A new approach based on intuitionistic fuzzy rough graphs for decision-making

Abstract

An intuitionistic fuzzy rough model is a hybrid model which is made by combining two mathematical models namely, intuitionistic fuzzy sets and rough sets. This hybrid model deals with soft computing and vagueness by using the lower and upper approximation spaces. In this research study, the concept of intuitionistic fuzzy rough sets is applied to graphs. The notion of intuitionistic fuzzy rough graphs and various methods of their construction are presented. An application of intuitionistic fuzzy rough graphs in decision-making problem is discussed. In particular, an efficient algorithm is developed to solve decision-making problems. Time complexity of proposed algorithm is computed.

Keywords

Intuitionistic fuzzy rough graphs algorithms decision-making

1 Introduction

Zadeh [32] introduced the concept of fuzzy sets as a method of representing uncertainty and vagueness. From thenceforth, the theory of fuzzy sets has been applied in different disciplines. Atanassov [5] introduced the concept of intuitionistic fuzzy sets as an extension of the fuzzy sets. It has been successfully applied in many fields for decision making and pattern recognition. Decision-making is an intellectual process which deals with selection of the best option among multiple available options, it begins when we need to do something but we do not know what. It involves single criteria decision making, multi criteria decision making and group decision making. Several researchers, including Liu et al. [15 –18] and Wei [25, 26] have studied different decision-making methods. The theory of rough sets was originally proposed by Pawlak [21] as a formal tool for modeling and processing incomplete information. Using the concepts of lower and upper approximations in rough set theory, knowledge hidden in information systems may be unraveled and expressed in the form of decision rules. This has lead to various other generalized rough set models (Bi et al. [7]; Liu [15]; Pei et al. [22]; Wu et al. [27]; Zhang et al. [34] and Zhang et al. [35]). In recent years the concept of fuzzy rough set [10] was worked to be more useful in decision making problems. The fuzziness in rough sets was also studied in [8].

Zhou et al. [36] studied the pair of lower and upper intuitionistic fuzzy rough approximation operators induced from an arbitrary intuitionistic fuzzy relation. Basic properties of the intuitionistic fuzzy rough approximation operators are then examined. By introducing cut sets of intuitionistic fuzzy sets, classical representations of intuitionistic fuzzy rough approximation operators are presented. The connections between special intuitionistic fuzzy relations and intuitionistic fuzzy rough approximation operators are further established [13]. An operator-oriented characterization of intuitionistic fuzzy rough sets is proposed, that is, intuitionistic fuzzy rough approximation operators are defined by the axioms. Different axiom sets of lower and upper intuitionistic fuzzy set-theoretic operators guarantee the existence of different types of intuitionistic fuzzy relations which produce the same operators. Cornelis et al. [9] described intuitionistic fuzzy rough sets. Mi et al. [19] applied composition in intuitionistic fuzzy rough approximation spaces. Xu et al. [29] considered intuitionistic fuzzy rough sets model based on operators. Wu also worked on intuitionistic fuzzy rough set in the papers [27, 28]. Yang et al. [30, 31] dealt bipolar fuzzy rough set model on two different universes. They also illustrated the application of the bipolar rough set model.

Graph theory is a useful tool to study different network models and pairwise relationships among objects. Intuitionistic fuzzy rough set has a vital role in vagueness. Fuzzy graph theory is finding an increasing number of applications in modeling real time systems where the level of information inherent in the system varies with different levels of precision. Fuzzy models are becoming useful because of their aim of reducing the differences between the traditional numerical models used in engineering and sciences and the symbolic models used in expert systems. Kaufmann [14] defined first fuzzy graphs. Rosenfeld [23] considered fuzzy graphs and obtained analogs of several graph theoretical concepts. Afterwards, Bhattacharya [6] provided some thoughts on fuzzy graphs. Mordeson and Peng [20] defined some operations on fuzzy graphs. Sunitha et al. [24] introduced the complement of a fuzzy graph. Gani et al. [11, 12] introduced several new ideas in fuzzy graph. Recently, Akram et al. [1 –4] has discussed several new concepts with applications. On the other hand, recent investigations have shown how fuzzy set, rough set and graph theory can be combined into a more flexible, more expressive framework for modeling and processing incomplete information in information systems. At the same time, intuitionistic fuzzy rough sets have been proposed as an attractive extension of fuzzy sets, enriching the latter with extra features to represent uncertainty (on top of vagueness). We intend to fill an obvious gap by introducing a new definition of intuitionistic fuzzy rough graph, as the most natural generalization of Pawlak’s original concept of rough sets. When dealing with decision-making problems, considering the relations among alternatives gives more valuable results. Therefore, it would be beneficial to use graphical approaches instead of set theory and logic. Fariha and Akram [33] have introduced rough fuzzy graphs. In this research paper, the concept of intuitionistic fuzzy rough sets is applied to graphs.

2 Preliminaries

Definition 2.1. [21] Let K be a nonempty finite set and L ⊆ K × K an indiscernibility relation on K. Further, consider L is an equivalence relation. A pair (K, L) is called an approximation space. Let Z be a subset of K and it can be characterized by a pair of lower and upper approximations: $\begin{matrix} L^{*} Z & = & {z \in K | [z]_{L} \cap Z \neg = \emptyset}, \\ L_{*} Z & = & {z \in K | [z]_{L} \subseteq Z}, \end{matrix}$ where [z] _L denotes the equivalence class containing z. A pair (L_∗Z, L^∗Z) is called a rough set, if L_∗Z ≠ L^∗Z .

Dubois and Prade [10] presented the concept of fuzzy rough sets.

Definition 2.2. [10] Let K be a finite set and L a fuzzy equivalence relation on K. Let Z ∈ F (K), where F (K) denotes the fuzzy power set. The lower and upper approximations of the fuzzy set Z, denoted by L_∗Z and L^∗Z respectively, are defined as fuzzy sets in K, ∀ z ∈ K such that: $\begin{matrix} (L^{*} Z) (z) & = & ⋁_{k \in K} (L (z, k) \land Z (k)), \\ (L_{*} Z) (z) & = & ⋀_{k \in K} ((1 - L (z, k)) \lor Z (k)) . \end{matrix}$

The pair LZ = (L_∗Z, L^∗Z) is called a fuzzy rough set if L_∗Z ≠ L^∗Z .

Definition 2.3. [36] Let K be a nonempty finite set and L ∈ IF (K × K). A pair J = (K, L) is called an intuitionistic fuzzy approximation space, where IF (K × K) is an intuitionistic fuzzy relation (IFR) on K. For any Z ∈ IF (K) , the upper and lower intuitionistic fuzzy rough approximations of Z w.r.t. (K, L) , denoted by L_∗Z and L^∗Z, are two intuitionistic fuzzy sets defined as follows: $L_{*} Z = (L_{*} Z^{+}, L_{*} Z^{-}), L^{*} Z = (L^{*} Z^{+}, L^{*} Z^{-}),$ where k ∈ K and $\begin{matrix} (L_{*} Z)^{+} (k) & = & ⋁_{z \in K} [L^{+} (k, z) \land Z^{+} (z)], \\ (L_{*} Z)^{-} (k) & = & ⋀_{z \in K} [L^{-} (k, z) \lor Z^{-} (z)], \\ (L^{*} Z)^{+} (k) & = & ⋀_{z \in K} [L^{-} (k, z) \lor Z^{+} (z)], \\ (L^{*} Z)^{-} (k) & = & ⋁_{z \in K} [L^{+} (k, z) \land Z^{-} (z)] . \end{matrix}$

Moreover, Z is an intuitionistic fuzzy rough set, if K_∗Z ≠ K^∗Z .

3 Intuitionistic fuzzy rough graphs

Definition 3.1. Let C^★ be a Boolean set and P an intuitionistic fuzzy relation on C^★. Let C be an intuitionistic fuzzy set on C^★ and PC = (P_∗C, P^∗C) an intuitionistic fuzzy rough set. Let D^★ ⊆ C^★ × C^★ and Q an intuitionistic fuzzy relation on D^★ such that, for all p₁p₂, q₁q₂ ∈ D^★, $\begin{matrix} μ_{Q}^{+} ((p_{1} p_{2}), (q_{1} q_{2})) & \leq & μ_{P}^{+} (p_{1}, q_{1}) \land μ_{P}^{+} (p_{2}, q_{2}), \\ μ_{Q}^{-} ((p_{1} p_{2}), (q_{1} q_{2})) & \leq & μ_{P}^{-} (p_{1}, q_{1}) \lor μ_{P}^{-} (p_{2}, q_{2}) . \end{matrix}$

Let D be an intuitionistic fuzzy set on D^★ such that, for all p, q ∈ C^★, $\begin{matrix} μ_{D}^{+} (pq) & \leq & min {μ_{P^{*} C}^{+} (p), μ_{P^{*} C}^{+} (q)}, \\ μ_{D}^{-} (pq) & \leq & max {μ_{P^{*} C}^{-} (p), μ_{P^{*} C}^{-} (q)} . \end{matrix}$

Then the lower and upper approximations of D, denoted by Q_∗D and Q^∗D respectively, are defined as intuitionistic fuzzy sets in C^★ × C^★ such that, for all pq ∈ D^★, Q_∗D = (Q_∗D⁺, Q_∗D^-) , Q^∗D = (Q^∗D⁺, Q^∗D^-) , where pq ∈ D^★ and $\begin{matrix} (Q^{*} D)^{+} (pq) & = & ⋁_{rs \in D^{★}} [Q^{+} (pq, rs) \land D^{+} (rs)], \\ (Q^{*} D)^{-} (pq) & = & ⋀_{rs \in D^{★}} [Q^{-} (pq, rs) \lor D^{-} (rs)], \\ (Q_{*} D)^{+} (pq) & = & ⋀_{rs \in D^{★}} [Q^{-} (pq, rs) \lor D^{+} (rs)], \\ (Q_{*} D)^{-} (pq) & = & ⋁_{rs \in D^{★}} [Q^{+} (pq, rs) \land D^{-} (rs)] . \end{matrix}$

A pair QD = (Q_∗D, Q^∗D) is called an intuitionistic fuzzy rough relation.

Definition 3.2. An intuitionistic fuzzy rough graph $G$ on a non-empty set C^★ is a 4-ordered tuple (P, PC, Q, QD) such that

P is an intuitionistic fuzzy relation on C^★,

Q is an intuitionistic fuzzy relation on D^★⊆ C^★ × C^★,

PC = (P_∗C, P^∗C) is an intuitionistic fuzzy rough set on C^★,

QD = (Q_∗D, Q^∗D) is an intuitionistic fuzzy rough relation on C^★ .

Thus $G = (G_{*}, G^{*}) = (PC, QD)$ is an intuitionistic fuzzy rough graph, where $G_{*} = (P_{*} C, Q_{*} D)$ and $G^{*} = (P^{*} C, Q^{*} D)$ are lower and upper approximate intuitionistic fuzzy graphs of $G$ such that, ∀ c, d ∈ C^★, $\begin{matrix} (Q_{*} D)^{+} (cd) & \leq & min {(P_{*} C)^{+} (c), (P_{*} C)^{+} (d)}, \\ (Q_{*} D)^{-} (cd) & \leq & max {(P_{*} C)^{-} (c), (P_{*} C)^{-} (d)}, \\ (Q^{*} D)^{+} (cd) & \leq & min {(P^{*} C)^{+} (c), (P^{*} C)^{+} (d)}, \\ (Q^{*} D)^{-} (cd) & \leq & max {(P^{*} C)^{-} (c), (P^{*} C)^{-} (d)} . \end{matrix}$

Example 3.1. Let C = {(c₁, 0.6, 0.1) , (c₂, 0.1, 0.3) , (c₃, 0.4, 0.4) , c₄, 0.3, 0.5) , (c₅, 0.0, 0.0)} be an intuitionistic fuzzy set on C^★ = {c₁, c₂, c₃, c₄, c₅} and P an intuitionistic fuzzy relation on C^★ as given in Table 1. PC = (P_∗C, P^∗C) is an intuitionistic fuzzy rough set, where P_∗C and P^∗C are lower and upper approximations of C, respectively, given below: $\begin{matrix} P^{*} C & = & {\frac{(0.6, 0.1)}{c_{1}}, \frac{(0.4, 0.1)}{c_{2}}, \frac{(0.6, 0.2)}{c_{3}} \frac{(0.5, 0.2)}{c_{4}}, \frac{(0.3, 0.3)}{c_{5}}}, \\ P_{*} C & = & {\frac{(0.1, 0.4)}{c_{1}}, \frac{(0.1, 0.5)}{c_{2}}, \frac{(0.3, 0.2)}{c_{3}} \frac{(0.2, 0.4)}{c_{4}}, \frac{(0.1, 0.5)}{c_{5}}} . \end{matrix}$

Table 1
Intuitionistic fuzzy relation on C^★

P c ₁ c ₂ c ₃ c ₄ c ₅

c ₁ (0.6,0.1) (0.5,0.2) (0.4,0.3) (0.3,0.2) (0.9,0.1)

c ₂ (0.4,0.2) (0.9,0.1) (1.0,0.0) (0.7,0.1) (0.1,0.1)

c ₃ (0.8,0.1) (0.1,0.8) (0.0,0.0) (0.2,0.0) (0.3,0.3)

c ₄ (0.5,0.1) (0.6,0.2) (0.6,0.4) (0.0,0.4) (0.0,0.9)

c ₅ (0.3,0.4) (0.1,0.0) (0.2,0.3) (0.6,0.1) (0.0,1.0)

P	c ₁	c ₂	c ₃	c ₄	c ₅
c ₁	(0.6,0.1)	(0.5,0.2)	(0.4,0.3)	(0.3,0.2)	(0.9,0.1)
c ₂	(0.4,0.2)	(0.9,0.1)	(1.0,0.0)	(0.7,0.1)	(0.1,0.1)
c ₃	(0.8,0.1)	(0.1,0.8)	(0.0,0.0)	(0.2,0.0)	(0.3,0.3)
c ₄	(0.5,0.1)	(0.6,0.2)	(0.6,0.4)	(0.0,0.4)	(0.0,0.9)
c ₅	(0.3,0.4)	(0.1,0.0)	(0.2,0.3)	(0.6,0.1)	(0.0,1.0)

Let D^★ = {c₁c₂, c₂c₅, c₃c₁, c₄c₃, c₅c₄} ⊆ C^★ × C^★ and Q be an intuitionistic fuzzy relation on D^★ defined in Table 2. Let D = {(c₁c₂, 0.3, 0.1) , (c₂c₅, 0.2, 0.3) , c₃c₁, 0.5, 0.1) , (c₄c₃, 0.4, 0.1) , (c₅c₄, 0.1, 0.3)} be an intuitionistic fuzzy set on D^★. Then, the upper and lower approximations of D are calculated as: $\begin{matrix} Q_{*} D & = & {\frac{(0.4, 0.1)}{c_{1} c_{2}}, \frac{(0.3, 0.1)}{c_{2} c_{5}}, \frac{(0.3, 0.1)}{c_{3} c_{1}}, \frac{(0.3, 0.1)}{c_{4} c_{3}}, \frac{(0.2, 0.1)}{c_{5} c_{4}}}, \\ Q^{*} D & = & {\frac{(0.1, 0.3)}{c_{1} c_{2}}, \frac{(0.1, 0.1)}{c_{2} c_{5}}, \frac{(0.1, 0.3)}{c_{3} c_{1}}, \frac{(0.1, 0.1)}{c_{4} c_{3}}, \frac{(0.1, 0.1)}{c_{5} c_{4}}} . \end{matrix}$

Table 2

Intuitionistic fuzzy relation on D^★

Q	c ₁ c ₂	c ₂ c ₅	c ₃ c ₁	c ₄ c ₃	c ₅ c ₄
c ₁ c ₂	(0.4,0.0)	(0.1,0.1)	(0.4,0.2)	(0.3,0.2)	(0.7,0.1)
c ₂ c ₅	(0.1,0.0)	(0.0,0.6)	(0.3,0.3)	(0.1,0.0)	(0.1,0.0)
c ₃ c ₁	(0.3,0.1)	(0.1,0.1)	(0.0,0.0)	(0.2,0.0)	(0.3,0.1)
c ₄ c ₃	(0.1,0.0)	(0.0,0.3)	(0.3,0.2)	(0.0,0.2)	(0.0,0.0)
c ₅ c ₄	(0.3,0.0)	(0.0,0.0)	(0.2,0.1)	(0.6,0.3)	(0.0,1.0)

Hence, (Q_∗D, Q^∗D) is an intuitionistic fuzzy rough relation on C^★. Thus, $G_{*} = (P_{*} C, Q_{*} D)$ and $G^{*} = (P_{*} C, Q_{*} D)$ are intuitionistic fuzzy graphs as shown in Fig. 1.

Fig.1

Intuitionistic Fuzzy Rough Graph $G = (G_{*}, G^{*})$ .

Now the algebraic operations are presented on intuitionistic fuzzy rough graphs.

Definition 3.3. The lexicographic product of $G_{1}$ and $G_{2}$ is an intuitionistic fuzzy rough graph $G = G_{1} ⊙ G_{2} = ({G_{1}}_{*} ⊙ {G_{2}}_{*}, G_{1}^{*} ⊙ G_{2}^{*}),$ where ${G_{1}}_{*} ⊙ {G_{2}}_{*} = (P_{*} C_{1} ⊙ P_{*} C_{2}, Q_{*} D_{1} ⊙ Q_{*} D_{2})$ and $G_{1}^{*} ⊙ G_{2}^{*} = (P^{*} C_{1} ⊙ P^{*} C_{2}, Q^{*} D_{1} ⊙ Q^{*} D_{2})$ are intuitionistic fuzzy graphs, respectively, such that

$(P_{*} C_{1}^{+} ⊙ P_{*} C_{2}^{+}) (p_{1}, p_{2}) = min {(P_{*} C_{1})^{+} (p_{1}), (P_{*} C_{2})^{+} (p_{2})},$ $(P_{*} C_{1}^{-} ⊙ P_{*} C_{2}^{-}) (p_{1}, p_{2}) = max {(P_{*} C_{1})^{-} (p_{1}), (P_{*} C_{2})^{-} (p_{2})},$ $(P^{*} C_{1}^{+} ⊙ P^{*} C_{2}^{+}) (p_{1}, p_{2}) = min {(P^{*} C_{1})^{+} (p_{1}), (P^{*} C_{2})^{+} (p_{2})},$ $(P^{*} C_{1}^{-} ⊙ P^{*} C_{2}^{-}) (p_{1}, p_{2}) = max {(P^{*} C_{1})^{-} (p_{1}), (P^{*} C_{2})^{-} (p_{2})}, \forall (p_{1}, p_{2}) \in P C_{1} ⊙ P C_{2}$ .

$(Q_{*} D_{1}^{+} ⊙ Q_{*} D_{2}^{+}) ((x, p_{2}) (x, q_{2})) = min {(P_{*} C_{1})^{+} (x), (Q_{*} D_{2})^{+} (p_{2} q_{2})},$ $(Q_{*} D_{1}^{-} ⊙ Q_{*} D_{2}^{-}) ((x, p_{2}) (x, q_{2})) = max {(P_{*} C_{1})^{-} (x), (Q_{*} D_{2})^{-} (p_{2} q_{2})},$ $(Q^{*} D_{1}^{+} ⊙ Q^{*} D_{2}^{+}) ((x, p_{2}) (x, q_{2})) = min {(P^{*} C_{1})^{+} (x), (Q^{*} D_{2})^{+} (p_{2} q_{2})},$ $(Q^{*} D_{1}^{-} ⊙ Q^{*} D_{2}^{-}) ((x, p_{2}) (x, q_{2})) = max {(P^{*} C_{1})^{-} (x), (Q^{*} D_{2})^{-} (p_{2} q_{2})}, \forall p_{2} q_{2} \in Q D_{2}, x \in P C_{1}$ .

$(Q_{*} D_{1}^{+} ⊙ Q_{*} D_{2}^{+}) ((p_{1}, p_{2}) (q_{1}, q_{2})) = min {(Q_{*} D_{1})^{+} (p_{1} q_{1}), (Q_{*} D_{2})^{+} (p_{2} q_{2})},$ $(Q_{*} D_{1}^{-} ⊙ Q_{*} D_{2}^{-}) ((p_{1}, p_{2}) (q_{1}, q_{2})) = max {(Q_{*} D_{1})^{-} (p_{1} q_{1}), (Q_{*} D_{2})^{-} (p_{2} q_{2})},$ $(Q^{*} D_{1}^{+} ⊙ Q^{*} D_{2}^{+}) ((p_{1}, p_{2}) (q_{1}, q_{2})) = min {(Q^{*} D_{1})^{+} (p_{1} q_{1}), (Q^{*} D_{2})^{+} (p_{2} q_{2})},$ $(Q^{*} D_{1}^{-} ⊙ Q^{*} D_{2}^{-}) ((p_{1}, p_{2}) (q_{1}, q_{2})) = max {(Q^{*} D_{1})^{-} (p_{1} q_{1}), (Q^{*} D_{2})^{-} (p_{2} q_{2})}, \forall p_{1} q_{1} \in {QD}_{1}, p_{2} q_{2} \in {QD}_{2} .$

Example 3.2. Let C^★ = {p₁, p₂, p₃} be a set. Let $G_{1} = ({G_{1}}_{*}, G_{1}^{*})$ and $G_{2} = ({G_{2}}_{*}, G_{2}^{*})$ be two intuitionistic fuzzy rough graphs on C^★, where ${G_{1}}_{*} = (P_{*} C_{1}, Q_{*} D_{1})$ and $G_{1}^{*} = (P^{*} C_{1}, Q^{*} D_{1})$ are intuitionistic fuzzy graphs as shown in Fig. 2. Thus

${G_{2}}_{*} = (P_{*} C_{2}, Q_{*} D_{2})$ and $G_{2}^{*} = (P^{*} C_{2}, Q^{*} D_{2})$ are intuitionistic fuzzy graphs as shown in Fig. 3. The lexicographic product of $G_{1}$ and $G_{2}$ is $G = G_{1} ⊙ G_{2} = ({G_{1}}_{*} ⊙ {G_{2}}_{*}, G_{1}^{*} ⊙ G_{2}^{*})$ , where ${G_{1}}_{*} ⊙ {G_{2}}_{*} = (P_{*} C_{1} ⊙ P_{*} C_{2}, Q_{*} D_{1} ⊙ Q_{*} D_{2})$ and $G_{1}^{*} ⊙ G_{2}^{*} = (P^{*} C_{1} ⊙ P^{*} C_{2}, Q^{*} D_{1} ⊙ Q^{*} D_{2})$ are intuitionistic fuzzy graphs as shown in the Fig. 4. Hence $G_{1} ⊙ G_{2}$ is an intuitionistic fuzzy rough graph.

Fig.2

Intuitionistic Fuzzy Rough graph $G_{1} = ({G_{1}}_{*}, G_{1}^{*})$ .

Fig.3

Intuitionistic Fuzzy Rough graph $G_{2} = ({G_{2}}_{*}, G_{2}^{*})$ .

Fig.4

Intuitionistic fuzzy Rough Graph $G_{1}$ ⊙ $G_{2}$ = $({G_{1}}_{*}$ ⊙ ${G_{2}}_{*},$ $G_{1}^{*}$ ⊙ $G_{2}^{*})$ .

Theorem 3.1. The lexicographic product of two intuitionistic fuzzy rough graphs is an intuitionistic fuzzy rough graph.

Proof. Let $G_{1} = ({G_{1}}_{*}, G_{1}^{*})$ and $G_{2} = ({G_{2}}_{*}, G_{2}^{*})$ be two intuitionistic fuzzy rough graphs. Let $G = G_{1} ⊙ G_{2} = ({G_{1}}_{*} ⊙ {G_{2}}_{*}, G_{1}^{*} ⊙ G_{2}^{*})$ be the lexicographic product of $G_{1}$ and $G_{2}$ , where ${G_{1}}_{*} ⊙ {G_{2}}_{*} = (P_{*} C_{1} ⊙ P_{*} C_{2}, Q_{*} D_{1} ⊙ Q_{*} D_{2})$ and $G_{1}^{*} ⊙ G_{2}^{*} = (P^{*} C_{1} ⊙ P^{*} C_{2}, Q^{*} D_{1} ⊙ Q^{*} D_{2}) .$ For proving $G = G_{1} ⊙ G_{2}$ is an intuitionistic fuzzy rough graph. It is enough to show that Q_∗D₁ ⊙ Q_∗D₂ and Q^∗D₁ ⊙ Q^∗D₂ are intuitionistic fuzzy relations on P_∗C₁ ⊙ P_∗C₂ and P^∗C₁ ⊙ P^∗C₂, respectively. First, Q_∗D₁ ⊙ Q_∗D₂ is shown as an intuitionistic fuzzy relation on P_∗C₁ ⊙ P_∗C₂.

If x ∈ P_∗C₁, p₂q₂ ∈ Q_∗D₂ then, $\begin{matrix} (Q_{*} D_{1}^{+} ⊙ Q_{*} D_{2}^{+}) ((x, p_{2}) (x, q_{2})) \\ = (P_{*} C_{1})^{+} (x) \land (Q_{*} D_{2})^{+} (p_{2} q_{2}) \\ \leq (P_{*} C_{1})^{+} (x) \land {(P_{*} C_{2})^{+} (p_{2}) \land (P_{*} C_{2})^{+} (q_{2})} \\ = {(P_{*} C_{1})^{+} (x) \land (P_{*} C_{2})^{+} (p_{2})} \\ \land {(P_{*} C_{1})^{+} (x) \land (P_{*} C_{2})^{+} (q_{2})} \\ = (P_{*} C_{1}^{+} ⊙ P_{*} C_{2}^{+}) (x, p_{2}) \\ \land (P_{*} C_{1}^{+} ⊙ P_{*} C_{2}^{+}) (x, q_{2}) . \end{matrix}$

Similarly, we can prove that, $\begin{matrix} (Q_{*} D_{1}^{-} ⊙ Q_{*} D_{2}^{-}) ((x, p_{2}) (x, q_{2})) \\ \leq (P_{*} C_{1}^{-} ⊙ P_{*} C_{2}^{-}) (x, p_{2}) \\ \lor (P_{*} C_{1}^{-} ⊙ P_{*} C_{2}^{-}) (x, q_{2}) . \end{matrix}$

If p₁q₁ ∈ Q_∗D₁, p₂q₂ ∈ Q_∗D₂ then, $\begin{matrix} (Q_{*} D_{1}^{+} ⊙ Q_{*} D_{2}^{+}) ((p_{1}, p_{2}) (q_{1}, q_{2})) \\ = (Q_{*} D_{1})^{+} (p_{1} q_{1}) \land (Q_{*} D_{2})^{+} (p_{2} q_{2}) \\ \leq {(P_{*} C_{1})^{+} (p_{1}) \land (P_{*} C_{1})^{+} (q_{1})} \\ \land {(P_{*} C_{2})^{+} (p_{2}) \land (P_{*} C_{2})^{+} (q_{2})} \\ = {(P_{*} C_{1})^{+} (p_{1}) \land (P_{*} C_{2})^{+} (p_{2})} \\ \land {(P_{*} C_{1})^{+} (q_{1}) \land (P_{*} C_{2})^{+} (q_{2})} \\ = (P_{*} C_{1}^{+} ⊙ P_{*} C_{2}^{+}) (p_{1}, p_{2}) \\ \land (P_{*} C_{1}^{+} ⊙ P_{*} C_{2}^{+}) (q_{1}, q_{2}) . \end{matrix}$

Similarly, it can be proved that $\begin{matrix} (Q_{*} D_{1}^{-} ⊙ Q_{*} D_{2}^{-}) ((p_{1}, p_{2}) (q_{1}, q_{2})) \\ \leq (P_{*} C_{1}^{-} ⊙ P_{*} C_{2}^{-}) (p_{1}, p_{2}) \\ \lor (P_{*} C_{1}^{-} ⊙ P_{*} C_{2}^{-}) (x, q_{2}) . \end{matrix}$

Thus, Q_∗D₁ ⊙ Q_∗D₂ is an intuitionistic fuzzy relation on P_∗C₁ ⊙ P_∗C₂. Similarly, Q^∗D₁ ⊙ Q^∗D₂ is an intuitionistic fuzzy relation on P^∗C₁ ⊙ P^∗C₂. Hence, $G$ is an intuitionistic fuzzy rough graph.

Definition 3.4. The strong product of $G_{1}$ and $G_{2}$ is an intuitionistic fuzzy rough graph $G = G_{1} \otimes G_{2} = ({G_{1}}_{*} \otimes {G_{2}}_{*}, G_{1}^{*} \otimes G_{2}^{*}),$ where ${G_{1}}_{*} \otimes {G_{2}}_{*} = (P_{*} C_{1} \otimes P_{*} C_{2}, Q_{*} D_{1} \otimes Q_{*} D_{2})$ and $G_{1}^{*} \otimes G_{2}^{*} = (P^{*} C_{1} \otimes P^{*} C_{2}, Q^{*} D_{1} \otimes Q^{*} D_{2})$ are intuitionistic fuzzy graphs, respectively, such that

$(P_{*} C_{1}^{+} \otimes P_{*} C_{2}^{+}) (p_{1}, p_{2}) = min {(P_{*} C_{1})^{+} (p_{1}), (P_{*} C_{2})^{+} (p_{2})},$ $(P_{*} C_{1}^{-} \otimes P_{*} C_{2}^{-}) (p_{1}, p_{2}) = max {(P_{*} C_{1})^{-} (p_{1}), (P_{*} C_{2})^{-} (p_{2})},$ $(P^{*} C_{1}^{+} \otimes P^{*} C_{2}^{+}) (p_{1}, p_{2}) = min {(P^{*} C_{1})^{+} (p_{1}), (P^{*} C_{2})^{+} (p_{2})},$ $(P^{*} C_{1}^{-} \otimes P^{*} C_{2}^{-}) (p_{1}, p_{2}) = max {(P^{*} C_{1})^{-} (p_{1}), (P^{*} C_{2})^{-} (p_{2})}, \forall (p_{1}, p_{2}) \in P C_{1} \otimes P C_{2}$ .

$(Q_{*} D_{1}^{+} \otimes Q_{*} D_{2}^{+}) ((x, p_{2}) (x, q_{2})) = min {(P_{*} C_{1})^{+} (x), (Q_{*} D_{2})^{+} (p_{2} q_{2})},$ $(Q_{*} D_{1}^{-} \otimes Q_{*} D_{2}^{-}) ((x, p_{2}) (x, q_{2})) = max {(P_{*} C_{1})^{-} (x), (Q_{*} D_{2})^{-} (p_{2} q_{2})},$ $(Q^{*} D_{1}^{+} \otimes Q^{*} D_{2}^{+}) ((x, p_{2}) (x, q_{2})) = min {(P^{*} C_{1})^{+} (x), (Q^{*} D_{2})^{+} (p_{2} q_{2})},$ $(Q^{*} D_{1}^{-} \otimes Q^{*} D_{2}^{-}) ((x, p_{2}) (x, q_{2})) = max {(P^{*} C_{1})^{-} (x), (Q^{*} D_{2})^{-} (p_{2} q_{2})}, \forall p_{2} q_{2} \in Q D_{2}, x \in P C_{1}$ .

$(Q_{*} D_{1}^{+} \otimes Q_{*} D_{2}^{+}) ((p_{1}, p_{2}) (q_{1}, z)) = min {(Q_{*} D_{1})^{+} (p_{1} q_{1}), (P_{*} C_{2})^{+} (z)},$ $(Q_{*} D_{1}^{-} \otimes Q_{*} D_{2}^{-}) ((p_{1}, p_{2}) (q_{1},)) = max {(Q_{*} D_{1})^{-} (p_{1} q_{1}), (P_{*} C_{2})^{-} (z)},$ $(Q^{*} D_{1}^{+} \otimes Q^{*} D_{2}^{+}) ((p_{1}, p_{2}) (q_{1},)) = min {(Q^{*} D_{1})^{+} (p_{1} q_{1}), (P^{*} C_{2})^{+} (z)},$ $(Q^{*} D_{1}^{-} \otimes Q^{*} D_{2}^{-}) ((p_{1}, p_{2}) (q_{1}, q_{2})) = max {(Q^{*} D_{1})^{-} (p_{1} q_{1}), (P^{*} C_{2})^{-} ({zq}_{2})}, \forall p_{1} q_{1} \in {QD}_{1}, z \in {PC}_{2} .$

$(Q_{*} D_{1}^{+} \otimes Q_{*} D_{2}^{+}) ((p_{1}, p_{2}) (q_{1}, q_{2})) = min {(Q_{*} D_{1})^{+} (p_{1} q_{1}), (Q_{*} D_{2})^{+} (p_{2} q_{2})}, (Q_{*} D_{1}^{-} \otimes Q_{*} D_{2}^{-}) ((p_{1}, p_{2}) (q_{1}, q_{2})) = max {(Q_{*} D_{1})^{-} (p_{1} q_{1}), (Q_{*} D_{2})^{-} (p_{2} q_{2})}, (Q^{*} D_{1}^{+} \otimes Q^{*} D_{2}^{+}) ((p_{1}, p_{2}) (q_{1}, q_{2})) = min {(Q^{*} D_{1})^{+} (p_{1} q_{1}), (Q^{*} D_{2})^{+} (p_{2} q_{2})}, (Q^{*} D_{1}^{-} \otimes Q^{*} D_{2}^{-}) ((p_{1}, p_{2}) (q_{1}, q_{2})) = max {(Q^{*} D_{1})^{-} (p_{1} q_{1}), (Q^{*} D_{2})^{-} (p_{2} q_{2})}, \forall p_{1} q_{1} \in {QD}_{1}, p_{2} q_{2} \in {QD}_{2} .$

Example 3.3. Consider the two intuitionistic fuzzy rough graphs $G_{1}$ and $G_{2}$ as shown in Figs. 2 and 3. The strong product of $G_{1}$ and $G_{2}$ is $G = G_{1} \otimes G_{2} = ({G_{1}}_{*} \otimes {G_{2}}_{*}, G_{1}^{*} \otimes G_{2}^{*})$ , where ${G_{1}}_{*} \otimes {G_{2}}_{*} = (P_{*} C_{1} \otimes P_{*} C_{2}, Q_{*} D_{1} \otimes Q_{*} D_{2})$ and $G_{1}^{*} \otimes G_{2}^{*} = (P^{*} C_{1} \otimes P^{*} C_{2}, Q^{*} D_{1} \otimes Q^{*} D_{2})$ are intuitionistic fuzzy graphs as shown in Fig. 5.

Fig.5

Intuitionistic fuzzy Rough Graph $G_{1}$ ⊗ $G_{2} = ({G_{1}}_{*}$ ⊗ ${G_{2}}_{*}$ , $G_{1}^{*}$ ⊗ $G_{2}^{*})$ .

Theorem 3.2. The strong product of two intuitionistic fuzzy rough graphs is an intuitionistic fuzzy rough graph.

Proof. Let $G_{1} = ({G_{1}}_{*}, G_{1}^{*})$ and $G_{2} = ({G_{2}}_{*}, G_{2}^{*})$ be two intuitionistic fuzzy rough graphs. Let $G = G_{1} \otimes G_{2} = ({G_{1}}_{*} \otimes {G_{2}}_{*}, G_{1}^{*} \otimes G_{2}^{*})$ be the strong product of $G_{1}$ and $G_{2}$ , where ${G_{1}}_{*} \otimes {G_{2}}_{*} = (P_{*} C_{1} \otimes P_{*} C_{2}, Q_{*} D_{1} \otimes Q_{*} D_{2})$ and $G_{1}^{*} \otimes G_{2}^{*} = (P^{*} C_{1} \otimes P^{*} C_{2}, Q^{*} D_{1} \otimes Q_{*} D_{2}) .$ For proving $G = G_{1} \otimes G_{2}$ is an intuitionistic fuzzy rough graph. It is enough to show that Q_∗D₁ ⊗ Q_∗D₂ and Q^∗D₁ ⊗ Q^∗D₂ are intuitionistic fuzzy relations on P_∗C₁ ⊗ P_∗C₂ and P^∗C₁ ⊗ P^∗C₂, respectively. First, Q_∗D₁ ⊗ Q_∗D₂ is shown as an intuitionistic fuzzy relation on P_∗C₁ ⊗ P_∗C₂. If x ∈ P_∗C₁, p₂q₂ ∈ Q_∗D₂ then, $\begin{matrix} (Q_{*} D_{1}^{+} \otimes Q_{*} D_{2}^{+}) ((x, p_{2}) (x, q_{2})) \\ = (P_{*} C_{1})^{+} (x) \land (Q_{*} D_{2})^{+} (p_{2} q_{2}) \\ \leq (P_{*} C_{1})^{+} (x) \land {(P_{*} C_{2})^{+} (p_{2}) \land (P_{*} C_{2})^{+} (q_{2})} \\ = {(P_{*} C_{1})^{+} (x) \land (P_{*} C_{2})^{+} (p_{2})} \\ \land {(P_{*} C_{1})^{+} (x) \land (P_{*} C_{2})^{+} (q_{2})} \\ = (P_{*} C_{1}^{+} \otimes P_{*} C_{2}^{+}) (x, p_{2}) \\ \land (P_{*} C_{1}^{+} \otimes P_{*} C_{2}^{+}) (x, q_{2}) . \\ (Q_{*} D_{1}^{-} \otimes Q_{*} D_{2}^{-}) ((x, p_{2}) (x, q_{2})) \\ = (P_{*} C_{1})^{-} (x) \lor (Q_{*} D_{2})^{-} (p_{2} q_{2}) \\ \leq (P_{*} C_{1})^{-} (x) \lor {(P_{*} C_{2})^{-} (p_{2}) \lor (P_{*} C_{2})^{-} (q_{2})} \\ = {(P_{*} C_{1})^{-} (x) \lor (P_{*} C_{2})^{-} (p_{2})} \\ \lor {(P_{*} C_{1})^{-} (x) \lor (P_{*} C_{2})^{-} (q_{2})} \\ = (P_{*} C_{1}^{-} \otimes P_{*} C_{2}^{-}) (x, p_{2}) \\ \lor (P_{*} C_{1}^{-} \otimes P_{*} C_{2}^{-}) (x, q_{2}) . \end{matrix}$

Other cases can be proved similarly.

Thus, Q_∗D₁ ⊗ Q_∗D₂ is an intuitionistic fuzzy relation on P_∗C₁ ⊗ P_∗C₂. Similarly, Q^∗D₁ ⊗ Q^∗D₂ is an intuitionistic fuzzy relation on P^∗C₁ ⊗ P^∗C₂. Hence, $G$ is an intuitionistic fuzzy rough graph. □

Definition 3.5. The composition of $G_{1}$ and $G_{2}$ is an intuitionistic fuzzy rough graph $G = G_{1} [G_{2}] = ({G_{1}}_{*} \times {G_{2}}_{*}, G_{1}^{*} \times G_{2}^{*}),$ where ${G_{1}}_{*} \times {G_{2}}_{*} = (P_{*} C_{1} \times P_{*} C_{2}, Q_{*} D_{1} \times Q_{*} D_{2})$ and $G_{1}^{*} \times G_{2}^{*} = (P^{*} C_{1} \times P^{*} C_{2}, Q^{*} D_{1} \times Q^{*} D_{2})$ are intuitionistic fuzzy graphs, respectively, such that

$(P_{*} C_{1}^{+} \times P_{*} C_{2}^{+}) (p_{1}, p_{2}) = min {(P_{*} C_{1})^{+} (p_{1}), (P_{*} C_{2})^{+} (p_{2})},$ $(P_{*} C_{1}^{-} \times P_{*} C_{2}^{-}) (p_{1}, p_{2}) = max {(P_{*} C_{1})^{-} (p_{1}), (P_{*} C_{2})^{-} (p_{2})},$ $(P^{*} C_{1}^{+} \times P^{*} C_{2}^{+}) (p_{1}, p_{2}) = min {(P^{*} C_{1})^{+} (p_{1}), (P^{*} C_{2})^{+} (p_{2})},$ $(P^{*} C_{1}^{-} \times P^{*} C_{2}^{-}) (p_{1}, p_{2}) = max {(P^{*} C_{1})^{-} (p_{1}), (P^{*} C_{2})^{-} (p_{2})}, \forall (p_{1}, p_{2}) \in P C_{1} \times P C_{2}$ .

$(Q_{*} D_{1}^{+} \times Q_{*} D_{2}^{+}) ((x, p_{2}) (x, q_{2})) = min {(P_{*} C_{1})^{+} (x), (Q_{*} D_{2})^{+} (p_{2} q_{2})},$ $(Q_{*} D_{1}^{-} \times Q_{*} D_{2}^{-}) ((x, p_{2}) (x, q_{2})) = max {(P_{*} C_{1})^{-} (x), (Q_{*} D_{2})^{-} (p_{2} q_{2})},$ $(Q^{*} D_{1}^{+} \times Q^{*} D_{2}^{+}) ((x, p_{2}) (x, q_{2})) = min {(P^{*} C_{1})^{+} (x), (Q^{*} D_{2})^{+} (p_{2} q_{2})},$ $(Q^{*} D_{1}^{-} \times Q^{*} D_{2}^{-}) ((x, p_{2}) (x, q_{2})) = max {(P^{*} C_{1})^{-} (x), (Q^{*} D_{2})^{-} (p_{2} q_{2})}, \forall p_{2} q_{2} \in Q D_{2}, x \in P C_{1}$ .

$(Q_{*} D_{1}^{+} \times Q_{*} D_{2}^{+}) ((p_{1}, p_{2}) (q_{1}, z)) = min {(Q_{*} D_{1})^{+} (p_{1} q_{1}), (P_{*} C_{2})^{+} (z)},$ $(Q_{*} D_{1}^{-} \times Q_{*} D_{2}^{-}) ((p_{1}, p_{2}) (q_{1},)) = max {(Q_{*} D_{1})^{-} (p_{1} q_{1}), (P_{*} C_{2})^{-} (z)},$ $(Q^{*} D_{1}^{+} \times Q^{*} D_{2}^{+}) ((p_{1}, p_{2}) (q_{1},)) = min {(Q^{*} D_{1})^{+} (p_{1} q_{1}), (P^{*} C_{2})^{+} (z)},$ $(Q^{*} D_{1}^{-} \times Q^{*} D_{2}^{-}) ((p_{1}, p_{2}) (q_{1}, q_{2})) = max {(Q^{*} D_{1})^{-} (p_{1} q_{1}), (P^{*} C_{2})^{-} ({zq}_{2})}, \forall p_{1} q_{1} \in {QD}_{1}, z \in {PC}_{2} .$

$(Q_{*} D_{1}^{+} \times Q_{*} D_{2}^{+}) ((p_{1}, p_{2}) (q_{1}, q_{2})) = min {(Q_{*} D_{1})^{+} (p_{1} q_{1}), (Q_{*} D_{2})^{+} (p_{2} q_{2})}, (Q_{*} D_{1}^{-} \times Q_{*} D_{2}^{-}) ((p_{1}, p_{2}) (q_{1}, q_{2})) = max {(Q_{*} D_{1})^{-} (p_{1} q_{1}), (Q_{*} D_{2})^{-} (p_{2} q_{2})}, (Q^{*} D_{1}^{+} \times Q^{*} D_{2}^{+}) ((p_{1}, p_{2}) (q_{1}, q_{2})) = min {(Q^{*} D_{1})^{+} (p_{1} q_{1}), (Q^{*} D_{2})^{+} (p_{2} q_{2})}, (Q^{*} D_{1}^{-} \times Q^{*} D_{2}^{-}) ((p_{1}, p_{2}) (q_{1}, q_{2})) = max {(Q^{*} D_{1})^{-} (p_{1} q_{1}), (Q^{*} D_{2})^{-} (p_{2} q_{2})}, \forall p_{1} q_{1} \in Q D_{1}, p_{2} \neq q_{2} .$

Example 3.4. Consider the two intuitionistic fuzzy rough graphs $G_{1}$ and $G_{2}$ as shown in Figs. 2 and 3. The composition of $G_{1}$ and $G_{2}$ is $G = G_{1} \times G_{2} = ({G_{1}}_{*} \times {G_{2}}_{*}, G_{1}^{*} \times G_{2}^{*})$ , where ${G_{1}}_{*} \times {G_{2}}_{*} = (P_{*} C_{1} \times P_{*} C_{2}, Q_{*} D_{1} \times Q_{*} D_{2})$ and $G_{1}^{*} \times G_{2}^{*} = (P^{*} C_{1} \times P^{*} C_{2}, Q^{*} D_{1} \times Q^{*} D_{2})$ are intuitionistic fuzzy graphs as shown in Fig. 6.

Fig.6

Intuitionistic fuzzy Rough Graph $G_{1}$ × $G_{2} = ({G_{1}}_{*}$ × ${G_{2}}_{*}$ , $G_{1}^{*}$ × $G_{2}^{*})$ .

Remark 3.1. In Definition 3.4, if there is an edge between ‘p₁p₂’ and ‘q₁q₂’ then there must be an edge between p₁ and q₁ in approximations graph $G_{1}$ and p₂ and q₂ in approximation graph $G_{2}$ . Whereas in Definition 3.5, if there is an edge between ‘p₁p₂’ and ‘q₁q₂’ then there must be an edge between p₁ and q₁ in approximations graph $G_{1}$ but not necessarily an edge between p₂ and q₂ in graph $G_{2}$ .

Theorem 3.3. The composition of two intuitionistic fuzzy rough graphs is an intuitionistic fuzzy rough graph.

Proof. Let $G_{1} = ({G_{1}}_{*}, G_{1}^{*})$ and $G_{2} = ({G_{2}}_{*}, G_{2}^{*})$ be two intuitionistic fuzzy rough graphs. Let $G = G_{1} \times G_{2} = ({G_{1}}_{*} \times {G_{2}}_{*}, G_{1}^{*} \times G_{2}^{*})$ be the composition of $G_{1}$ and $G_{2}$ , where ${G_{1}}_{*} \times {G_{2}}_{*} = (P_{*} C_{1} \times P_{*} C_{2}, Q_{*} D_{1} \times Q_{*} D_{2})$ and $G_{1}^{*} \times G_{2}^{*} = (P^{*} C_{1} \times P^{*} C_{2}, Q^{*} D_{1} \times Q^{*} D_{2}) .$ For proving $G = G_{1} \times G_{2}$ is an intuitionistic fuzzy rough graph. It is enough to show that Q_∗D₁ × Q_∗D₂ and Q^∗D₁ × Q^∗D₂ are intuitionistic fuzzy relations on P_∗C₁ × P_∗C₂ and P^∗C₁ × P^∗C₂, respectively. First, Q_∗D₁ × Q_∗D₂ is shown as an intuitionistic fuzzy relation on P_∗C₁ × P_∗C₂ . If x ∈ P_∗C₁, p₂q₂ ∈ Q_∗D₂ then, $\begin{matrix} (Q_{*} D_{1}^{+} ⊙ Q_{*} D_{2}^{+}) ((x, p_{2}) (x, q_{2})) \\ = (P_{*} C_{1})^{+} (x) \land (Q_{*} D_{2})^{+} (p_{2} q_{2}) \\ \leq (P_{*} C_{1})^{+} (x) \land {(P_{*} C_{2})^{+} (p_{2}) \land (P_{*} C_{2})^{+} (q_{2})} \\ = {(P_{*} C_{1})^{+} (x) \land (P_{*} C_{2})^{+} (p_{2})} \\ \land {(P_{*} C_{1})^{+} (x) \land (P_{*} C_{2})^{+} (q_{2})} \\ = (P_{*} C_{1}^{+} ⊙ P_{*} C_{2}^{+}) (x, p_{2}) \\ \land (P_{*} C_{1}^{+} ⊙ P_{*} C_{2}^{+}) (x, q_{2}) . \\ (Q_{*} D_{1}^{-} ⊙ Q_{*} D_{2}^{-}) ((x, p_{2}) (x, q_{2})) \\ = (P_{*} C_{1})^{-} (x) \lor (Q_{*} D_{2})^{-} (p_{2} q_{2}) \\ \leq (P_{*} C_{1})^{-} (x) \lor {(P_{*} C_{2})^{-} (p_{2}) \lor (P_{*} C_{2})^{-} (q_{2})} \\ = {(P_{*} C_{1})^{-} (x) \lor (P_{*} C_{2})^{-} (p_{2})} \\ \lor {(P_{*} C_{1})^{-} (x) \lor (P_{*} C_{2})^{-} (q_{2})} \\ = (P_{*} C_{1}^{-} ⊙ P_{*} C_{2}^{-}) (x, p_{2}) \\ \lor (P_{*} C_{1}^{-} ⊙ P_{*} C_{2}^{-}) (x, q_{2}) . \end{matrix}$

Similarly, we can prove other cases.

Thus, Q_∗D₁ × Q_∗D₂ is an intuitionistic fuzzy relation on P_∗C₁ × P_∗C₂. Similarly, Q^∗D₁ × Q^∗D₂ is an intuitionistic fuzzy relation on P^∗C₁ × P^∗C₂. Hence, $G$ is an intuitionistic fuzzy rough graph. □

Definition 3.6. Let $G$ be an intuitionistic fuzzy rough graph. The μ-complement of $G$ is defined as $G^{μ} = ((PC)^{μ}, (QD)^{μ})$ , where (PC) ^μ = PC = (P_∗C, P^∗C) and (QD) ^μ = ((Q_∗D) ^μ, (Q^∗D) ^μ) and for all a, b ∈ C^★ such that

When (QD) (ab) ≠0, $\begin{matrix} {(Q_{*} D)^{μ}}^{+} (ab) & = & (P_{*} C)^{+} (a) \land (P_{*} C)^{+} (b) \\ - (Q_{*} D)^{+} (ab), \\ {(Q_{*} D)^{μ}}^{-} (ab) & = & (P_{*} C)^{-} (a) \lor (P_{*} C)^{-} (b) \\ - (Q_{*} D)^{-} (ab), \\ {(Q^{*} D)^{μ}}^{+} (ab) & = & (P^{*} C)^{+} (a) \land (P^{*} C)^{+} (b) \\ - (Q^{*} D)^{+} (ab), \\ {(Q^{*} D)^{μ}}^{-} (ab) & = & (P^{*} C)^{-} (a) \lor (P^{*} C)^{-} (b) \\ - (Q^{*} D)^{-} (ab) . \end{matrix}$

When (QD) (ab) =0, $\begin{matrix} {(Q_{*} D)^{μ}}^{+} (ab) & = & 0, {(Q_{*} D)^{μ}}^{-} (ab) = 0, \\ {(Q^{*} D)^{μ}}^{+} (ab) & = & 0, {(Q^{*} D)^{μ}}^{-} (ab) = 0 . \end{matrix}$

Example 3.5. Consider an intuitionistic fuzzy rough graph $G$ as shown in Fig. 1. The μ-complement of $G$ is $G^{μ} = (G_{*}^{μ}, {G^{μ}}^{*})$ , where $G_{*}^{μ} = ((P_{*} C)^{μ}, (Q_{*} D)^{μ}), {G^{μ}}^{*} = ((P^{*} C)^{μ}, (Q^{*} D)^{μ})$ are intuitionistic fuzzy graphs as shown in Fig. 7”.

Fig.7

Intuitionistic Fuzzy Rough Graph $G^{μ} = ({G_{*}}^{μ}, {G^{*}}^{μ})$ .

Fig.8

Strong Intuitionistic Fuzzy Rough Graph $G = (PC, QD)$ .

Definition 3.7. An intuitionistic fuzzy rough graph $G$ is called strong intuitionistic fuzzy rough graph if ∀ pq ∈ D, $\begin{matrix} (Q_{*} D)^{+} (pq) & = & min {(P_{*} C)^{+} (p), (P_{*} C)^{+} (q)}; \\ (Q_{*} D)^{-} (pq) & = & max {(P_{*} C)^{-} (p), (P_{*} C)^{-} (q)}; \\ (Q^{*} D)^{+} (pq) & = & min {(P^{*} C)^{+} (p), (P^{*} C)^{+} (q)}; \\ (Q^{*} D)^{-} (pq) & = & max {(P^{*} C)^{-} (p), (P^{*} C)^{-} (q)} . \end{matrix}$

Example 3.6. Consider a graph $G$ such that C^★ = {p, q, r} and D^★ = {pq, qr, rp} . Let C be an intuitionistic fuzzy rough subset of C^★ and D an intuitionistic fuzzy rough subset of D^★ defined in the Table 3.

Table 3

Intuitionistic Fuzzy Rough Sets on C^★ and D^★

C ^★	P^∗ (C)	P_∗ (C)
p	(0.9, 0.1)	(0.8, 0.1)
q	(0.6, 0.1)	(0.2, 0.3)
r	(0.4, 0.1)	(0.3, 0.6)
D ^★	Q^∗ (D)	Q_∗ (D)
pq	(0.6, 0.1)	(0.2, 0.3)
qr	(0.4, 0.1)	(0.2, 0.6)
rp	(0.4, 0.1)	(0.3, 0.6)

Hence, $G = (PC, QD)$ is a strong intuitionistic fuzzy rough graph.

Definition 3.8. An intuitionistic fuzzy rough graph $G$ is called complete intuitionistic fuzzy rough graph if ∀ p, q ∈ C^★, $\begin{matrix} (Q_{*} D)^{+} (pq) & = & min {(P_{*} C)^{+} (p), (P_{*} C)^{+} (q)}; \\ (Q_{*} D)^{-} (pq) & = & max {(P_{*} C)^{-} (p), (P_{*} C)^{-} (q)}; \\ (Q^{*} D)^{+} (pq) & = & min {(P^{*} C)^{+} (p), (P^{*} C)^{+} (q)}; \\ (Q^{*} D)^{-} (pq) & = & max {(P^{*} C)^{-} (p), (P^{*} C)^{-} (q)} . \end{matrix}$

Example 3.7. Consider a graph $G$ such that C^★ = {p₁, p₂, p₃, p₄, p₅} and D^★ = {p₁p₂, p₁p₃, p₁p₄, p₁p₅, p₂p₁, p₂p₃, p₂p₄, p₂p₅, p₃p₁, p₃p₂, p₃p₄, p₃p₅, p₄p₁, p₄p₂, p₄p₃, p₄p₅, p₅p₁, p₅p₂, p₅p₃, p₅p₄} . Let $G$ be an intuitionistic fuzzy rough graph of C^★ as shown in Fig. 9.

Fig.9

Complete intuitionistic Fuzzy Rough Graph $G = (PC, QD)$ .

Definition 3.9. An intuitionistic fuzzy rough graph $G$ is isolated, if ∀ p, q ∈ PC $\begin{matrix} (Q_{*} D)^{+} (pq) & = & 0; (Q_{*} D)^{-} (pq) = 0; \\ (Q^{*} D)^{+} (pq) & = & 0; (Q^{*} D)^{-} (pq) = 0 . \end{matrix}$

Definition 3.10. A strong intuitionistic fuzzy rough graph $G$ is called self complementary if $G^{c} \approx G .$

Example 3.8. The graph $G$ in Fig. 9 is not only strong but also self complementary. Moreover, The complement and μ-complement of the graph are isolated as shown in Fig. 10.

Fig.10

Complement and μ-complement of Intuitionistic Fuzzy Rough Graph $G = (PC, QD)$ .

Theorem 3.4. Let $G$ be an intuitionistic fuzzy rough graph then $G^{μ}$ has isolated nodes if and only if $G$ is a strong intuitionistic fuzzy rough graph.

Proof. Let $G$ be a strong intuitionistic fuzzy rough graph. When ∀ pq ∈ D^★,

$\begin{matrix} (Q_{*} D)^{+} (pq) & = & (P_{*} C)^{+} (p) \land (P_{*} C)^{+} (q), \\ (Q_{*} D)^{-} (pq) & = & (P_{*} C)^{-} (p) \lor (P_{*} C)^{-} (q), \\ (Q^{*} D)^{+} (pq) & = & (P^{*} C)^{+} (p) \land (P^{*} C)^{+} (q), \\ (Q^{*} D)^{-} (pq) & = & (P^{*} C)^{-} (p) \lor (P^{*} C)^{-} (q) . \end{matrix}$ (1)

Otherwise, $\begin{matrix} {(Q_{*} D)^{μ}}^{+} (ab) & = & 0, {(Q_{*} D)^{μ}}^{-} (ab) = 0, \\ {(Q^{*} D)^{μ}}^{+} (ab) & = & 0, {(Q^{*} D)^{μ}}^{-} (ab) = 0 . \end{matrix}$

By applying Definition 3.6 Equation 1. Therefore, $G^{μ}$ is an isolated intuitionistic fuzzy rough graph. Conversely, assume $G^{μ}$ is an isolated intuitionistic fuzzy rough graph. It follows that $(QD)^{μ} (pq) = 0, \forall pq \in C^{★} \times C^{★} .$

By Definition 3.6, for an edge pq ∈ D^★, $\begin{matrix} {(Q_{*} D)^{μ}}^{+} (pq) & = & (P_{*} C)^{+} (p) \land (P_{*} C)^{+} (q) \\ - (Q_{*} D)^{+} (pq), \\ {(Q_{*} D)^{μ}}^{-} (pq) & = & (P_{*} C)^{-} (p) \lor (P_{*} C)^{-} (q) \\ - (Q_{*} D)^{-} (pq), \\ {(Q^{*} D)^{μ}}^{+} (pq) & = & (P^{*} C)^{+} (p) \land (P^{*} C)^{+} (q) \\ - (Q^{*} D)^{+} (pq), \\ ({Q^{*} D)^{μ}}^{-} (pq) & = & (P^{*} C)^{-} (p) \lor (P^{*} C)^{-} (q) \\ - (Q^{*} D)^{-} (pq) . \end{matrix}$

Using Definition 3.9, ∀ pq ∈ D^★, $\begin{matrix} (P_{*} C)^{+} (p) \land (P_{*} C)^{+} (q) - (Q_{*} D)^{+} (pq) & = & 0, \\ (P_{*} C)^{-} (p) \lor (P_{*} C)^{-} (q) - (Q_{*} D)^{-} (pq) & = & 0, \\ (P^{*} C)^{+} (p) \land (P^{*} C)^{+} (q) - (Q^{*} D)^{+} (pq) & = & 0, \\ (P^{*} C)^{-} (p) \lor (P^{*} C)^{-} (q) - (Q^{*} D)^{-} (pq) & = & 0 . \end{matrix}$

Hence, $G^{μ}$ has isolated nodes if and only if $G$ is a strong intuitionistic fuzzy rough graph. □

Definition 3.11. Let $G_{1}$ and $G_{2}$ are intuitionistic fuzzy rough graph. A homomorphism of an intuitionistic fuzzy rough graphs $ℙ : G_{1} \to G_{2}$ is a map $ℙ : C_{1}^{★} \to C_{2}^{★}$ which satisfies $(P_{1} C_{1}) (a) \leq (P_{2} C_{2}) (ℙ (a)), \forall a \in C_{1}^{★}$ and $(Q_{1} D_{1}) (ab) \leq (Q_{2} D_{2}) (ℙ (a) ℙ (b)), \forall ab \in D_{1}^{★},$ i.e., $\begin{matrix} ({P_{1}}_{*} C_{1})^{+} (a) & \leq & ({P_{2}}_{*} C_{2})^{+} (ℙ (a)), \\ ({P_{1}}_{*} C_{1})^{-} (a) & \geq & ({P_{2}}_{*} C_{2})^{-} (ℙ (a)), \\ ({P_{1}}^{*} C_{1})^{+} (a) & \leq & ({P_{2}}^{*} C_{2})^{+} (ℙ (a)), \\ ({P_{1}}^{*} C_{1})^{-} (a) & \geq & ({P_{2}}^{*} C_{2})^{-} (ℙ (a)), \\ ({Q_{1}}_{*} D_{1})^{+} (ab) & \leq & ({Q_{2}}_{*} D_{2})^{+} (ℙ (a) ℙ (b)), \\ ({Q_{1}}_{*} D_{1})^{-} (ab) & \geq & ({Q_{2}}_{*} D_{2})^{-} (ℙ (a) ℙ (b)), \\ ({Q_{1}}^{*} D_{1})^{+} (ab) & \leq & ({Q_{2}}^{*} D_{2})^{+} (ℙ (a) ℙ (b)), \\ ({Q_{1}}^{*} D_{1})^{-} (ab) & \geq & ({Q_{2}}^{*} D_{2})^{-} (ℙ (a) ℙ (b)) . \end{matrix}$

Definition 3.12. Let $G_{1}$ and $G_{2}$ are intuitionistic fuzzy rough graph. A isomorphism of an intuitionistic fuzzy rough graphs $ℙ : G_{1} \to G_{2}$ is a map $ℙ : C_{1}^{★} \to C_{2}^{★}$ which satisfies $(P_{1} C_{1}) (a) \leq (P_{2} C_{2}) (ℙ (a)), \forall a \in C_{1}^{★}$ and $(Q_{1} D_{1}) (ab) \leq (Q_{2} D_{2}) (ℙ (a) ℙ (b)), \forall ab \in D_{1}^{★},$ i.e., $\begin{matrix} ({P_{1}}_{*} C_{1})^{+} (a) & = & ({P_{2}}_{*} C_{2})^{+} (ℙ (a)), \\ ({P_{1}}_{*} C_{1})^{-} (a) & = & ({P_{2}}_{*} C_{2})^{-} (ℙ (a)), \\ ({P_{1}}^{*} C_{1})^{+} (a) & = & ({P_{2}}^{*} C_{2})^{+} (ℙ (a)), \\ ({P_{1}}^{*} C_{1})^{-} (a) & = & ({P_{2}}^{*} C_{2})^{-} (ℙ (a)), \\ ({Q_{1}}_{*} D_{1})^{+} (ab) & = & ({Q_{2}}_{*} D_{2})^{+} (ℙ (a) ℙ (b)), \\ ({Q_{1}}_{*} D_{1})^{-} (ab) & = & ({Q_{2}}_{*} D_{2})^{-} (ℙ (a) ℙ (b)), \\ ({Q_{1}}^{*} D_{1})^{+} (ab) & = & ({Q_{2}}^{*} D_{2})^{+} (ℙ (a) ℙ (b)), \\ ({Q_{1}}^{*} D_{1})^{-} (ab) & = & ({Q_{2}}^{*} D_{2})^{-} (ℙ (a) ℙ (b)) . \end{matrix}$

Example 3.9. Let $G_{1} = ({G_{1}}_{*}, G_{1}^{*})$ and $G_{2} = ({G_{2}}_{*}, G_{2}^{*})$ be two intuitionistic fuzzy rough graphs on C^★ = {a, b, c, d}, where ${G_{1}}_{*} = ({P_{1}}_{*} C_{1}, {Q_{1}}_{*} D_{1})$ , $G_{1}^{*} = ({P_{1}}^{*} C_{1}, {Q_{1}}^{*} D_{1})$ and ${G_{2}}_{*} = ({P_{2}}_{*} C_{2}, {Q_{2}}_{*} D_{2})$ , $G_{2}^{*} = ({P_{2}}^{*} C_{2}, {Q_{2}}^{*} D_{2})$ are intuitionistic fuzzy graphs as shown in Figs. 11 and 12, respectively. The graphs $G_{1}$ and $G_{2}$ are isomorphic under $ℙ$ . A mapping $ℙ : C_{1}^{★} \to C_{2}^{★}$ is defined as $ℙ (a) = d, ℙ (b) = a, ℙ (c) = b, ℙ (d) = c .$

Fig.11

Intuitionistic Fuzzy Rough Graph $G_{1} = ({G_{1}}_{*}, G_{1}^{*})$ .

Fig.12

Intuitionistic Fuzzy Rough Graph $G_{2} = ({G_{2}}_{*}, G_{2}^{*})$ .

Definition 3.13. Let $G_{1}$ and $G_{2}$ be two intuitionistic fuzzy rough graphs. A weak isomorphism is a bijective homomorphism of an intuitionistic fuzzy rough graph $ℙ : G_{1} \to G_{2}$ is a map $ℙ : C_{1}^{★} \to C_{2}^{★}$ that satisfies $(P_{1} C_{1}) (a) = (P_{2} C_{2}) (ℙ (a)), \forall a \in C_{1}^{★},$ i.e., $\begin{matrix} ({P_{1}}_{*} C_{1})^{+} (a) & = & ({P_{2}}_{*} C_{2})^{+} (ℙ (a)), \\ ({P_{1}}_{*} C_{1})^{-} (a) & = & ({P_{2}}_{*} C_{2})^{-} (ℙ (a)), \\ ({P_{1}}^{*} C_{1})^{+} (a) & = & ({P_{2}}^{*} C_{2})^{+} (ℙ (a)), \\ ({P_{1}}^{*} C_{1})^{-} (a) & = & ({P_{2}}^{*} C_{2})^{-} (ℙ (a)) . \end{matrix}$

Definition 3.14. Let $G_{1}$ and $G_{2}$ be two intuitionistic fuzzy rough graphs. A co-weak isomorphism is a bijective homomorphism of an intuitionistic fuzzy rough graph $ℙ : G_{1} \to G_{2}$ is a map $ℙ : C_{1}^{★} \to C_{2}^{★}$ that satisfies $(Q_{1} D_{1}) (ab) = (Q_{2} D_{2}) (ℙ (a) ℙ (b)), \forall ab \in D^{★},$ i.e., $\begin{matrix} ({Q_{1}}_{*} D_{1})^{+} (ab) & = & ({Q_{2}}_{*} D_{2})^{+} (ℙ (a) ℙ (b)), \\ ({Q_{1}}_{*} D_{1})^{-} (ab) & = & ({Q_{2}}_{*} D_{2})^{-} (ℙ (a) ℙ (b)), \\ ({Q_{1}}^{*} D_{1})^{+} (ab) & = & ({Q_{2}}^{*} D_{2})^{+} (ℙ (a) ℙ (b)), \\ ({Q_{1}}^{*} D_{1})^{-} (ab) & = & ({Q_{2}}^{*} D_{2})^{-} (ℙ (a) ℙ (b)) . \end{matrix}$

Example 3.10. Consider the two intuitionistic fuzzy rough graphs $G_{1}$ and $G_{2}$ as shown in Figs. 2 and 3 on a set C^★ = {p₁, p₂, p₃}. Let $G_{1} = ({G_{1}}_{*}, G_{1}^{*})$ and $G_{2} = ({G_{2}}_{*}, G_{2}^{*})$ be two intuitionistic fuzzy rough graphs on C^★, where ${G_{1}}_{*} = (P_{*} C_{1}, Q_{*} D_{1})$ and $G_{1}^{*} = (P^{*} C_{1}, Q^{*} D_{1})$ are intuitionistic fuzzy graphs. Also ${G_{2}}_{*} = (P_{*} C_{2}, Q_{*} D_{2})$ and $G_{2}^{*} = (P^{*} C_{2}, Q^{*} D_{2})$ are intuitionistic fuzzy graphs. $G_{1}$ and $G_{2}$ are neither homomorphism under P nor weak homomorphism nor co-weak homomorphism. In any case either $ℙ : C_{1}^{★} \to C_{2}^{★}$ or $ℙ : C_{2}^{★} \to C_{1}^{★}$ can not be defined because the above conditions are not satisfied.

Definition 3.15. Given an intuitionistic fuzzy rough graph $G$ with the underlying set C^★. The order of $G$ is defined and denoted as $\begin{matrix} O (G) & = & (\sum_{a \in C^{★}} {(P_{*} C)^{+} (a) + (P^{*} C)^{+} (a)}, \\ \sum_{\begin{matrix} a \in C^{★} \end{matrix}} {(P_{*} C)^{-} (a) + (P^{*} C)^{-} (a)}) . \end{matrix}$

The size of $G$ is defined and denoted as $\begin{matrix} S (G) & = & (\sum_{a, b \in C^{★}} {(Q_{*} D)^{+} (ab) + (Q^{*} D)^{+} (ab) \\ - (Q_{*} D)^{+} (aa) - (Q^{*} D)^{+} (aa)}, \\ \sum_{a, b \in C^{★}} {(Q_{*} D)^{-} (ab) + (Q^{*} D)^{-} (ab) \\ - (Q_{*} D)^{-} (aa) - (Q^{*} D)^{-} (aa)}) . \end{matrix}$

Definition 3.16. In an intuitionistic fuzzy rough graph, the sum of upper and lower approximations of arcs directed away from the vertex a is called the outdegree of vertex a ∈ C^★, it is denoted by $\begin{matrix} od (a) & = & (\sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} {(Q_{*} D)^{+} (ab) + (Q^{*} D)^{+} (ab)}, \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} {(Q_{*} D)^{-} (ab) + (Q^{*} D)^{-} (ab)}) . \end{matrix}$

Definition 3.17. The sum of upper and lower approximations of arcs directed to the vertex a is called indegree of vertex a ∈ C^★, and defined by $\begin{matrix} id (a) & = & (\sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} {(Q_{*} D)^{+} (ba) + (Q^{*} D)^{+} (ba)}, \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} {(Q_{*} D)^{-} (ba) + (Q^{*} D)^{-} (ba)}) . \end{matrix}$

Definition 3.18. The degree of a vertex a ∈ C^★ in an intuitionistic fuzzy rough graph is defined as $\begin{matrix} d (a) & = & od (a) + id (a) \\ = & (\sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} {(Q_{*} D)^{+} (ab) + (Q^{*} D)^{+} (ab) \\ + (Q_{*} D)^{+} (ba) + (Q^{*} D)^{+} (ba)}, \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} {(Q_{*} D)^{-} (ab) + (Q^{*} D)^{-} (ab) \\ + (Q_{*} D)^{-} (ba) + (Q^{*} D)^{-} (ba)}) . \end{matrix}$

The ordered pair (od (a) , id (a)) is called the degree pair of a.

Definition 3.19. The total degree of a vertex a ∈ C^★ in an intuitionistic fuzzy rough graph is defined as $\begin{matrix} td (a) & = & d (a) + P_{*} C (a) + P^{*} C (a) \\ = & (\sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} {(Q_{*} D)^{+} (ab) + (Q^{*} D)^{+} (ab) \\ + (Q_{*} D)^{+} (ba) + (Q^{*} D)^{+} (ba) \\ + (P_{*} C)^{+} (a) + (P^{*} C)^{+} (a)}, \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} {(Q_{*} D)^{-} (ab) + (Q^{*} D)^{-} (ab) \\ + (Q_{*} D)^{-} (ba) + (Q^{*} D)^{-} (ba) \\ + (P_{*} C)^{-} (a) + (P^{*} C)^{-} (a)}) . \end{matrix}$

Definition 3.20. The minimum degree of $G$ is $\begin{matrix} θ (G) \\ = (min_{\begin{matrix} a \in C^{★} \end{matrix}} {\sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} ((Q_{*} D)^{+} (ab) + (Q^{*} D)^{+} (ab) \\ + (Q_{*} D)^{+} (ba) + (Q^{*} D)^{+} (ba))}, \\ max_{\begin{matrix} a \in C^{★} \end{matrix}} {\sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} ((Q_{*} D)^{-} (ab) + (Q^{*} D)^{-} (ab) \\ + (Q_{*} D)^{-} (ba) + (Q^{*} D)^{-} (ba))}) . \end{matrix}$

The maximum degree of $G$ is $\begin{matrix} ϑ (G) \\ = (max_{\begin{matrix} a \in C^{★} \end{matrix}} {\sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} ((Q_{*} D)^{+} (ab) + (Q^{*} D)^{+} (ab) \\ + (Q_{*} D)^{+} (ba) + (Q^{*} D)^{+} (ba))}, \\ min_{a \in C^{★}} {\sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} ((Q_{*} D)^{-} (ab) + (Q^{*} D)^{-} (ab) \\ + (Q_{*} D)^{-} (ba) + (Q^{*} D)^{-} (ba))}) . \end{matrix}$

Definition 3.21. An edge ab ∈ D^★ in an intuitionistic fuzzy rough graph is called an effective edge, $\begin{matrix} (Q_{*} D)^{+} (ab) & = & (P_{*} C)^{+} (a) \land (P_{*} C)^{+} (b), \\ (Q_{*} D)^{-} (ab) & = & (P_{*} C)^{-} (a) \lor (P_{*} C)^{-} (b), \\ (Q^{*} D)^{+} (ab) & = & (P^{*} C)^{+} (a) \land (P^{*} C)^{+} (b), \\ (Q^{*} D)^{-} (ab) & = & (P^{*} C)^{-} (a) \lor (P^{*} C)^{-} (b) . \end{matrix}$

Definition 3.22. A node a ∈ C^★ in an intuitionistic fuzzy rough graph $G$ is called a busy node if $\begin{matrix} (P_{*} C)^{+} (a) & \leq & min {\sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{+} (ab), \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{+} (ba)}, \\ (P_{*} C)^{-} (a) & \geq & max {\sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{-} (ab), \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{-} (ba)}, \\ (P^{*} C)^{+} (a) & \leq & min {\sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{+} (ab), \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{+} (ba)}, \\ (P^{*} C)^{-} (a) & \geq & max {\sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{-} (ab), \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{-} (ba)} . \end{matrix}$

Otherwise, it is called a free node.

Example 3.11. Consider an intuitionistic fuzzy rough graphs $G$ as shown in Fig. 1. The order $O (G)$ , size $O (G)$ of $G$ and degree of vertices c₁, c₂, c₃, c₄, c₅ are respectively, $\begin{matrix} O (G) & = & (3.1, 2.9), S (G) = (2.0, 1.4), \\ d (c_{1}) & = & (0.9, 1.1), d (c_{2}) = (0.9, 0.6), \\ d (c_{3}) & = & (0.8, 0.6), d (c_{4}) = (0.7, 0.4), \\ d (c_{5}) & = & (0.7, 0.4) . \end{matrix}$

$G$ has busy nodes i.e c₄, c₅. Moreover, $G$ has no effective edge but $G^{*}$ has an effective edge, that is; c₁c₂ .

Remark 3.2. Let $G$ be an intuitionistic fuzzy rough graph.

$G$ is a self μ- complementary intuitionistic fuzzy rough graph, if $S (G) = (α + β, γ + δ)$ where $\begin{matrix} α & = & \frac{1}{2} \sum_{x \neq y} {(P^{*} C)^{+} (x) \land (P^{*} C)^{+} (y)}, \\ β & = & \frac{1}{2} \sum_{x \neq y} {(P_{*} C)^{+} (x) \land (P_{*} C)^{+} (y)}, \\ γ & = & \frac{1}{2} \sum_{x \neq y} {(P^{*} C)^{-} (x) \lor (P^{*} C)^{-} (y)}, \\ δ & = & \frac{1}{2} \sum_{x \neq y} {(P_{*} C)^{-} (x) \lor (P_{*} C)^{-} (y)} . \end{matrix}$

A graph $G$ is a self weak μ- complementary intuitionistic fuzzy rough graph, if $G$ weak isomorphism with $G^{μ}$ .

$G$ is a self weak μ- complementary intuitionistic fuzzy rough graph, if $S (G) = (α + β, γ + δ)$ where $\begin{matrix} α & = & \frac{1}{2} \sum_{x \neq y} {(P^{*} C)^{+} (x) \land (P^{*} C)^{+} (y)}, \\ β & = & \frac{1}{2} \sum_{x \neq y} {(P_{*} C)^{+} (x) \land (P_{*} C)^{+} (y)}, \\ γ & = & \frac{1}{2} \sum_{x \neq y} {(P^{*} C)^{-} (x) \lor (P^{*} C)^{-} (y)}, \\ δ & = & \frac{1}{2} \sum_{x \neq y} {(P_{*} C)^{-} (x) \lor (P_{*} C)^{-} (y)} . \end{matrix}$

If $S (G) = (α + β, γ + δ)$ where $\begin{matrix} α & = & \frac{1}{2} \sum_{x \neq y} {(P^{*} C)^{+} (x) \land (P^{*} C)^{+} (y)}, \\ β & = & \frac{1}{2} \sum_{x \neq y} {(P_{*} C)^{+} (x) \land (P_{*} C)^{+} (y)}, \\ γ & = & \frac{1}{2} \sum_{x \neq y} {(P^{*} C)^{-} (x) \lor (P^{*} C)^{-} (y)}, \\ δ & = & \frac{1}{2} \sum_{x \neq y} {(P_{*} C)^{-} (x) \lor (P_{*} C)^{-} (y)} . \end{matrix}$

Then $G$ may not be self μ- complementary intuitionistic fuzzy rough graph but it is self weak μ- complementary.

Definition 3.23. An intuitionistic fuzzy rough graph $G = (G_{*}, G^{*})$ is called regular, if in each approximation is regular intuitionistic fuzzy graph. Equivalently, an intuitionistic fuzzy rough graph $G$ is called regular intuitionistic fuzzy rough graph if in each approximation, each vertex has same indegree and outdegree.

If each vertex in $G_{*}$ and $G^{*}$ has same indegree k and outdegree k in each approximation space then $G$ is said to be a regular intuitionistic fuzzy rough graph. In other words if the degree pair of a vertex a are same is called k-regular intuitionistic fuzzy rough graph. Both approximation graphs have a vertex of same innerdegree as well as outerdegree. $\begin{matrix} \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{+} (ab) & = & k_{1} = \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{+} (ba), \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{-} (ab) & = & k_{2} = \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{-} (ba), \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{+} (ab) & = & k_{3} = \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{+} (ba), \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{-} (ab) & = & k_{4} = \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{-} (ba), \end{matrix}$ where k_i are constants ∀ i = 1, 2, 3, 4 .

Example 3.12. Let C = {(c₁, 0.2, 0.2) , (c₂, 0.2, 0.2) , (c₃, 0.2, 0.2)} be an intuitionistic fuzzy set on C^★ = {c₁, c₂, c₃} , and P be an intuitionistic fuzzy relation on C^★ as given in Table 4. PC = (P_∗C, P^∗C) an intuitionistic fuzzy rough set, where P_∗C and P^∗C are lower and upper approximations of C, respectively given below: $\begin{matrix} P^{*} C & = & {\frac{(0.2, 0.3)}{c_{1}}, \frac{(0.2, 0.3)}{c_{2}}, \frac{(0.2, 0.3)}{c_{3}}}, \\ P_{*} C & = & {\frac{(0.3, 0.2)}{c_{1}}, \frac{(0.3, 0.2)}{c_{2}}, \frac{(0.3, 0.2)}{c_{3}}} . \end{matrix}$

Table 4

Intuitionistic fuzzy relation on C^★

P	c ₁	c ₂	c ₃
c ₁	(0.6,0.3)	(0.6,0.3)	(0.6,0.3)
c ₂	(0.6,0.3)	(0.6,0.3)	(0.6,0.3)
c ₃	(0.6,0.3)	(0.6,0.3)	(0.6,0.3)

Let D^★ = {c₁c₂, c₂c₃, c₃c₁} ⊆ C^★ × C^★ . Let Q be an intuitionistic fuzzy relation on D^★ defined s given in Table 5. Let D = {(c₁c₂, 0.2, 0.1) , (c₂c₅, 0.2, 0.1) , (c₃c₁, 0.2, 0.1)} be an intuitionistic fuzzy set on D^★. Then by definition, the upper and lower approximation relations are calculated as $\begin{matrix} Q^{*} D & = & {\frac{(0.2, 0.3)}{c_{1} c_{2}}, \frac{(0.2, 0.3)}{c_{2} c_{3}}, \frac{(0.2, 0.3)}{c_{3} c_{1}}}, \\ Q_{*} D & = & {\frac{(0.3, 0.1)}{c_{1} c_{2}}, \frac{(0.3, 0.1)}{c_{2} c_{3}}, \frac{(0.3, 0.1)}{c_{3} c_{1}}} . \end{matrix}$

Table 5

Intuitionistic fuzzy relation on D^★

Q	c ₁ c ₂	c ₂ c ₃	c ₃ c ₁
c ₁ c ₂	(0.6,0.3)	(0.6,0.3)	(0.6,0.3)
c ₂ c ₃	(0.6,0.3)	(0.6,0.3)	(0.6,0.3)
c ₃ c ₁	(0.6,0.3)	(0.6,0.3)	(0.6,0.3)

Hence, (Q_∗D, Q^∗D) is an intuitionistic fuzzy rough relation on C^★. Thus, $G_{*} = (P_{*} C, Q_{*} D)$ and $G^{*} = (P^{*} C, Q^{*} D)$ are intuitionistic fuzzy graphs as shown in Fig. 13. It is a regular and total regular intuitionistic fuzzy rough graph.

Fig.13

Intuitionistic Fuzzy Rough Graph $G = (G_{*}, G^{*})$ .

Definition 3.24. An intuitionistic fuzzy rough graph $G = (G_{*}, G^{*})$ is called totally regular, if in each approximation space is an intuitionistic fuzzy graph with same total degree. Equivalently, if in each approximation is a totally regular intuitionistic fuzzy graph, i.e., $\begin{matrix} \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{+} (ab) + (P_{*} C)^{+} (a) = t_{1}, \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{-} (ab) + (P_{*} C)^{-} (a) = t_{2}, \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{+} (ab) + (P^{*} C)^{+} (a) = t_{3}, \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{-} (ab) + (P^{*} C)^{-} (a) = t_{4}, \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{+} (ba) + (P_{*} C)^{+} (a) = t_{1}, \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{-} (ba) + (P_{*} C)^{-} (a) = t_{2}, \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{+} (ba) + (P^{*} C)^{+} (a) = t_{3}, \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{-} (ba) + (P^{*} C)^{-} (a) = t_{4}, \end{matrix}$ where t_i are constants ∀i = 1, 2, 3, 4 .

Theorem 3.5. Let $G$ be an intuitionistic fuzzy rough graph. If $G$ is a regular (totally regular) and PC = (P_∗C, P^∗C) is a constant function, then $G$ is a totally regular (regular) intuitionistic fuzzy rough graph.

Proof. Let $G = (G_{*}, G^{*})$ be an intuitionistic fuzzy rough graph, where $G_{*} = (P_{*} C, Q_{*} D)$ and $G^{*} = (P^{*} C, Q^{*} D) .$ If $G$ is regular and PC is constant, then for any a ∈ PC, (ab) ∈ QD, $\begin{matrix} \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{+} (ab) & = & k_{1} = \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{+} (ba), \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{-} (ab) & = & k_{2} = \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{-} (ba), \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{+} (ab) & = & k_{3} = \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{+} (ba), \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{-} (ab) & = & k_{4} = \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{-} (ba), \end{matrix}$ $\begin{matrix} (P_{*} C)^{+} (a) = m_{1}, (P_{*} C)^{-} (a) = m_{2}, \\ (P^{*} C)^{+} (a) = m_{3}, (P^{*} C)^{-} (a) = m_{4}, \end{matrix}$ where k_i and m_i are constants ∀ i = 1, 2, 3, 4.

Therefore, $\begin{matrix} \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{+} (ab) + (P_{*} C)^{+} (a) = k_{1} + m_{1}, \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{-} (ab) + (P_{*} C)^{-} (a) = k_{2} + m_{2}, \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{+} (ab) + (P^{*} C)^{+} (a) = k_{3} + m_{3}, \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{-} (ab) + (P^{*} C)^{-} (a) = k_{4} + m_{4}, \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{+} (ab) + (P_{*} C)^{+} (a) = k_{1} + m_{1}, \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q_{*} D)^{-} (ab) + (P_{*} C)^{-} (a) = k_{2} + m_{2}, \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{+} (ab) + (P^{*} C)^{+} (a) = k_{3} + m_{3}, \\ \sum_{\begin{matrix} \forall b \in C^{★} \end{matrix}} (Q^{*} D)^{-} (ab) + (P^{*} C)^{-} (a) = k_{4} + m_{4} . \end{matrix}$

Hence, $G$ is totally regular. □

Theorem 3.6. If an intuitionistic fuzzy rough graph $G$ is regular and PC = (P_∗C, P^∗C) is a constant function, then $G^{c}$ and $G^{μ}$ are regular.

Proof. By using the same approach of the proof in Theorem 3.5 and definition of complement of $G$ , $\begin{matrix} (Q_{*} D)^{+} (ab) & = & (P_{*} C)^{+} (a) \land (P_{*} C)^{+} (b) \\ - (Q_{*} D)^{+} (ab), \\ (Q_{*} D)^{-} (ab) & = & (P_{*} C)^{-} (a) \lor (P_{*} C)^{-} (b) \\ - (Q_{*} D)^{-} (ab), \\ (Q^{*} D)^{+} (ab) & = & (P^{*} C)^{+} (a) \land (P^{*} C)^{+} (b) \\ - (Q^{*} D)^{+} (ab), \\ (Q^{*} D)^{-} (ab) & = & (P^{*} C)^{-} (a) \lor (P^{*} C)^{-} (b) \\ - (Q^{*} D)^{-} (ab) . \end{matrix}$

Therefore, G^c is regular. By Definition 3.6 of G. When (QD) (ab) ≠0, $\begin{matrix} {(Q_{*} D)^{μ}}^{+} (ab) & = & (P_{*} C)^{+} (a) \land (P_{*} C)^{+} (b) \\ - (Q_{*} D)^{+} (ab), \\ {(Q_{*} D)^{μ}}^{-} (ab) & = & (P_{*} C)^{-} (a) \lor (P_{*} C)^{-} (b) \\ - (Q_{*} D)^{-} (ab), \\ {(Q^{*} D)^{μ}}^{+} (ab) & = & (P^{*} C)^{+} (a) \land (P^{*} C)^{+} (b) \\ - (Q^{*} D)^{+} (ab), \\ {(Q^{*} D)^{μ}}^{-} (ab) & = & (P^{*} C)^{-} (a) \lor (P^{*} C)^{-} (b) \\ - (Q^{*} D)^{-} (ab) . \end{matrix}$

When (QD) (ab) =0, $\begin{matrix} {(Q_{*} D)^{μ}}^{+} (ab) & = & 0, {(Q_{*} D)^{μ}}^{-} (ab) = 0, \\ {(Q^{*} D)^{μ}}^{+} (ab) & = & 0, {(Q^{*} D)^{μ}}^{-} (ab) = 0 . \end{matrix}$

Hence, μ-complement of $G$ is regular. □

Theorem 3.7. If an intuitionistic fuzzy rough graph $G$ is totally regular and PC is a constant function, then $G^{c}$ and $G^{μ}$ are totally regular.

Proof. Let $G = (G_{*}, G^{*})$ be an intuitionistic fuzzy rough graph, where $G_{*} = (P_{*} C, Q_{*} D)$ and $G^{*} = (P^{*} C, Q^{*} D) .$ If $G$ is totally regular and PC is constant, then by Theorems 3.5, G^c is regular. Therefore, by Theorem 3.6 G^c and G^μ are regular. Hence, G^c and G^μ are totally regular. □

Definition 3.25. Let $G$ be an intuitionistic fuzzy rough graph. Then $G$ is irregular, if there is a vertex which is adjacent to vertices with distinct indegree and outdegree in each approximation spaces.

Definition 3.26 Let $G$ be a connected intuitionistic fuzzy rough graph. Then $G$ is said to be a neighbourly irregular intuitionistic fuzzy rough graph if every two adjacent vertices of $G$ have distinct degree pair in each approximation spaces.

Definition 3.27. Let $G$ be an intuitionistic fuzzy rough graph. Then $G$ is totally irregular, if there is a vertex which is adjacent to vertices with distinct total degrees in each approximation.

Example 3.13. Let C^★ = {p, q, r} be a set. Let $G = (G_{*}, G^{*})$ be an intuitionistic fuzzy rough graphs on C^★, where $G_{*} = (P_{*} C, Q_{*} D)$ and $G^{*} = (P^{*} C, Q^{*} D)$ are intuitionistic fuzzy graphs as shown in Fig. 8. The graph $G$ is an neighbourly and totally irregular intuitionistic fuzzy rough graph.

Definition 3.28. Let $G$ be a connected intuitionistic fuzzy rough graph. $G$ is said to be a highly irregular intuitionistic fuzzy rough graph if every vertex of $G$ is adjacent to vertices with distinct degree pair.

Remark 3.3. A highly irregular intuitionistic fuzzy rough graph need not be a neighbourly irregular intuitionistic fuzzy rough graph.

Example 3.14. Let C^★ = {a, b, c, d} be a set. Let $G = (G_{*}, G^{*})$ be an intuitionistic fuzzy rough graph on C^★, where $G_{*} = (P_{*} C, Q_{*} D)$ and $G^{*} = (P^{*} C, Q^{*} D)$ are intuitionistic fuzzy graphs as shown in Fig. 14. The graph is a highly irregular intuitionistic fuzzy rough graph. As degree pair of c and d are same: od (c) = (0.0, 0.1) = od (d) , id (c) = (0.0, 0.1) = id (c) in lower approximation graph and od (c) = (0.1, 0.1) = od (d) , id (c) = (0.1, 0.1) = id (c) in upper approximation graph, therefore, it is not neighbourly irregular intuitionistic fuzzy rough graph.

Fig.14

Intuitionistic Fuzzy Rough Graph $G = (G_{*}, G^{*})$ .

Theorem 3.8. Let $G$ be an intuitionistic fuzzy rough graph. $G$ is highly irregular intuitionistic fuzzy rough graph and neighbourly irregular intuitionistic fuzzy rough graph if and only if the degree pair of all vertices of $G$ are distinct.

Proof. Let $G$ be an intuitionistic fuzzy rough graph with n vertices u₁, u₂, u₃, ⋯ , u_n . Assume that $G_{*}$ is highly irregular and neighbourly irregular intuitionistic fuzzy graph. Let the adjacent vertices of u₁ be u₂, u₃, ⋯ , u_n with outdegree and indegree (k₂, l₂) , (k₃, l₃) , ⋯ , (k_n, l_n) and (p₂, q₂) , (p₃, q₃) , ⋯ , (p_n, q_n) respectively. Since, $G_{*}$ is highly irregular, $\begin{matrix} (k_{2}, l_{2}) \neq (k_{3}, l_{3}) \neq \dots \neq (k_{n}, l_{n}) \\ (p_{2}, q_{2}) \neq (p_{3}, q_{3}) \neq \dots \neq (p_{n}, q_{n}) . \end{matrix}$

The outdegree od (u₁) of u₁ cannot be either of (k₂, l₂) , (k₃, l₃) , ⋯ , (k_n, l_n) and the indegree id (u₁) of u₁ not either of (p₂, q₂) , (p₃, q₃) , ⋯ , (p_n, q_n) as $G_{*}$ is neighbourly irregular. Therefore, the degree pair of all vertices of $G_{*}$ are distinct. Similarly, the degree pairs of all vertices of $G^{*}$ are distinct. Hence, the degree pair of all vertices of $G$ are distinct. Converse part is obvious. □

Example 3.15. Let C^★ = {c₁, c₂, c₃, c₄, c₅} be a set. Let $G = (G_{*}, G^{*})$ be an intuitionistic fuzzy rough graphs on C^★, where $G_{*} = (P_{*} C, Q_{*} D)$ and $G^{*} = (P^{*} C, Q^{*} D)$ are intuitionistic fuzzy graphs as shown in Fig. 1.

The graph is a highly irregular intuitionistic fuzzy rough graph. As degree pair of c₄ and c₅ are same: od (c₅) = (0.1, 0.1) = od (c₄) , id (c₅) = (0.1, 0.1) = id (c₄) in lower approximation graph, therefore, it is not neighbourly irregular intuitionistic fuzzy rough graph.

Theorem 3.9. Let $G$ be an intuitionistic fuzzy rough graph. If G is neighbourly irregular and PC is a constant function, then $G$ is a neighbourly total irregular intuitionistic fuzzy rough graph.

Proof. Assume that $G$ is a neighbourly irregular intuitionistic fuzzy rough graph, i.e., the degrees of every two adjacent vertices are distinct in each approximation. Consider two adjacent vertices u₁ and u₂ with distinct degree pair, as out degree (k₁, l₁) and (k₂, l₂) and indegree (p₁, q₁) and (p₂, q₂) respectively, in $G_{*},$ i.e., od (u₁) = (k₁, l₁) , id (u₁) = (p₁, q₁) and od (u₂) = (k₂, l₂) , id (u₂) = (p₂, q₂) where k₁ ≠ k₂, l₁ ≠ l₂, p₁ ≠ p₂, q₁ ≠ q₂, and d (u₁) ≠ d (u₂) . Also assume that (P_∗C) (u₁) = (c₁, c₂) = (P_∗C) (u₂), a constant c_i ∈ [0, 1] , i = 1, 2.

Therefore td (u₁) d (u₁) + (P_∗C) (u₁) = (k₁, l₁) + (p₁, q₁) + (c₁, c₂) ; td (u₂) = d (u₂) + (PC) (u₂) = (k₂, l₂) + (p₂, q₂) + (c₁, c₂) To prove: td (u₁) ≠ td (u₂). Suppose td (u1) = td (u₂) ; d (u₁) + (P_∗C) (u₁) = d (u₂) + P_∗C (u₁) d (u₁) = d (u₂) , a contradiction to d (u₁) ≠ d (u₂) . Therefore, td (u₁) ≠ td (u₂) in $G_{*}$ , i.e., for any two adjacent vertices u₁ and u₂ with distinct degrees, its total degrees are also distinct, provided PC is a constant function. Thus, $G_{*}$ is a neighbourly total irregular intuitionistic fuzzy graph. Similarly, $G^{*}$ is a neighbourly total irregular intuitionistic fuzzy graph. The above argument is true for every pair of adjacent vertices in $G$ . □

Theorem 3.10. Let $G$ be a intuitionistic fuzzy rough graph. If $G$ is both neighbourly irregular and neighbourly total irregular intuitionistic fuzzy rough graph, then PC need not be a constant function.

Proof. This proof follows from Theorems 3.5, 3.6 and 3.9.

4 Application to decision making

Decision-making plays an important role in our daily life. Same decisions are very important that they can change the course of our lives. The process of decision-making yields a choice among different alternatives. Decision-making is considered very useful in gathering as much information from different sources and evaluating all possible alternatives to the problem or situation at hand. Going through this whole process we arrive at the best possible solution for the problem. Here we present some applications of decision-making from our real World. The given decision-making method can be used to evaluate upper and lower approximations to develop deep considerations of the problem. The presented algorithms can be applied to avoid lengthy calculations when dealing with large number of objects. This method can be applied in various domains for multi-criteria selection of objects. The amazing collection of bridal dresses has been introduced in many fashion shows by the best and leading fashion designers of our country. Usually the brides take too much interest in different types of gowns for wedding so this collection also contains beautiful gowns. Different famous fashion designers took part in many fashion weeks related to wedding so that these outfits are worked with different reactivities. If these bridal dresses are compared with the Pakistani dresses which were delivered, they are totally different from them as they hold their own value. The outfits for bridles are according to the traditions and mayoon, mehndi, baraat and reception outfits are including in this collection. Let a new fashion designer wants to make a bridal dress. The brides look more traditional in lehngas and shararas made by using different types of embroidery such as Dabka, Beats, Naqshi, Mukesh, Pearls, Crystals, Studded Squins, Motifs and Work of Zari so they are also available for all the girls. He decides to make the bridal dresses which are customary and traditional along with the modern twist. He uses a set F^★ consisting of f₁ = Dabka, f₂ = Mukesh, f₃ = Pearls, f₄ = Crystals and f₅ = Work of Zari, on a red tradition color with coffee color.

F = {(f₁, 0.8, 0.1) , (f₂, 0.7, 0.2) , (f₃, 0.6, 0.1) , (f₄, 0.8, 0.1) , (f₅, 0.8, 0.1)} is an intuitionistic fuzzy set on F^★, shows the quality of above embroideries and P an intuitionistic fuzzy relation on F^★ defined as given in Table 6.

Table 6
Intuitionistic fuzzy relation on F^★

P f ₁ f ₂ f ₃ f ₄ f ₅

f ₁ (0.9,0.0) (0.7,0.1) (0.6,0.2) (0.5,0.1) (0.3,0.2)

f ₂ (0.8,0.1) (0.4,0.4) (0.8,0.1) (0.7,0.1) (1.0,0.0)

f ₃ (0.7,0.2) (0.3,0.1) (0.0,0.6) (0.2,0.2) (0.6,0.2)

f ₄ (0.6,0.1) (0.5,0.5) (0.4,0.4) (0.7,0.2) (0.3,0.4)

f ₅ (0.9,0.0) (0.0,1.0) (0.1,0.1) (0.5,0.5) (0.3,0.3)

P	f ₁	f ₂	f ₃	f ₄	f ₅
f ₁	(0.9,0.0)	(0.7,0.1)	(0.6,0.2)	(0.5,0.1)	(0.3,0.2)
f ₂	(0.8,0.1)	(0.4,0.4)	(0.8,0.1)	(0.7,0.1)	(1.0,0.0)
f ₃	(0.7,0.2)	(0.3,0.1)	(0.0,0.6)	(0.2,0.2)	(0.6,0.2)
f ₄	(0.6,0.1)	(0.5,0.5)	(0.4,0.4)	(0.7,0.2)	(0.3,0.4)
f ₅	(0.9,0.0)	(0.0,1.0)	(0.1,0.1)	(0.5,0.5)	(0.3,0.3)

PF = (P_∗F, P^∗F) an intuitionistic fuzzy rough set, where P_∗F and P^∗F are lower and upper approximations of F, respectively, as pursued: $\begin{matrix} P^{*} F & = & {\frac{(0.8, 0.1)}{f_{1}}, \frac{(0.8, 0.1)}{f_{2}}, \frac{(0.7, 0.2)}{f_{3}}, \\ \frac{(0.7, 0.1)}{f_{4}}, \frac{(0.8, 0.1)}{f_{5}}} \\ P_{*} F & = & {\frac{(0.6, 0.2)}{f_{1}}, \frac{(0.6, 0.2)}{f_{2}}, \frac{(0.6, 0.2)}{f_{3}}, \\ \frac{(0.6, 0.2)}{f_{4}}, \frac{(0.6, 0.1)}{f_{5}}} . \end{matrix}$

F is an intuitionistic fuzzy rough set as $\vec{P} F - \overset{\leftarrow}{P} F \neq \emptyset$ . Let D^★ = {f₁f₂, f₁f₅, f₂f₃, f₂f₅, f₃f₄, f₃f₅, f₄f₁, f₅f₂, f₅f₄} ⊆ F^★ × F^★ and Q an intuitionistic fuzzy relation on D^★ defined as given in Table 7.

Table 7

Intuitionistic fuzzy relation on D^★

Q	f ₁ f ₂	f ₁ f ₅	f ₂ f ₃	f ₂ f ₅	f ₃ f ₄
f ₁ f ₂	(0.4,0.2)	(0.9,0.0)	(0.6,0.1)	(0.7,0.1)	(0.4,0.0)
f ₁ f ₅	(0.0,0.0)	(0.3,0.2)	(0.0,0.0)	(0.3,0.3)	(0.4,0.5)
f ₂ f ₃	(0.3,0.1)	(0.5,0.0)	(0.0,0.2)	(0.4,0.3)	(0.2,0.2)
f ₂ f ₅	(0.0,0.3)	(0.3,0.0)	(0.0,0.4)	(0.0,0.3)	(0.1,0.1)
f ₃ f ₄	(0.4,0.2)	(0.2,0.0)	(0.2,0.4)	(0.3,0.1)	(0.0,0.1)
f ₃ f ₅	(0.0,0.0)	(0.2,0.0)	(0.1,0.1)	(0.1,0.0)	(0.0,0.4)
f ₄ f ₁	(0.6,0.0)	(0.2,0.0)	(0.5,0.0)	(0.2,0.0)	(0.4,0.0)
f ₅ f ₂	(0.0,0.3)	(0.3,0.0)	(0.0,0.4)	(0.0,0.3)	(0.1,0.1)
f ₅ f ₄	(0.5,0.4)	(0.3,0.3)	(0.0,0.9)	(0.0,0.8)	(0.1,0.1)
Q	f ₃ f ₅	f ₄ f ₁	f ₅ f ₂	f ₅ f ₄
f ₁ f ₂	(0.5,0.2)	(0.4,0.1)	(0.3,0.4)	(0.1,0.1)
f ₁ f ₅	(0.3,0.1)	(0.4,0.0)	(0.0,0.6)	(0.2,0.4)
f ₂ f ₃	(0.6,0.1)	(0.7,0.1)	(0.2,0.0)	(0.2,0.0)
f ₂ f ₅	(0.1,0.1)	(0.5,0.1)	(0.0,0.3)	(0.3,0.3)
f ₃ f ₄	(0.0,0.1)	(0.2,0.1)	(0.5,0.1)	(0.4,0.1)
f ₃ f ₅	(0.0,0.4)	(0.2,0.1)	(0.0,1.0)	(0.2,0.2)
f ₄ f ₁	(0.3,0.0)	(0.7,0.0)	(0.3,0.0)	(0.3,0.0)
f ₅ f ₂	(0.1,0.1)	(0.5,0.1)	(0.0,0.3)	(0.3,0.3)
f ₅ f ₄	(0.1,0.3)	(0.5,0.0)	(0.3,0.4)	(0.3,0.2)

Let D = {(f₁f₂, 0.6, 0.1) , (f₁f₅, 0.5, 0.0) , (f₂f₃, 0.7, 0.0) , (f₂f₅, 0.7, 0.0) , (f₃f₄, 0.6, 0.1) , (f₃f₅, 0.5, 0.2) , (f₄f₁, 0.7, 0.0) , (f₅f₂, 0.7, 0.0) , (f₅f₄, 0.7, 0.1)} be an intuitionistic fuzzy set on D^★, represents the enhancement of beauty in the structure combined the type of embroidery and QD = (Q_∗D, Q^∗D) an intuitionistic fuzzy relation, where Q_∗D and Q^∗D are lower and upper approximations of D, respectively, as follows in Table 8.

Table 8

Intuitionistic fuzzy rough edge set

QD	f ₁ f ₂	f ₁ f ₅	f ₂ f ₃	f ₂ f ₅
Q ^∗ D	(0.7,0.0)	(0.4,0.0)	(0.7,0.0)	(0.5,0.0)
Q _∗ D	(0.5,0.2)	(0.5,0.2)	(0.5,0.2))	(0.5,0.1)
QD	f ₃ f ₄	f ₃ f ₅	f ₄ f ₁	f ₅ f ₂	f ₅ f ₄
Q ^∗ D	(0.5,0.0)	(0.2,0.0)	(0.7,0.0)	(0.5,0.0)	(0.5,0.0)
Q _∗ D	(0.5,0.1)	(0.5,0.1)	(0.5,0.2)	(0.5,0.1)	(0.5,0.1)

Thus, $G_{*} = (P_{*} C, Q_{*} D)$ and $G^{*} = (P^{*} C, Q^{*} D)$ are fuzzy digraphs as shown in Fig. 15. Score value is given in Table 9. Since f₁ has maximum value, designer uses dabka in his work to enhance the beauty in the dress. The algorithm with time complexity (TC) of this application is shown in Table 10.

Fig.15

Intuitionistic Fuzzy Rough Graph $G = (G_{*}, G^{*})$ .

Table 9

$T_{i} = \sum_{j} \sqrt{(Q^{*} D^{+} e_{ij} - Q_{*} D^{+} e_{ij})^{2} + (Q^{*} D^{-} e_{ij} - Q_{*} D^{-} e_{ij})^{2}}$

F ^★	f ₁	f ₂	f ₃	f ₄	f ₅
T _i	0.5064	0.3828	0.4162	0.2828	0.2

Table 10

Algorithm of Decision-making problem

Algorithm	TC
1. Begin	O(1)
2. Input the vertex set C^★ of embroidery f₁, f₂, …, f_n.	O(n)
3. Input an intuitionistic fuzzy relation P on C^★.	O(n²)
4. Input an intuitionistic fuzzy set C on C^★.	O(n)
5. Input the edge set D^★ of relations l₁, l₂, …, l_r where, l_i = f_jf_k, for some j, k ∈ {1, 2, …, n}.	O(r)
6. Input an intuitionistic fuzzy relation Q on D^★ ⊆ C^★ × C^★.	O(r²)
7. Input an intuitionistic fuzzy set D on D^★.	O(r)
8. doi from 1 to n	O(n)
9. read*, C (f_i)	O(n)
10. end do	O(n)
11. doi from 1 to n	O(n)
12. (P^∗C) ⁺ (f_i) =0	O(n)
13. (P^∗C) ^- (f_i) =1	O(n)
14. (P_∗C) ⁺ (f_i) =1	O(n)
15. (P_∗C) ^- (f_i) =0	O(n)
16. doj from 1 to n	O(n²)
17. $T_{*}^{+} (f_{i}) = max {P^{-} (f_{i}, f_{j}), C^{+} (f_{i})}$	O(n²)
18. $T_{*}^{-} (f_{i}) = min {P^{+} (f_{i}, f_{j}), C^{-} (f_{i})}$	O(n²)
19. T^∗⁺ (f_i) = min {P⁺ (f_i, f_j) , C⁺ (f_i)}	O(n²)
20. T^∗^- (f_i) = max {P^- (f_i, f_j) , C^- (f_i)}	O(n²)
21. $(P_{} C)^{+} (f_{i}) = min {(P_{} C)^{+} (f_{i}), T_{*}^{+} (f_{i})}$	O(n²)
22. $(P_{} C)^{-} (f_{i}) = max {(P_{} C)^{-} (f_{i}), T_{*}^{-} (f_{i})}$	O(n²)
23. (P^∗C) ⁺ (f_i) = max {(P^∗C) ⁺ (f_i) , T^∗⁺ (f_i)}	O(n²)
24. (P^∗C) ^- (f_i) ^- = min {(P^∗C) ^- (f_i) , T^∗^- (f_i)}	O(n²)
25. end do	O(n²)
26. end do	O(n)
27. doi from 1 to r	O(r)
28. read*, D (l_i)	O(r)
29. end do	O(r)
30. doi from 1 to r	O(r)
31. (Q^∗D) ⁺ (l_i) =0	O(r)
32. (Q^∗D) ^- (l_i) =1	O(r)
33. (Q_∗D) ⁺ (l_i) =1	O(r)
34. (Q_∗D) ^- (l_i) =0	O(r)
35. doj from 1 to r	O(r²)
36. $F_{*}^{+} (l_{i}) = max {Q^{-} (l_{i}, l_{j}), D^{+} (l_{i})}$	O(r²)
37. $F_{*}^{-} (l_{i}) = min {Q^{+} (l_{i}, l_{j}), D^{-} (l_{i})}$	O(r²)
38. F^∗⁺ (l_i) = min {Q⁺ (l_i, l_j) , D⁺ (l_i)}	O(r²)
39. F^∗^- (l_i) = max {Q^- (l_i, l_j) , D^- (l_i)}	O(r²)
40. $(Q_{} D)^{+} (l_{i}) = min {(Q_{} D)^{+} (l_{i}), F_{*}^{+} (l_{i})}$	O(r²)
41. $(Q_{} D)^{-} (l_{i}) = max {(Q_{} D)^{-} (l_{i}), F_{*}^{-} (l_{i})}$	O(r²)
42. (Q^∗D) ⁺ (l_i) = max {(Q^∗D) ⁺ (l_i) , F^∗⁺ (l_i)}	O(r²)
43. (Q^∗D) ^- (l_i) ^- = min {(Q^∗D) ^- (l_i) , F^∗^- (l_i)}	O(r²)
44. end do	O(r²)
45. end do	O(r)
46. doi from 1 to n	O(n)
47. $T_{i} = \sum_{j} \sqrt{(Q^{} D^{+} l_{ij} - Q_{} D^{+} l_{ij})^{2} + (Q^{} D^{-} l_{ij} - Q_{} D^{-} l_{ij})^{2}}$	O(n)
48. end do	O(n)
49. End	O(n²) or
O(r²)

5 Conclusions

Rough set theory is a mathematical tool to deal with incomplete and vague information. An intuitionistic fuzzy set theory deals the problem of how to understand and manipulate imperfect knowledge. Present researchers have been shown that these two theories can be combined into a more flexible and expressive framework for modeling and processing incomplete information in information systems. In this research article, the concept of intuitionistic fuzzy rough relation and intuitionistic fuzzy rough graphs have been introduced. Some algebraic operations on intuitionistic fuzzy rough graphs are studied. An applications of intuitionistic fuzzy rough graphs has been presented in decision making problems. This research work can be extended to intuitionistic fuzzy rough hypergraphs.

Footnotes

Acknowledgment

The authors are very grateful to the Associate Editor and the reviewers.

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