OR and AND-products of ifp-intuitionistic fuzzy soft sets and their applications in decision making

Abstract

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.

Keywords

Fuzzy set intuitionistic fuzzy set soft set intuitionistic fuzzy soft set decision making

1 Introduction

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.

2 Preliminaries

In this section, we present some basic definitions and operations required in next sections. Throughout this paper U, E and $P (U)$ 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 $F (X)$ . If μ (x) =0 for all x ∈ X, then μ is denoted by $\bar{0}$ .

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 $A = {〈 x, μ_{A} (x), ν_{A} (x) 〉 : x \in X},$ where the functions μ_A : X → [0, 1] and ν_A : 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, $0 \leq μ_{X} (u) + ν_{X} (u) \leq 1 .$

In addition, for all x ∈ X, $\tilde{1} = {〈 x, 1, 0 〉 : x \in X}$ and $\tilde{0} = {〈 x, 0, 1 〉 : x \in X}$ .

Note that the set of all the if-sets over X will be denoted by $IF (X)$ .

In this paper, we will prefer the notationA = {(μ_A (x) , ν_A (x))/x : x ∈ X} instead of Atanassov’s notation [1] for if-sets.

Definition 2.3. [1] Let $A, B \in IF (X)$ . Then, fundamental set operations inclusion, union, intersection and complement of if-sets are defined as follow, respectively:

A ⊆ B iff μ_A (x) ≤ μ_B (x) and ν_A (x) ≥ ν_B (x) for all x ∈ X

A ∪ B = {〈x, max {μ_A (x) , μ_B (x)} , min {ν_A (x) , ν_B (x)} 〉 : x ∈ X}

A ∩ B = {〈x, min {μ_A (x) , μ_B (x)} , max {ν_A (x) , ν_B (x)} 〉 : x ∈ X }

A^c = {〈x, ν_A (x) , μ_A (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 $F : A \to P (U)$ .

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 $P (U)$ . It can be written a set of ordered pairs $F = {(x, F (x)) : x \in E} .$

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 $\tilde{F}$ over U is a function from E into $IF (U)$ . Therefore it can written as $\tilde{F} = {(x, \tilde{F} (x)) : x \in E}$ where, the value $\tilde{F} (x)$ is an if-set over U, that is, $\tilde{F} (x) = {(μ_{\tilde{F} (x)} (u), ν_{\tilde{F} (x)} (u)) / u : u \in U}$ for all x ∈ X.

Note that, the set of all ifs-set over U is denoted by $IFS (U)$ .

It must be noted that the sets of all fpfs-sets over U will be denoted by $FPFS (U)$ .

Definition 2.7. [13] Let A be an if-set over E and Ω_A (x) be an if-set over U for all x ∈ E. Then, an intuitionistic fuzzy parameterized ifs-set Ω_A (or namely ifpifs-set) is defined by $Ω_{A} = {((μ_{A} (x), ν_{A} (x)) / x, Ω_{A} (x)) : x \in E}$ where $Ω_{A} (x) = {(μ_{Ω_{A} (x)} (u), ν_{Ω_{A} (x)} (u)) / u : u \in U}$ for all x ∈ E. If μ_A (x) =0 and ν_A (x) =1 for x ∈ E, then $Ω_{A} (x) = \tilde{0}$ and such elements are not appeared in Ω_A.

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, $\begin{matrix} Ω_{A} \\ = {((0.5, 0.2) / x_{1}, {(0.7, 0.2) / u_{1}, (0.5, 0.4) / u_{2}, \\ (0.6, 0.2) / u_{3},}), ((0.6, 0.3) / x_{3}, {(0.4, 0.3) / u_{1}, \\ (0.8, 0.1) / u_{2}, (0.6, 0.3) / u_{3}}), ((1.0, 0.0) / x_{4}, \tilde{1})} \end{matrix}$ is an ifpifs-set over U.

Definition 2.9. [13] Let Ω_A, Ω_B ∈ Ω (U). Then, union of Ω_A and Ω_B, denoted by Ω_A ⊔ Ω_B, is defined by $\begin{matrix} Ω_{A} ⊔ Ω_{B} \\ = {((max {μ_{A} (x), μ_{B} (x)}, min {ν_{A} (x), \\ ν_{B} (x)}) / x, Ω_{A} (x) \cup Ω_{B} (x)) : x \in E} . \end{matrix}$

Definition 2.10. [13] Let Ω_A, Ω_B ∈ Ω (U). Then, intersection of Ω_A and Ω_B, denoted by Ω_A ⊓ Ω_B, is defined by $\begin{matrix} Ω_{A} ⊓ Ω_{B} \\ = {((min {μ_{A} (x), μ_{B} (x)}, max {ν_{A} (x), \\ ν_{B} (x)}) / x, Ω_{A} (x) \cap Ω_{B} (x)) : x \in E} . \end{matrix}$

3 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 Ω_A, Ω_B ∈ Ω (U). Then, ∨-product of two ifpifs-setsΩ_A and Ω_B, denoted by Ω_A ∨ Ω_B, is defined by $\begin{matrix} Ω_{C} & = & {((μ_{C} (x, y), ν_{C} (x, y)) / (x, y), Ω_{C} (x, y)) \\ : x, y \in E} \end{matrix}$ where Ω_C = Ω_A ∨ Ω_B and $\begin{matrix} μ_{C} (x, y) & = & max {μ_{A} (x), μ_{B} (y)} \\ ν_{C} (x, y) & = & min {ν_{A} (x), ν_{B} (y)} \\ Ω_{C} (x, y) & = & Ω_{A} (x) \cup Ω_{B} (y) \end{matrix}$ for all x, y ∈ E.

Definition 3.2. Let Ω_A, Ω_B ∈ Ω (U). Then, ∧-product of two ifpifs-setsΩ_A and Ω_B, denoted by Ω_A ∧ Ω_B, is defined by $\begin{matrix} Ω_{C} & = & {((μ_{C} (x, y), ν_{C} (x, y)) / (x, y), Ω_{C} (x, y)) \\ : x, y \in E} \end{matrix}$ where Ω_C = Ω_A ∧ Ω_B and

$\begin{matrix} μ_{C} (x, y) & = & min {μ_{A} (x), μ_{B} (y)} \\ ν_{C} (x, y) & = & max {ν_{A} (x), ν_{B} (y)} \\ Ω_{C} (x, y) & = & Ω_{A} (x) \cap Ω_{B} (y) \end{matrix}$ for all x, y ∈ E.

Example 3.3. Let U = {u₁, u₂, u₃, u₄, u₅} and E = {x₁, x₂, x₃, x₄}. Assume that

$\begin{matrix} A & = & {(0.7, 0.2) / x_{1}, (0.0, 1.0) / x_{2}, (0.6, 0.3) / x_{3}, \\ (0.0, 1.0) / x_{4}} \\ B & = & {(0.0, 1.0) / x_{1}, (0.8, 0.1) / x_{2}, (0.0, 1.0) / x_{3}, \\ (0.4, 0.5) / x_{4}} \end{matrix}$ are two intuitionistic fuzzy sets over E. Let us consider ifpifs-sets given as follow:

$\begin{matrix} Ω_{A} & = & {((0.7, 0.2) / x_{1}, {(0.0, 1.0) / u_{1}, (0.4, 0.5) / u_{2}, \\ (0.6, 0.4) / u_{3}, (0.0, 1.0) / u_{4}, (0.8, 0.1) / u_{5}}), \\ ((0.6, 0.3) / x_{3}, {(0.5, 0.2) / u_{1}, (0.0, 1.0) / u_{2}, \\ (0.0, 1.0) / u_{3}, (0.8, 0.1) / u_{4}, (0.0, 1.0) / u_{5}})} \\ Ω_{B} & = & {((0.8, 0.1) / x_{2}, {(0.6, 0.3) / u_{1}, (0.6, 0.2) / u_{2}, \\ (0.5, 0.2) / u_{3}, (0.0, 1.0) / u_{4}, (0.0, 1.0) / u_{5}}), \\ ((0.4, 0.5) / x_{4}, {(0.7, 0.2) / u_{1}, (0.0, 1.0) / u_{2}, \\ (0.8, 0.1) / u_{3}, (0.0, 1.0) / u_{4}, (0.0, 1.0) / u_{5}})} . \end{matrix}$

Then, $\begin{matrix} Ω_{A} \lor Ω_{B} \\ = {((0.8, 0.1) / (x_{1}, x_{2}), {(0.6, 0.3) / u_{1}, \\ (0.6, 0.2) / u_{2}, (0.6, 0.2) / u_{3}, (0.0, 1.0) / u_{4}, \\ (0.8, 0.1) / u_{5}}), ((0.7, 0.2) / (x_{1}, x_{4}), \\ {(0.7, 0.2) / u_{1}, (0.4, 0.5) / u_{2}, (0.8, 0.1) / u_{3}, \\ (0.0, 1.0) / u_{4}, (0.8, 0.1) / u_{5}}), \\ ((0.8, 0.1) / (x_{3}, x_{2}), {(0.6, 0.2) / u_{1}, \\ (0.6, 0.2) / u_{2}, (0.5, 0.2) / u_{3}, (0.8, 0.1) / u_{4}, \\ (0.0, 1.0) / u_{5}}), ((0.6, 0.3) / (x_{3}, x_{4}), \\ {(0.7, 0.2) / u_{1}, (0.0, 1.0) / u_{2}, (0.8, 0.1) / u_{3}, \\ (0.8, 0.1) / u_{4}, (0.0, 1.0) / u_{5}})} \end{matrix}$ and $\begin{matrix} Ω_{A} \land Ω_{B} \\ = {((0.7, 0.2) / (x_{1}, x_{2}), {(0.0, 1.0) / u_{1}, \\ (0.4, 0.5) / u_{2}, (0.5, 0.4) / u_{3}, (0.0, 1.0) / u_{4}, \\ (0.0, 1.0) / u_{5}}), ((0.4, 0.5) / (x_{1}, x_{4}), \\ {(0.0, 1.0) / u_{1}, (0.0, 1.0) / u_{2}, (0.6, 0.4) / u_{3}, \\ (0.0, 1.0) / u_{4}, (0.0, 1.0) / u_{5}}), \\ ((0.6, 0.3) / (x_{3}, x_{2}), {(0.5, 0.3) / u_{1}, \\ (0.0, 1.0) / u_{2}, (0.0, 1.0) / u_{3}, (0.0, 1.0) / u_{4}, \\ (0.0, 1.0) / u_{5}}), ((0.4, 0.5) / (x_{3}, x_{4}), \\ {(0.5, 0.2) / u_{1}}, (0.0, 1.0) / u_{2}, (0.0, 1.0) / u_{3}, \\ (0.0, 1.0) / u_{4}, (0.0, 1.0) / u_{5})} . \end{matrix}$

For the simplicity, we can denote the Ω_A, Ω_B, Ω_A ∨ Ω_B and Ω_A ∧ Ω_B ifpifs-sets as in Tables 1, 2, 3 and 4, respectively.

Proposition 3.4. If Ω_A, Ω_B, Ω_C ∈ Ω (U), then

Ω_A ∨ Ω_A = Ω_A

Ω_A ∧ Ω_A = Ω_A

(Ω_A ∨ Ω_B) ∨ Ω_C = Ω_A ∨ (Ω_B ∨ Ω_C)

(Ω_A ∧ Ω_B) ∧ Ω_C = Ω_A ∧ (Ω_B ∧ Ω_C)

Proof. The proofs of 1. and 2. are easily obtained from Definition 3.3.

3. Then, we have

$\begin{matrix} (Ω_{A} & \lor & Ω_{B}) \lor Ω_{C} \\ = & {((max {max {μ_{A} (x), μ_{B} (y)}, μ_{C} (z)}, \\ min {min {ν_{A} (x), ν_{B} (y)}, ν_{C} (z)}) / (x, y, z), \\ (Ω_{A} (x) \cup Ω_{B} (y)) \cup Ω_{C} (z)) : x, y, z \in E} \\ = & {(max {μ_{A} (x), μ_{B} (y), μ_{C} (z)}, \\ min {ν_{A} (x), ν_{B} (y), ν_{C} (z)) / (x, y, z), \\ Ω_{A} (x) \cup Ω_{B} (y) \cup Ω_{C} (z)) : x, y, z \in E} \\ = & {((max {μ_{A} (x), max {μ_{B} (y), μ_{C} (z)}}, \\ min {ν_{A} (x), min {ν_{B} (y), ν_{C} (z)}) / (x, y, z), \\ Ω_{A} (x) \cup (Ω_{B} (y) \cup Ω_{C} (z))) : x, y, z \in E} \\ = & Ω_{A} \lor (Ω_{B} \lor Ω_{C}) \end{matrix}$

4. Likewise, the proof can be made in a similar way in 3.

Proposition 3.5. If Ω_A, Ω_B, Ω_C ∈ Ω (U), then

Ω_A ∨ (Ω_B ∧ Ω_C) = (Ω_A ∨ Ω_B) ∧ (Ω_A ∨ Ω_C)

Ω_A ∧ (Ω_B ∨ Ω_C) = (Ω_A ∧ Ω_B) ∨ (Ω_A ∧ Ω_C)

Proof. Let Ω_A, Ω_B, Ω_C ∈ Ω (U).

Then, we have that $\begin{matrix} Ω_{A} & \lor & (Ω_{B} \land Ω_{C}) \\ = & {((max {μ_{A} (x), min {μ_{B} (y), μ_{C} (z)}}, \\ min {ν_{A} (x), max {ν_{B} (y), ν_{C} (z)}}), \\ Ω_{A} (x) \cup (Ω_{B} (y) \cap Ω_{C} (z))) : x, y, z \in E} \\ = & {((min {max {μ_{A} (x), μ_{B} (y)} / (x, y), \\ max {μ_{A} (x), μ_{C} (z)} / (x, z)}, \\ max {min {ν_{A} (x), ν_{B} (y)} / (x, y), \\ min {ν_{A} (x), ν_{C} (z)} / (x, z)}), \\ (Ω_{A} (x) \cup Ω_{B} (y)) \cap (Ω_{A} (x) \cup Ω_{C} (z))) : \\ x, y, z \in E}, \\ = & {(min {(max {μ_{A} (x), μ_{B} (y)}, min {ν_{A} (x), \\ ν_{B} (y)}) / (x, y), (max {μ_{A} (x), μ_{C} (z)}, \\ min {ν_{A} (x), ν_{C} (z)}) / (x, z)}, \\ (Ω_{A} (x) \cup Ω_{B} (y)) \cap (Ω_{A} (x) \cup Ω_{C} (z))) : \\ x, y, z \in E} \\ = & (Ω_{A} \lor Ω_{B}) \land (Ω_{A} \lor Ω_{C}) \end{matrix}$

The proof can be made similar way 1.

Definition 3.6. Let Ω_{A
_i} ∈ Ω (U) such that i = 1, 2, …, n. Then,

The OR-product of ifpifs-setsΩ_{A
_i} is defined by $\begin{matrix} ⋁_{i = 1}^{n} Ω_{A_{i}} & = & {((max_{1 \leq i \leq n} μ_{A_{i}} (x_{i}), min_{1 \leq i \leq n} ν_{A_{i}} (x_{i})) / \\ (x_{1}, x_{2}, \dots, x_{n}), Ω_{A_{1}} (x_{1}) \cup Ω_{A_{2}} (x_{2}) \\ \cup \dots \cup Ω_{A_{n}} (x_{n})) : x_{1}, x_{2}, \dots, x_{n} \in E} . \end{matrix}$

The AND-product of ifpifs-setsΩ_{A
_i} is defined by $\begin{matrix} ⋀_{i = 1}^{n} Ω_{A_{i}} & = & {((min_{1 \leq i \leq n} μ_{A_{i}} (x_{i}), max_{1 \leq i \leq n} ν_{A_{i}} (x_{i})) / \\ (x_{1}, x_{2}, \dots, x_{n}), Ω_{A_{1}} (x_{1}) \cap Ω_{A_{2}} (x_{2}) \\ \cap \dots \cap Ω_{A_{n}} (x_{n})) : x_{1}, x_{2}, \dots, x_{n} \in E} . \end{matrix}$

Definition 3.7. Let Ω_A ∈ Ω (U). Then the complement of an ifpifs-set Ω_A, denoted by $Ω_{A}^{\tilde{c}}$ , is defined by $Ω_{A}^{\tilde{c}} = {((ν_{A} (x), μ_{A} (x)) / x, Ω_{A}^{c} (x)) : x \in E}$ where $Ω_{A}^{c} (x)$ is complement of the if-set Ω_A (x), for every x ∈ E.

Proposition 3.8. Let Ω_A, Ω_B ∈ Ω (U). Then,

$(Ω_{A} \lor Ω_{B})^{\tilde{c}} = Ω_{A}^{\tilde{c}} \land Ω_{B}^{\tilde{c}}$

$(Ω_{A} \land Ω_{B})^{\tilde{c}} = Ω_{A}^{\tilde{c}} \lor Ω_{B}^{\tilde{c}}$

Proof. Let Ω_A, Ω_B ∈ Ω (U). Then,

From Definition 3.7 we have $\begin{matrix} (Ω_{A} & \lor & Ω_{B})^{\tilde{c}} \\ = & {((max {μ_{A} (x), μ_{B} (y)}, \\ min {ν_{A} (x), ν_{B} (y)}) / (x, y), \\ Ω_{A} (x) \cup Ω_{B} (y)) : x, y \in E}^{\tilde{c}} \\ = & {((min {ν_{A} (x), ν_{B} (y)}, \\ max {μ_{A} (x), μ_{B} (y)}) / (x, y), \\ Ω_{A} (x)^{c} \cap Ω_{B} (y)^{c}) : x, y \in E} \\ = & {{((ν_{A} (x), μ_{A} (x)) / x, Ω_{A}^{c} (x))} \\ \lor {((ν_{B} (y), μ_{B} (y)) / y, Ω_{B}^{c} (y))} \\ = & Ω_{A}^{\tilde{c}} \land Ω_{B}^{\tilde{c}} \end{matrix}$

The proof can be made similar way 1.

Proposition 3.9. Let Ω_{A
_i} ∈ Ω (U) such that i = 1, 2, …, n. Then,

$(⋁_{i = 1}^{n} Ω_{A_{i}})^{\tilde{c}} = ⋀_{i = 1}^{n} Ω_{A_{i}}^{\tilde{c}}$

$(⋀_{i = 1}^{n} Ω_{A_{i}})^{\tilde{c}} = ⋁_{i = 1}^{n} Ω_{A_{i}}^{\tilde{c}}$

Proof. The proof clear from Definition 3.7, 2.9 and 2.10.

4 ∧-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 Ω_A, Ω_B ∈ Ω (U), |E| = n and |U| = m. Then Ω-aggregation operator, denoted by Ω_aggr, is a function $Ω_{aggr} : Ω (U) \times Ω (U) \to IF (U)$ and $Ω_{aggr} (Ω_{C}) = Ω_{C}^{*}$ for Ω_A ∧ Ω_B = Ω_C. Where $\begin{matrix} Ω_{C}^{*} \\ = {(μ^{*} (u_{k}), ν^{*} (u_{k})) / u_{k} : u_{k} \in U, k = 1, 2, \dots, m} \end{matrix}$ which is an if-set over U. The $Ω_{C}^{*}$ is called aggregate if-set of the Ω_C. Here, μ^* (u_k) and ν^* (u_k) are defined as follows: $\begin{matrix} μ^{*} (u_{k}) & = & \frac{1}{| E^{*} |} \sum_{x_{i} \in E^{*}} [μ_{x_{i}} \times μ_{x_{i}} (u_{k})] \\ ν^{*} (u_{k}) & = & \frac{1}{| E^{*} |} \sum_{x_{i} \in E^{*}} [ν_{x_{i}} \times ν_{x_{i}} (u_{k})] \end{matrix}$ where $\begin{matrix} E^{*} & = & {x_{i} \in E : μ_{A} (x_{i}) \neq 0} \\ μ_{x_{i}} & = & min_{x_{j} \in E^{*}} {μ_{C} (x_{i}, x_{j})} \\ ν_{x_{i}} & = & max_{x_{j} \in E^{*}} {ν_{C} (x_{i}, x_{j})} \\ μ_{x_{i}} (u_{k}) & = & min_{x_{j} \in E^{*}} {μ_{Ω_{C} (x_{i}, x_{j})} (u_{k})} \\ ν_{x_{i}} (u_{k}) & = & max_{x_{j} \in E^{*}} {ν_{Ω_{C} (x_{i}, x_{j})} (u_{k})} \end{matrix}$ for all x_i ∈ E^* and u_k ∈ U. Also, μ^* (u_k) and ν^* (u_k) are membership degree and non membership degree of u_k in if-set $Ω_{C}^{*}$ , respectively. Furthermore μ_{Ω_C(x_i,x_j)} (u_k) and ν_{Ω_C(x_i,x_j)} (u_k) are membership degree and non-membership degree of u_k ∈ U in if-set Ω_C (x_i, x_j), respectively.

We can construct an Ω-decision making method by the following algorithm.

5 Algorithm

In this section we will give an algorithm for decision making.

Step 1: Input the ifpifs-setsΩ_A and Ω_B,

Step 2: Find the Ω_A ∧ Ω_B = Ω_C,

Step 3: Compute the aggregate if-set $Ω_{C}^{*}$

Step 4: Find $α = ν_{k}^{*}$ and $β = ν_{s}^{*}$ . Here $ν_{k}^{*}$ and $ν_{s}^{*}$ are non-membership degrees corresponding to u_k ∈ U such that $μ_{k}^{*} = max_{i \in {1, 2, . . ., | U |}} μ_{i}^{*}$ and corresponding to u_s ∈ U \ {u_k} such that $μ_{s}^{*} = max_{i \in {1, 2, . . ., | U |} \ {k}} μ_{i}^{*}$ , respectively.

Step 5: Find $\frac{1 - 2 α}{2} = α^{'}$ and $\frac{1 - 2 β}{2} = β^{'}$

Step 6: Opportune element of U is denoted by Opp (U) and it is chosen as follow $Opp (U) = {\begin{matrix} u_{k}, & if α^{'} \geq β^{'} \\ u_{s}, & if α^{'} < β^{'} \end{matrix}$

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; $\begin{matrix} A & = & {(0.5, 0.2) / x_{1}, (0.7, 0.3) / x_{2}, (0.6, 0.2) / x_{3}, \\ (0.6, 0.4) / x_{4}, (0.0, 1.0) / x_{5}, (0.0, 1.0) / x_{6}, \\ (0.0, 1.0) / x_{7}} \\ B & = & {(0.0, 1.0) / x_{1}, (0.0, 1.0) / x_{2}, (0.0, 1.0) / x_{3}, \\ (0.0, 1.0) / x_{4}, (0.6, 0.1) / x_{5}, (0.4, 0.2) / x_{6}, \\ (0.7, 0.3) / x_{7}} \end{matrix}$

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-setsΩ_A and Ω_B are determined by decision makers as in Tables 5 and 6.

Step 2:Ω_A ∧ Ω_B = Ω_C is obtained as in Table 7.Ω_C can be written as $\begin{matrix} Ω_{C} & = & {((μ_{C} (x_{i}, x_{j}), ν_{C} (x_{i}, x_{j})) / (x_{i}, x_{j}), \\ Ω_{C} (x_{i}, x_{j})) : x_{i}, x_{j} \in E, i = 1, 2, 3, 4 \\ and j = 5, 6, 7} \end{matrix}$ where Ω_C (x_i, x_j) = {(μ_{Ω_C(x_i,x_j)} (u_k) , ν_{Ω_C(x_i,x_j)} (u_k))/u_k} for all i = 1, 2, 3, 4, j = 5, 6, 7 and k = 1, 2, 3, 4, 5.

Step 3: Thus, $\begin{matrix} μ_{x_{1}} & = & min_{5 \leq j \leq 7} {μ_{C} (x_{1}, x_{j})} = 0.4 \\ μ_{x_{2}} & = & min_{5 \leq j \leq 7} {μ_{C} (x_{2}, x_{j})} = 0.4 \\ μ_{x_{3}} & = & min_{5 \leq j \leq 7} {μ_{C} (x_{3}, x_{j})} = 0.3 \\ μ_{x_{4}} & = & min_{5 \leq j \leq 7} {μ_{C} (x_{4}, x_{j})} = 0.4 \end{matrix}$ and $\begin{matrix} ν_{x_{1}} & = & max_{5 \leq j \leq 7} {ν_{C} (x_{1}, x_{j})} = 0.4 \\ ν_{x_{2}} & = & max_{5 \leq j \leq 7} {ν_{C} (x_{2}, x_{j})} = 0.3 \\ ν_{x_{3}} & = & max_{5 \leq j \leq 7} {ν_{C} (x_{3}, x_{j})} = 0.3 \\ ν_{x_{4}} & = & max_{5 \leq j \leq 7} {ν_{C} (x_{4}, x_{j})} = 0.4 \end{matrix}$ hence, by using $μ_{x_{i}} (u_{k}) = min_{5 \leq j \leq 7} {μ_{Ω_{C} (x_{i}, x_{j})} (u_{k})}$ and $ν_{x_{i}} (u_{k}) = max_{5 \leq j \leq 7} {μ_{Ω_{C} (x_{i}, x_{j})} (u_{k})}$ for i = 1, 2, 3, 4, we obtain Table 8, Then, by using $μ_{k}^{*} = \frac{1}{4} \sum_{i = 1}^{4} [μ_{x_{i}} \times μ_{x_{i}} (u_{k})]$ and $ν_{k}^{*} = \frac{1}{4} \sum_{i = 1}^{4} [ν_{x_{i}} \times ν_{x_{i}} (u_{k})]$ for k = 1, 2, 3, 4, 5, we have $\begin{matrix} Ω_{C}^{*} & = & {(0.180, 0.120) / u_{1}, (0.105, 0.190) / u_{2}, \\ (0.177, 0.160) / u_{3}, (0.140, 0.195) / u_{4}, \\ (0.187, 0.175) / u_{5}} . \end{matrix}$

Step 4: Therefore $α = ν_{5}^{*} = 0.175 and β = ν_{1}^{*} = 0.120$

Step 5: Here, $\frac{1 - 2 \times 0.175}{2} = 0.325 = α^{'}$ and $\frac{1 - 2 \times 0.120}{2} = 0.380 = β^{'}$ .

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.

5 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

Atanassov

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems20 (1986), 87–96.

Ali

M.I.

, Feng

, Liu

, Min

W.K.

and Shabir

, On some new operations in soft set theory, Computer and Mathematics with Application57 (2009), 1547–1553.

Çağman

and Enginoğlu

, Soft set theory and uni-int decision making, European Journal of Operational Research207 (2010), 848–855.

Çağman

, Enginoğlu

and Çitak

, Fuzzy soft set theory and its applications, Iranian Joournal of Fuzzy Systems8(3) (2011), 137–147.

Çağman

, Çitak

and Enginoğlu

, Fuzzy parameterized fuzzy soft set theory and its applications, Turkish Journal of Fuzzy System1(1) (2010), 21–35.

Çağman

and Karataş

, Intuitionistic fuzzy soft set theory and its decision making, Journal of Intelligent and Fuzzy Systems24(4) (2013), 829–836.

Çağman

, Contributions to the theory of soft sets, Journal of New Results in Science (4) (2014), 33–41.

Deli

and Çağman

, Intuitionistic fuzzy parametrized soft set theory and its decision making, Applied Soft Computing28 (2015), 109–113.

Deli

and Karataş

, Interval valued intuitionistic fuzzy parametrized soft set theory and its decision making, Journal of Intelligent Fuzzy Systems, (Preprint)1–10.

10.

Feng

, Li

, Davvaz

and Ali

M.I.

, Soft sets combined with fuzzy sets and rough sets, a tentative approach, Soft Computingi14(9) (2010), 899–911.

11.

Jiang

, Tang

, Chen

, Liu

and Tang

, Interval-valued intuitionistic fuzzy soft sets and their properties, Computer and Mathematics with Application60(3) (2010), 906–918.

12.

Jiang

, Tang

and Chen

, An adjustable approach to intuitionistic fuzzy soft sets based decision making, Applied Mathematical Modelling35 (2011), 824–836.

13.

Karaaslan

, 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.

Maji

P.K.

, Biswas

and Roy

A.R.

, Fuzzy Soft Sets, Journal of Fuzzy Mathematics9(3) (2001), 589–602.

15.

Maji

P.K.

, Biswas

and Roy

A.R.

, Intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics9(3) (2001), 677–692.

16.

Maji

P.K.

, Biswas

and Roy

A.R.

, Soft set theory, Computer and Mathematics with Application45 (2003), 555–562.

17.

Molodtsov

D.A.

, Soft set theory-first results, Computer and Mathematics with Application37 (1999), 19–31.

18.

Majumdar

and Samanta

S.K.

, Generalised fuzzy soft sets, Computer and Mathematics with Application59 (2010), 1425–1432.

19.

Pawlak

, Rough sets, Int J Comput Inform Sci11 (1982), 341–356.

20.

Zadeh

L.A.

, Fuzzy Sets, Inform Control8 (1965), 338–353.