩA ∈ Ω (U). Then the complement of an ifpifs-set ΩA, denoted by , is defined by
where is complement of the if-set ΩA (x), for every x ∈ E.
Proposition 3.8. Let ΩA, ΩB ∈ Ω (U). Then,
Proof. Let ΩA, ΩB ∈ Ω (U). Then,
From Definition 3.7 we have
The proof can be made similar way 1.
Proposition 3.9.Let ΩAi ∈ Ω (U) such that i = 1, 2, …, n. Then,
Proof. The proof clear from Definition 3.7, 2.9 and 2.10.
∧-aggr decision making method
In this section, we define Ω-aggregate operator and decision function for the ∧-product to construct a ∧-aggr decision making method. Throughout this section, we suppose that cardinalities of |E| and |U| are finite.
Definition 4.1. Let ΩA, ΩB ∈ Ω (U), |E| = n and |U| = m. Then Ω-aggregation operator, denoted by Ωaggr, is a function and for ΩA ∧ ΩB = ΩC. Where
which is an if-set over U. The is called aggregate if-set of the ΩC. Here, μ* (uk) and ν* (uk) are defined as follows:
where
for all xi ∈ E* and uk ∈ U. Also, μ* (uk) and ν* (uk) are membership degree and non membership degree of uk in if-set , respectively. Furthermore μΩC(xi,xj) (uk) and νΩC(xi,xj) (uk) are membership degree and non-membership degree of uk ∈ U in if-set ΩC (xi, xj), respectively.
We can construct an Ω-decision making method by the following algorithm.
Algorithm
In this section we will give an algorithm for decision making.
Step 1: Input the ifpifs-setsΩA and ΩB,
Step 2: Find the ΩA ∧ ΩB = ΩC,
Step 3: Compute the aggregate if-set
Step 4: Find and . Here and are non-membership degrees corresponding to uk ∈ U such that and corresponding to us ∈ U \ {uk} such that , respectively.
Step 5: Find and
Step 6: Opportune element of U is denoted by Opp (U) and it is chosen as follow
Example 5.1. Assume that a company wants to fill a position. There are 5 candidates. There are two decision makers; one of them is from the department of human resources and the other one is from the board of direction. They want to interview the candidates, but it is difficult to make it all of them. Thus, they want to do a questionary which consisting of five question. Assume that the set of candidates U = {u1, u2, u3, u4, u5} which may be characterized by a set of questions (parameters) E = {x1, x2, x3, x4, x5, x6, x7}. The question {x1, x2, x3, x4} is related to human resources and others related to direction. Each question (parameters) is evaluated from point of view of goals and constrain according to chosen intuitionistic fuzzy sets A and B;
It can be seen clearly that the decision makers consider sets of parameters {x1, x2, x3, x4} and {x5, x6, x7}, respectively, to evaluate the candidates.
Step 1: ifpifs-setsΩA and ΩB are determined by decision makers as in Tables 5 and 6.
Step 2:ΩA ∧ ΩB = ΩC is obtained as in Table 7.ΩC can be written as
where ΩC (xi, xj) = {(μΩC(xi,xj) (uk) , νΩC(xi,xj) (uk))/uk} for all i = 1, 2, 3, 4, j = 5, 6, 7 and k = 1, 2, 3, 4, 5.
Step 3: Thus,
and
hence, by using and for i = 1, 2, 3, 4, we obtain Table 8, Then, by using
and
for k = 1, 2, 3, 4, 5, we have
Step 4: Therefore
Step 5: Here, and .
Step 6: Since α′ < β′, the element that belonging to the largest membership degree can be chosen as u1. Hence he is selected for the job.
Conclusion
In this paper, firstly we have defined OR andAND-products of ifpifs-sets. Then we have presented a decision making method based on the AND-product. Finally, we have provided an example that demonstrating that this method can successfully work. It can be applied to problems of many fields that contain uncertainty. Next, decision making method based on OR-product can be constructed. Also OR and AND-products of finite number ifpifs-sets can be defined and applied more complex problems.
References
1.
AtanassovK., Intuitionistic fuzzy sets, Fuzzy Sets and Systems20 (1986), 87–96.
2.
AliM.I., FengF., LiuX., MinW.K. and ShabirM., On some new operations in soft set theory, Computer and Mathematics with Application57 (2009), 1547–1553.
3.
ÇağmanN. and EnginoğluS., Soft set theory and uni-int decision making, European Journal of Operational Research207 (2010), 848–855.
4.
ÇağmanN., EnginoğluS. and ÇitakF., Fuzzy soft set theory and its applications, Iranian Joournal of Fuzzy Systems8(3) (2011), 137–147.
5.
ÇağmanN., ÇitakF. and EnginoğluS., Fuzzy parameterized fuzzy soft set theory and its applications, Turkish Journal of Fuzzy System1(1) (2010), 21–35.
6.
ÇağmanN. and KarataşS., Intuitionistic fuzzy soft set theory and its decision making, Journal of Intelligent and Fuzzy Systems24(4) (2013), 829–836.
7.
ÇağmanN., Contributions to the theory of soft sets, Journal of New Results in Science (4) (2014), 33–41.
8.
DeliI. and ÇağmanN., Intuitionistic fuzzy parametrized soft set theory and its decision making, Applied Soft Computing28 (2015), 109–113.
9.
DeliI. and KarataşS., Interval valued intuitionistic fuzzy parametrized soft set theory and its decision making, Journal of Intelligent Fuzzy Systems, (Preprint)1–10.
10.
FengF., LiC., DavvazB. and AliM.I., Soft sets combined with fuzzy sets and rough sets, a tentative approach, Soft Computingi14(9) (2010), 899–911.
11.
JiangY., TangY., ChenQ., LiuH. and TangJ., Interval-valued intuitionistic fuzzy soft sets and their properties, Computer and Mathematics with Application60(3) (2010), 906–918.
12.
JiangY., TangY. and ChenQ., An adjustable approach to intuitionistic fuzzy soft sets based decision making, Applied Mathematical Modelling35 (2011), 824–836.
13.
KaraaslanF., Intuitionistic fuzzy parameterized intuitionistic fuzzy soft sets with applications in decision making,(in Press), Annals of Fuzzy Mathematics and Informatics11(4) (2016), 607–619.
14.
MajiP.K., BiswasR. and RoyA.R., Fuzzy Soft Sets, Journal of Fuzzy Mathematics9(3) (2001), 589–602.
15.
MajiP.K., BiswasR. and RoyA.R., Intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics9(3) (2001), 677–692.
16.
MajiP.K., BiswasR. and RoyA.R., Soft set theory, Computer and Mathematics with Application45 (2003), 555–562.
17.
MolodtsovD.A., Soft set theory-first results, Computer and Mathematics with Application37 (1999), 19–31.
18.
MajumdarP. and SamantaS.K., Generalised fuzzy soft sets, Computer and Mathematics with Application59 (2010), 1425–1432.
19.
PawlakZ., Rough sets, Int J Comput Inform Sci11 (1982), 341–356.
