Z -soft rough fuzzy graphs: A new approach to decision making

Abstract

The aim of this paper is to discuss the uncertainty in soft graphs and a new type of graphs, called Z-soft rough fuzzy graphs is introduced. The notions of Z-soft rough vertex graphs, Z-soft rough edge graphs and Z-soft rough fuzzy graphs are given, and their related properties are explored. The theory of graph approximation space is developed. To validate the applicability of Z-soft rough fuzzy graphs, we put forward an innovative approach to multicriteria group decision making problems.

Keywords

rough sets soft graphs decision making

1 Introduction

Graph theory was developed with Leonhard Euler’s paper [13] in which he settled a famous unsolved problem known as Kōnigsberg Bridges Problem. The theory has greatly contributed to many developing fields, due to its applications in programming, civil engineering, communications theory, network theory, coding theory, switching circuits, psychology, computer science involving computations and algorithms, operation research(scheduling) and economics. Many extensions and applications of the subject of graph theory can be seen in [4 , 51]. A graph can also be used to discuss the relationship between the objects where each object can be represented by a vertex and the relationship between the objects can be represented by an edge. Network problems, like sensor network problems and automatic channel allocation networks can also be tackled by using graphs and the related structures.

In real life, most of the problems involve complexity which arises from uncertainty in the form of ambiguity. Usually the data collected from real life situations consists of uncertain, imprecise and incomplete information in which the classical mathematics is not absolutely prosperous and not successful. Most of our conventional mathematical tools of modeling, reasoning and computing are crisp, deterministic and precise in character. Different approaches have been developed to solve the problems of uncertainty and vagueness. For example probability theory, fuzzy set theory, rough set theory, soft set theory and blend of these approaches have been used. The theory of probability has been an efficient tool to handle the problems of uncertainty, but this theory can be applied only to the situations whose characteristics are based only on random processes.

In 1965, Lotfi. A. Zadeh [55] originated the theory of fuzzy sets in his seminal paper which is another excellent effective tool to handle the problems involving uncertainty. Fuzzy set theory can be used to get, simulate and even reason the fuzziness in many practical material of information by mainly examining the fuzzy information granularity. Due to the wide applications of fuzzy sets, almost all the areas of mathematics, medicine, engineering etc., notions have been redefined using fuzzy sets. Fuzzy sets are described approximately through the membership degree of sets with respect to objects.

Rough sets was originally proposed by Z. Pawlak [33] as a better mathematics tool for replacing the probability and fuzzy set theory, and can be used to model and process incomplete information in an information system. The rough set approach seems to be of fundamental significance to artificial intelligence and cognitive sciences, particularly in the field of knowledge acquisition, machine learning, decision analysis, pattern recognition, expert systems and knowledge discovery from databases. Many applications of rough sets including data mining, pattern recognition, machine learning and knowledge discovery can be seen in [9 , 49].

Molodtsov [30] introduced the notion of soft sets as a new mathematics tool for solving the problems involving vagueness and uncertainty. Since a soft set involve many parameters, so it is free from those difficulties which the previous theories possesses. Soft set theory has been modified and extended in some aspects to tackle many problems [6 , 38]. A number of applications have been established and applied regarding decision making and multi-attributes modeling [10 , 46]. A possible fusion has been established by Feng et al. [15] by considering approximation of a fuzzy set in soft approximation space and initiated a new concept called soft rough fuzzy sets, which is the generalization of Dubois and Prade’s rough fuzzy sets [12]. In their soft rough fuzzy model, a soft set is used to granulate the universe of discourse instead of equivalence class taken by Pawlak [33]. Meng et al. [28] further discussed the combination of rough sets, fuzzy sets and soft sets, and employed fuzzy soft sets to granulate the universe of discourse. Furthermore, they established a more general model called soft fuzzy rough model than the previous. Zhan et al. [60] introduced the notion of Z-soft rough fuzzy sets and gave a corresponding decision making. A number of applications and utilizations has been presented by many researchers in problems of multi-attributes, problems of reduction of attributes, knowledge base system, data mining and data labeling problems [5 , 67]

It can be seen that soft sets, fuzzy sets and rough sets are closely linked, and some extensions of these are available in [3 , 65]. An amalgamation of rough sets with fuzzy sets and referring them to theory of graphs can be found in [11 , 45]. Akram [1] introduced the notion soft graphs which is an approach to handle the problems of multi-attributes related with graph theory. A number of extensions and generalizations of soft graphs are available in [2 , 48]. The present article aims at providing a new approach to combine graphs, soft sets and rough sets which initiates the notion of Z-soft rough fuzzy graphs. This approach will not only enhance the applicability of soft graphs but will also use to discuss the uncertainty in soft graphs. The present paper is organized asbreak follows:

In Section 2 some basic concepts are revised. Section 3 is about the basic definitions and characterization of Z- soft rough fuzzy graphs, where the notions of lower (upper) Z- soft vertex approximations, lower (upper) Z- soft edge approximations and Z-soft rough fuzzy graph are defined to develop the basic theory with the help of examples. Section 4 is developed for the applications of Z- soft rough fuzzy graphs. An algorithm is proposed to compute the effectiveness of some diseases amongst some animals kept in a farm house in which the vertices are denoted by 20 animals and the edges are used to describe the interaction between the animals. This interaction may cause the spreadness of diseases amongst the animals. This is the motivation of the proposed theory. Conclusion of the article is presented in Section 5.

2 Preliminaries

In this section, some basic notions are given. These will be very helpful in later sections.

Definition 1. [8, 51] A graph G^* = (V, E) consists of set of finite objects V = {v₁, v₂, v₃......., v_n} called vertices (also called points or nodes) and the set E = {e₁, e₂, e₃....... e_m} whose element are called edges (also called lines or arcs). Let {v₁, v₂} be an edge of G^*, then it is more convenient to represent this edge by v₁v₂. If e = v₁v₂ is an edges of a graph G^*, then we say that v₁ and v₂ are adjacent in G^* and that e joins v₁ and v₂. If every edge in a graph has direction then such a graph is called a digraph. An edge whose endpoints are equal, is called a loop and multiple edges are edges having the same pair of endpoints. A graph G^* is called simple if it has no loops or multiple edges.

Definition 2. [24] Let A be the set of parameters. A pair (k, A) is called a soft set over the set U of universe, where k: A → P (U) is a set valued mapping. A soft set can be represented by a table having values 0 and 1, where 1 represent that x ∈ k (ρ) and 0 represent that x ∉ k (ρ) , for ρ ∈ A and x ∈ U.

Definition 3. [1] Let G^* = (V, E) be a given simple graph and A be the set of parameters. A quadruple $G = (G^{*}, G, H, A)$ is said to be a soft graph on G^* if

$G = (δ, A)$ is a soft set over V ;

$H = (μ, A)$ is a soft set over E ;

(δ (ρ) , μ (ρ)) is a subgraph of G^* = (V, E) , for all ρ ∈ A.

Definition 4. [16] Let S = (k, A) be a soft set over U. Then, the pair $D = (U, S)$ is called soft approximation space. The following two sets, $\begin{matrix} {\underline{appr}}_{D} (X) \\ = & {u \in U : \exists ρ \in A, [u \in k (ρ) \subseteq X]}, \\ {\bar{appr}}_{D} (X) \\ = & {u \in U : \exists ρ \in A, [u \in k (ρ), k (ρ) \cap X \neq φ]} \end{matrix}$ assigning to any set X ⊆ U, are called soft $D$ - lower approximation and soft $D$ - upper approximation of X, respectively. Also, the sets $\begin{matrix} Pos (X) & = & {\underline{appr}}_{D} (X), \\ Neg (X) & = & - {\underline{appr}}_{D} (X), \end{matrix}$ and $Bnd (X) = {\bar{appr}}_{D} (X) - {\underline{appr}}_{D} (X)$ are called respectively, the soft $D$ -positive region, the soft $D -$ negative region, and the soft $D$ -boundary region of X. If ${\bar{appr}}_{D} (X) = {\underline{appr}}_{D} (X),$ then X is said to be soft $D$ -definable; otherwise X is called a soft $D$ -rough set.

Definition 5. [15] Let S = (k, A) be a full soft set over U i.e., ⋃_ρ∈Rk (ρ) = U and $D = (U, S)$ is a soft approximation space. Let $F (U)$ be the set of all fuzzy subsets of U. For a fuzzy set μ∈ $F (U),$ the lower and upper soft rough approximation space of μ with respect to $D$ are denoted and defined as, $\begin{matrix} {\underline{s a p}}_{D} (μ) (x) \\ = ∧ {μ (y) : \exists ρ \in A [{x, y} \subseteq k (ρ) %]}, \\ {\bar{s a p}}_{D} (μ) (x) \\ = ∨ {μ (y) : \exists ρ \in A [{x, y} \subseteq k (ρ)]} \end{matrix}$ for all x ∈ U, the operators ${\underline{sap}}_{D}$ and ${\bar{sap}}_{D},$ are called the lower and upper soft rough approximation operators on fuzzy sets. If ${\underline{sap}}_{D} (μ) =$ ${\bar{sap}}_{D} (μ)$ then μ is said to be soft definable; otherwise μ is called a soft rough fuzzy set.

3 Z-soft rough fuzzy graphs

In this section, Z-soft rough fuzzy graphs are defined. The notion of lower/upper Z-soft rough vertex approximation, lower/upper Z-soft rough edge approximation, Z-soft rough fuzzy vertex graph, Z-soft rough fuzzy edge graph is introduced. Basic theory of soft rough fuzzy graphs is evolved and some fundamental properties are interrogated.

Definition 6. Let $G = (G^{*}, ψ, ξ, A)$ be a soft graph and λ: V → P (A) be a map such that λ (x) = {ρ ∈ A: x ∈ ψ (ρ)}. Denote by Z_λ = (V, λ) and call it Z-soft vertex approximation space. Let $F (V)$ be the set of all fuzzy subsets of V. Based on Z_λ = (V, λ) and for a fuzzy subset h of V, the lower Z- soft rough vertex approximation and upper Z- soft rough vertex approximation of h are denoted by ${\underline{appr}}_{Z_{λ}} (h) (x)$ and ${\bar{appr}}_{Z_{λ}} (h) (x)$ respectively, and are defined by $\begin{matrix} {\underline{appr}}_{Z_{λ}} (h) (x) \\ = & ⋀ {h (y) : y \in V, λ (x) = λ (y)}, \\ {\bar{appr}}_{Z_{λ}} (h) (x) \\ = & ⋁ {h (y) : y \in V, λ (x) = λ (y)} \end{matrix}$ for all x ∈ V. The sets ${\underline{appr}}_{Z_{λ}} (h)$ and ${\bar{appr}}_{Z_{λ}} (h)$ are fuzzy sets in V. If ${\bar{appr}}_{Z_{λ}} (h) = {\underline{appr}}_{Z_{λ}} (h),$ then G_{Z
_λ}: = (V, E) is called Z- soft vertex definable graph, otherwise G_{Z
_λ} is called Z- soft rough fuzzy vertex graph. Denote and define the lower and upper Z- soft vertex approximations of G_{Z
_λ} by $\underline{G_{Z_{λ}}} = ({\underline{appr}}_{Z_{λ}} (h), E)$ and $\bar{G_{Z_{λ}}} = ({\bar{appr}}_{Z_{λ}} (h), E)$ for $h \in F (V) .$

Example 1. Let a simple graph be G^∗ = (V, E), where V ={ x₁, x₂, x₃, x₄, x₅, x₆, x₇ } is the set of vertices, E ={ e₁, e₂, e₃, e₄, e₅, e₆, e₇, e₈, e₉, e₁₀ } is the edge set as shown in Figure 1, given in Appendix. Let be the set of parameters and $G = (G^{*}, ψ, ξ, A)$ be a soft graph. A soft set (ψ, A) over V is given in Table 1. and Z_λ = (V, λ) is a Z-soft vertex approximation space, a map λ: V → P (A) is defined as λ (x) = {ρ ∈ A: x ∈ ψ (ρ)} such that λ (x₁) = {ρ₁,ρ₂,ρ₄} = λ (x₃) , λ (x₂) = {ρ₃}, λ (x₄) = {ρ₂,ρ₃, ρ₄} = λ (x₆) , λ (x₅) = {ρ₁}.

Table 1
Tabular representation of soft set (ψ, A)

V x ₁ x ₂ x ₃ x ₄ x ₅ x ₆ x ₇

ρ ₁ 1 0 1 0 1 0 1

ρ ₂ 1 0 1 1 0 1 1

ρ ₃ 0 1 0 1 0 1 0

ρ ₄ 1 0 1 1 0 1 0

V	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆	x ₇
ρ ₁	1	0	1	0	1	0	1
ρ ₂	1	0	1	1	0	1	1
ρ ₃	0	1	0	1	0	1	0
ρ ₄	1	0	1	1	0	1	0

Then for $h \in F (V),$ where $h = {\frac{x_{1}}{0.2}, \frac{x_{2}}{0.4}, \frac{x_{3}}{0.1}, \frac{x_{4}}{0.5}, \frac{x_{5}}{0.6}, \frac{x_{6}}{0.9}, \frac{x_{7}}{0.8}},$ and one can calculate $\begin{matrix} {\underline{appr}}_{Z_{λ}} (h) (x) \\ = & {\frac{x_{1}}{0.1}, \frac{x_{2}}{0.4}, \frac{x_{3}}{0.1}, \frac{x_{4}}{0.5}, \frac{x_{5}}{0.6}, \frac{x_{6}}{0.5}, \frac{x_{7}}{0.8}} \end{matrix}$ and $\begin{matrix} {\bar{appr}}_{Z} (h) (x) \\ = & {\frac{x_{1}}{0.2}, \frac{x_{2}}{0.4}, \frac{x_{3}}{0.2}, \frac{x_{4}}{0.9}, \frac{x_{5}}{0.6}, \frac{x_{6}}{0.9}, \frac{x_{7}}{0.8}} . \end{matrix}$ Since ${\bar{appr}}_{Z} (h)$ ≠ ${\underline{appr}}_{Z} (h),$ So G_{Z
_λ}: = (V, E) is Z- soft rough fuzzy vertex graph, where $\begin{matrix} \underline{G_{Z_{λ}}} & = & ({\underline{appr}}_{Z_{λ}} (h), E) \\ = & (\begin{matrix} {\frac{x_{1}}{0.1}, \frac{x_{2}}{0.4}, \frac{x_{3}}{0.1}, \frac{x_{4}}{0.5}, \frac{x_{5}}{0.6}, \frac{x_{6}}{0.5}, \frac{x_{7}}{0.8}}, \\ {e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}, e_{7}, e_{8}, e_{9}, e_{10}} \end{matrix}), \end{matrix}$ $\begin{matrix} \bar{G_{Z_{λ}}} & = & ({\bar{appr}}_{Z_{λ}} (h), E) \\ = & (\begin{matrix} {\frac{x_{1}}{0.2}, \frac{x_{2}}{0.4}, \frac{x_{3}}{0.2}, \frac{x_{4}}{0.9}, \frac{x_{5}}{0.6}, \frac{x_{6}}{0.9}, \frac{x_{7}}{0.8}}, \\ {e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}, e_{7}, e_{8}, e_{9}, e_{10}} \end{matrix}) . \end{matrix}$

Definition 7. Let $G = (G^{*}, ψ, ξ, A)$ be a soft graph and η: E → P (A) be a map such that η (e) = {ρ ∈ A: e ∈ ξ (ρ)}. Denote by Z_η = (E, η) and call it Z-soft edge approximation space. Let $F (E)$ be the set of all fuzzy subsets of E. Based on Z_η = (E, η) and for a fuzzy subset k of E, the lower Z- soft rough edge approximation and upper Z-soft rough edge approximation of k are denoted by ${\underline{appr}}_{Z_{η}} (k) (e)$ and ${\bar{appr}}_{Z_{η}} (k) (e)$ respectively, and are defined by $\begin{matrix} {\underline{appr}}_{Z_{η}} (k) (e) \\ = & ⋀ {k (f) : f \in E, η (e) = η (f)}, \end{matrix}$ $\begin{matrix} {\bar{appr}}_{Z_{η}} (k) (e) \\ = & ⋁ {k (f) : f \in E, η (e) = η (f)} . \end{matrix}$ for all e ∈ E. The sets ${\underline{appr}}_{Z_{η}} (k)$ and ${\bar{appr}}_{Z_{η}} (k)$ are fuzzy sets in E. If ${\bar{appr}}_{Z_{η}} (k) = {\underline{appr}}_{Z_{η}} (k),$ then G_{Z
_η}: = (V, E) is called Z- soft edge definable graph, otherwise G_{Z
_η} is called Z- soft rough fuzzy edge graph. Denote and define the lower and upper Z- soft edge approximations of G_{Z
_η} by $\begin{matrix} \underline{G_{Z_{η}}} & = & ({\underline{appr}}_{Z_{η}} (h), E) and \\ \bar{G_{Z_{η}}} & = & ({\bar{appr}}_{Z_{η}} (h), E) for k \in F (E) . \end{matrix}$

Definition 8. A soft graph $G = (G^{*}, ψ, ξ, A)$ is called Z-soft definable if (1) h is Z- soft vertex definable i.e., ${\bar{appr}}_{Z_{λ}} (h) = {\underline{appr}}_{Z_{λ}} (h)$ and (2) k is Z- soft edge definable i.e., $\bar{app r_{Z η}} (k) = {\underline{app r}}_{z η} (k)$ .

Definition 9. A soft graph $G = (G *, ψ, ξ, A)$ is called Z-soft rough fuzzy graph if (1) h is Z-soft rough fuzzy vertex set i.e., ${\bar{a p p r}}_{Z_{λ}} (h) \neq {\underline{a p p r}}_{Z_{λ}} (h)$ (2) k is Z-soft rough fuzzy edge set i.e., ${\bar{a p p r}}_{Z_{η}} (k) \neq {\underline{a p p r}}_{Z_{η}} (k)$ A Z-soft rough fuzzy graph is denoted by

Definition 10. Let be a Z-soft rough fuzzy graph. Then $\underline{a p p r} (G) = ({\underline{a p p r}}_{Z_{λ}} (h) {\underline{a p p r}}_{Z_{η}} (k))$ and $\bar{a p p r} (G) = ({\bar{a p p r}}_{Z_{λ}} (h) {\bar{a p p r}}_{Z_{η}} (k))$ (k)) respectively, are called the lower and upper Z-soft approximations of $G = (G_{Z_{λ}}, G_{Z_{η}})$ , for $h \in F (V)$ and $k \in F (E)$

Definition 11. Let $G = (G^{*}, ψ, ξ, A)$ be a soft graph over the simple graph G^∗ = (V, E). Then $G$ is called

Full soft vertex graph if ⋃_ρ∈Aψ (ρ) = V and

Full soft edge graph if ⋃_ρ∈Aξ (ρ) = E.

A soft graph $G = (G^{*}, ψ, ξ, A)$ is called a full soft graph if G^∗ = (⋃ _ρ∈Aψ (ρ) , ⋃ _ρ∈Aξ (ρ).

Proposition 1. Let $G = (G^{*}, ψ, ξ, A)$ be a full soft graph then for $h \in F (V)$ and $k \in F (E),$ (1)

$\underline{appr} (G) = (⋀_{x \in ψ (ρ)} ⋀_{y \in ψ (ρ)} h (y), ⋀_{e \in ξ (ρ)} ⋀_{f \in ξ (ρ)} k (f))$ and (2) $\bar{appr} (G) = (⋁_{x \in ψ (ρ)} ⋁_{y \in ψ (ρ)} h (y), ⋁_{e \in ξ (ρ)} ⋁_{f \in ξ (ρ)} k (f)) .$

Proof. (1) Since $\underline{a p p r} (G) = ({\underline{a p p r}}_{Z_{λ}} (h), {\underline{a p p r}}_{Z_{η}} (k))$ (k)) so it is necessary to show only

${\underline{appr}}_{Z_{λ}} (h) (x) = ⋀_{x \in ψ (ρ)} ⋀_{y \in ψ (ρ)} h (y)$ and

${\underline{a p p r}}_{Z_{λ}} (h) (x) = \underset{x \in ψ (ρ)}{\land} \underset{y \in ψ (ρ)}{\land} h (y) a n d {\underline{a p p r}}_{Z_{η}} (k) (e) = \underset{e \in ξ (ρ)}{\land} \underset{f \in ξ (ρ)}{\land} k (f)$ As $G = (G *, ψ, ξ, A)$ be a full soft graph so for x ∈ V, e ∈ E there exists ρ ∈ A such that x ∈ ψ (ρ) and e ∈ ξ (ρ). But ${\underline{a p p r}}_{Z_{λ}} (h) (x) = \land {h (y) : y \in V, λ (x) = λ (y)}$ and ${\underline{a p p r}}_{Z_{η}} (k) (e) = \land {k (f) : f \in E, η (e) = η (f)}$ So $h (y) \geq {\underline{a p p r}}_{Z_{λ}} (h) (x)$ and $k (f) \geq {\underline{a p p r}}_{Z_{η}} (k) (e)$ Which implies $\underset{y \in ψ (ρ)}{\land} h (y) \geq {\underline{a p p r}}_{Z_{λ}} (h) (x)$ and $\underset{f \in ξ (ρ)}{\land} k (f) \geq {\underline{a p p r}}_{Z_{η}} (k) (e)$ or $\underset{x \in ψ (ρ)}{\land} \underset{y \in ψ (ρ)}{\land} h (y) \geq {\underline{a p p r}}_{Z_{λ}} (h) (x)$ and $\underset{e \in ξ (ρ)}{\land} \underset{f \in ξ (ρ)}{\land} k (f) \geq {\underline{a p p r}}_{Z_{η}} (k) (e)$ Let λ (x) = λ (q) and η (e) = η (l) , so ⋀{ h (q): q ∈ V, λ (x) = λ (q) } ≥ ⋀ _x∈ψ(ρ) ⋀ _y∈ψ(ρ)h (y) and ⋀ { k (l): l ∈ E, η (e) = η (l)} ≥ ⋀ _e∈ξ(ρ) ⋀ _f∈ξ(ρ)k (f) showing ${\underline{appr}}_{Z_{λ}} (h) (x) \geq ⋀_{x \in ψ (ρ)} ⋀_{y \in ψ (ρ)} h (y)$ and ${\underline{appr}}_{Z_{η}} (k) (e) \geq ⋀_{e \in ξ (ρ)} ⋀_{f \in ξ (ρ)} k (f) .$ Since $⋀_{x \in ψ (ρ)} ⋀_{y \in ψ (ρ)} h (y) \geq {\underline{appr}}_{Z_{λ}} (h) (x),$ appr_{Z
_λ} (h) (x) ≥ ⋀ _x∈ψ(ρ) ⋀ _y∈ψ(ρ)h (y) and $⋀_{e \in ξ (ρ)} ⋀_{f \in ξ (ρ)} k (f) \geq {\underline{appr}}_{Z_{η}} (k) (e),$ ${\underline{appr}}_{Z_{η}} (k) (e) \geq ⋀_{e \in ξ (ρ)} ⋀_{f \in ξ (ρ)} k (f) .$ Therefore ${\underline{appr}}_{Z_{λ}} (h) (x) = ⋀_{x \in ψ (ρ)} ⋀_{y \in ψ (ρ)} h (y)$ and ${\underline{appr}}_{Z_{η}} (k) (e) = ⋀_{e \in ξ (ρ)} ⋀_{f \in ξ (ρ)} k (f) .$ Hence (1) is proved. (2) As $\bar{appr} (G) = ({\bar{appr}}_{Z_{λ}} (h), {\bar{appr}}_{Z_{η}} (h)),$ so it is only to show ${\bar{appr}}_{Z_{λ}} (h) (x) = ⋁_{x \in ψ (ρ)} ⋁_{y \in ψ (ρ)} h (y)$ and ${\bar{appr}}_{Z_{η}} (k) (e) = ⋁_{e \in ξ (ρ)} ⋁_{f \in ξ (ρ)} k (f) .$ Given $G = (G^{*},$ ψ, ξ, A) is a full soft graph so for x ∈ V, e ∈ E there exists ρ ∈ A such that x ∈ ψ (ρ) and e ∈ ξ (ρ). But ${\bar{appr}}_{Z_{λ}} (h) (x) = ⋁ {h (y) : y \in V, λ (x) = λ (y)}$ and ${\bar{appr}}_{Z_{η}} (h) (e) = ⋁ {k (f) : f \in E, η (e) = η (f)} .$

Therefore $h (y) \leq {\bar{appr}}_{Z_{λ}} (h) (x)$ and $k (f) \leq {\bar{appr}}_{Z_{η}} (h) (e),$ which gives $⋁_{y \in ψ (ρ)} h (y) \leq {\bar{appr}}_{Z_{λ}} (h) (x)$ and ⋁_f∈ξ(ρ) $k (f) \leq {\bar{appr}}_{Z_{η}} (h) (e) .$ So $⋁_{x \in ψ (ρ)} ⋁_{y \in ψ (ρ)} h (y) \leq {\bar{appr}}_{Z_{λ}} (h) (x)$ and $⋁_{e \in ξ (ρ)} ⋁_{f \in ξ (ρ)} k (f) \leq {\bar{appr}}_{Z_{η}} (h) (e) .$ Now let λ (x) = λ (q) and η (e) = η (l) , so h (q) ≤ ⋁ _x∈ψ(ρ) ⋁ _y∈ψ(ρ)h (y) and k (l) ≤ ⋁ _e∈ξ(ρ) ⋁ _f∈ξ(ρ) k (f) , ⋁ { k (l): l ∈ E, η (e) = η (l) } ≤ ⋁ _e∈ξ(ρ) ⋁ _f∈ξ(ρ) k (f) showing that ${\bar{appr}}_{Z_{λ}} (h) (x) \leq ⋁_{x \in ψ (ρ)} ⋁_{y \in ψ (ρ)} h (y)$ and ${\bar{appr}}_{Z_{η}} (k) (e) \leq ⋁_{e \in ξ (ρ)} ⋁_{f \in ξ (ρ)} k (f) .$ As $⋁_{x \in ψ (ρ)} ⋁_{y \in ψ (ρ)} h (y) \geq {\bar{appr}}_{Z_{λ}} (h) (x),$ ${\bar{appr}}_{Z_{λ}} (h) (x) \leq ⋁_{x \in ψ (ρ)} ⋁_{y \in ψ (ρ)} h (y)$

and $⋁_{e \in ξ (ρ)} ⋁_{f \in ξ (ρ)} k (f) \geq {\bar{appr}}_{Z_{η}} (k) (e),$

${\bar{appr}}_{Z_{η}} (k) (e) \leq ⋁_{e \in ξ (ρ)} ⋁_{f \in ξ (ρ)} k (f) .$ So ${\bar{appr}}_{Z_{λ}} (h) (x) = ⋁_{x \in ψ (ρ)} ⋁_{y \in ψ (ρ)} h (y)$ and

${\bar{appr}}_{Z_{η}} (k) (e) = ⋁_{e \in ξ (ρ)} ⋁_{f \in ξ (ρ)} k (f) .$ Which completes the proof.□

Proposition 2. Let $G = (G^{*}, ψ, ξ, A)$ be a full soft graph then for $h \in F (V)$ and $k \in F (E),$ we have $(1) \underline{appr} (G) \subseteq \bar{appr} (G)$ $(2) \underline{appr} (φ) = \bar{appr} (φ) = φ$ $(3) \underline{appr} (G^{*}) = \bar{appr} (G^{*}) = G^{*}$ (4) ${(\bar{appr} (G))}^{c} = \underline{appr} (G^{c}),$ ${(\underline{appr} (G))}^{c} = \bar{appr} (G^{c})$ $(5) \underline{appr} (G_{1} \cap G_{2}) = \underline{appr} (G_{1}) \cap \underline{appr} (G_{2})$ (6) $\underline{appr} (G_{1} \cup G_{2}) \supseteq \underline{appr} (G_{1}) \cup \underline{appr} (G_{2})$ (7) $\bar{appr} (G_{1} \cup G_{2}) = \bar{appr} (G_{1}) \cup \bar{appr} (G_{2})$ (8) $\bar{appr} (G_{1} \cap G_{2}) \subseteq \bar{appr} (G_{1}) \cap \bar{appr} (G_{2}) .$

Proof. (1), (2) and (3) are straightforward. (4) Since $\bar{appr} (G) = ({\bar{appr}}_{Z_{λ}} (h), {\bar{appr}}_{Z_{η}} (k)),$ so we will prove only ${({\bar{appr}}_{Z_{λ}} (h))}^{c} = {\underline{appr}}_{Z_{λ}} (h^{c})$ and ${({\bar{appr}}_{Z_{η}} (k))}^{c} = {\underline{appr}}_{Z_{η}} (k^{c}) .$

Since ${\bar{appr}}_{Z_{λ}} (h) (x) = ⋁ {h (y) : y \in V, λ (x) = λ (y)}$

and ${\bar{appr}}_{Z_{η}} (k) (e) = ⋁ {k (f) : f \in E, η (e) =$ η (f) } . So $h (y) \leq {\bar{appr}}_{Z_{λ}} (h) (x)$ and $k (f) \leq {\bar{appr}}_{Z_{η}} (k) (e) .$ $1 - h (y) \geq 1 - {\bar{appr}}_{Z_{λ}} (h) (x)$ and $1 - k (f) \geq 1 - {\bar{appr}}_{Z_{η}} (k) (e) .$ $h^{c} (y) \geq {({\bar{appr}}_{Z_{λ}} (h))}^{c} (x)$ and $k^{c} (f) \geq {({\bar{appr}}_{Z_{η}} (k))}^{c} (e) .$ Or $⋀ {h^{c} (y) : y \in V, λ (x) = λ (y)} \supseteq {\bar{appr}}_{Z_{λ}} {(h)}^{c} (x)$ and $⋀ {k^{c} (f) : f \in E, η (e) = η (f)} \supseteq {({\bar{appr}}_{Z_{η}} (k))}^{c} (e) .$ Which shows

${\underline{appr}}_{Z_{λ}} (h^{c}) (x) \geq {({\bar{appr}}_{Z_{λ}} (h))}^{c} (x)$

and ${\underline{appr}}_{Z_{η}} (k^{c}) (e) \geq {({\bar{appr}}_{Z_{η}} (k))}^{c} (e) .$

So ${\underline{appr}}_{Z_{λ}} (h^{c}) \supseteq {({\bar{appr}}_{Z_{λ}} (h))}^{c} (x)$

and ${\underline{appr}}_{Z_{η}} (k^{c}) (e) \supseteq {({\bar{appr}}_{Z_{η}} (k))}^{c} (e) .$

Now let $r \in F (V)$ and $m \in F (E)$

such that ${\underline{appr}}_{Z_{λ}} (r) (x) = ⋀ {r (y) : y \in V,$ λ (x) = λ (y)} and ${\underline{appr}}_{Z_{η}} (m) (e) = ⋀ {m (f) : f \in E, η (e) =$ η (f) } .

Then $r (y) \geq {\underline{appr}}_{Z_{λ}} (r) (x)$ and $m (f) \geq {\underline{appr}}_{Z_{η}}$ (m) (e) ,

or $1 - r (y) \leq 1 - {\underline{a p p r}}_{Z_{λ}} (r) (x)$ and $1 - m (f) \leq 1 - {\underline{appr}}_{Z_{η}} (m) (e) .$ $r^{c} (y) \leq {({\underline{appr}}_{Z_{λ}} (r))}^{c} (x)$ and $m^{c} (f) \leq ({\underline{appr}}_{Z_{η}}$ (m)) ^c (e) ⋁{ r^c (y): y ∈ V, λ (x) = λ (y) }

$r^{c} (y) \leq {({\underline{a p p r}}_{Z_{λ}} (r))}^{c} (x)$ and ⋁{ m^c (f): f ∈ E, η (e) = η (f) }

So $\lor {r^{c} (y) : y \in V, λ (x) = λ (y)} \leq {({\underline{a p p r}}_{Z_{λ}} (r))}^{c} (x) a n d \lor {m^{c} (f) : f \in E, η (e) = η (f)} \leq {({\bar{a p p r}}_{Z_{η}} (m))}^{c} (e)$ Replacing r by h^c and m by n^c, we get ${\bar{a p p r}}_{Z_{λ}} {(h^{c})}^{c} (x) \leq {({\underline{a p p r}}_{Z_{λ}} (h^{c}))}^{c} (x)$ and ${\bar{a p p r}}_{Z_{η}} (n^{c})^{c} (e) \leq {({\underline{a p p r}}_{Z_{η}} (n^{c}))}^{c} (e)$ ${\bar{a p p r}}_{Z_{η}} {(n^{c})}^{c} (e) \leq {({\underline{a p p r}}_{Z_{η}} (n^{c}))}^{c} (e)$ and ${\bar{a p p r}}_{Z_{λ}} {(h^{c})}^{c} (x) \subseteq {({\underline{a p p r}}_{Z_{λ}} (h^{c}))}^{c} (x)$ So ${\bar{a p p r}}_{Z_{η}} (n) (e) \subseteq {({\underline{a p p r}}_{Z_{η}} (n^{c}))}^{c} (e)$ and ${\underline{a p p r}}_{Z_{λ}} (h^{c}) \subseteq {({\bar{a p p r}}_{Z_{η}} (n))}^{c}$ Since ${\underline{a p p r}}_{Z_{η}} (n^{c}) \subseteq {({\bar{a p p r}}_{Z_{η}} (n))}^{c}$ , ${\underline{a p p r}}_{Z_{η}} (n^{c}) \subseteq {({\bar{a p p r}}_{Z_{η}} (n))}^{c}$ and ${\underline{a p p r}}_{Z_{η}} (n^{c}) \subseteq {({\bar{a p p r}}_{Z_{η}} (n))}^{c}$ , ${\underline{a p p r}}_{Z_{λ}} (h^{c}) \subseteq {({\bar{a p p r}}_{Z_{λ}} (h))}^{c}$ . Therefore ${\underline{a p p r}}_{Z_{η}} (n^{c}) \subseteq {({\bar{a p p r}}_{Z_{η}} (n))}^{c}$ and ${({\bar{a p p r}}_{Z_{λ}} (h))}^{c} = {\underline{a p p r}}_{Z_{λ}} (h^{c})$ . Hence ${({\bar{a p p r}}_{Z_{η}} (n))}^{c} = {\underline{a p p r}}_{Z_{η}} (n^{c})$ Similarly one can show that ${(\bar{a p p r} (G))}^{c} = \underline{a p p r} (G^{c})$ (5) Suppose ${(\underline{a p p r} (G))}^{c} = \bar{a p p r} (G^{c})$ and $\underline{a p p r} (G_{1} \cap G_{2}) = ({\underline{a p p r}}_{Z_{λ}} (h_{1} \cap h_{2}), {\underline{a p p r}}_{Z_{η}} (K_{1} \cap k_{2}))$ where h_i∈ $\underline{a p p r} (G_{i}) = ({\underline{a p p r}}_{Z_{λ}} (h_{i}), {\underline{a p p r}}_{Z_{η}} (k_{i}))$ and $h_{i} \in F (V)$ for i = 1, 2. Using ${\underline{appr}}_{Z_{λ}} (h_{1} \cap h_{2}) (x)$ =⋀ { h₁ (y) ∧ h₂ (y): y ∈ V, λ (x) = λ (y) } and ${\underline{appr}}_{Z_{η}} (k_{1} \cap k_{2}) (e)$ = ⋀ { k₁ ((f)) ∧ k₂ (f): f ∈ E, η (e) = η (f) } , we have, ${\underline{appr}}_{Z_{λ}} (h_{1} \cap h_{2}) (x) \leq h_{1} (y) \land h_{2} (y) \leq h_{1} (y)$ and ${\underline{appr}}_{Z_{η}} (k_{1} \cap k_{2}) (e_{i}) \leq k_{1} ((f)) \land k_{2} (f) \leq k_{1} (f) .$ Which shows ${\underline{a p p r}}_{Z_{λ}} (h_{1} \cap^{} h_{2}) (x)$

≤⋀ { h₁ (y): y ∈ V, λ (x) = λ (y) }

$k_{i} \in F (E)$ ${\underline{appr}}_{Z_{η}} (k_{1} \cap k_{2}) (e)$

≤⋀ { m₁ (f): f ∈ E, η (e) = η (f) }

$= {\underline{appr}}_{Z_{η}} (m_{1}) (e) .$

It can also be seen that ${\underline{a p p r}}_{Z_{λ}} (h_{1} \cap^{} h_{2}) (x) \leq {\underline{a p p r}}_{Z_{λ}} (h_{2}) (x)$

and ${\underline{appr}}_{Z_{η}} (k_{1} \cap k_{2}) (e) \leq {\underline{appr}}_{Z_{η}} (k_{2}) (e) .$

As ${\underline{appr}}_{Z_{λ}} (h_{1} \cap h_{2}) (x) \leq {\underline{appr}}_{Z_{λ}} (h_{1}) (x)$

and ${\underline{appr}}_{Z_{λ}} (h_{1} \cap h_{2}) (x) \leq {\underline{appr}}_{Z_{λ}} (h_{2}) (x)$

so ${\underline{appr}}_{Z_{λ}} (h_{1} \cap h_{2}) (x) \leq {\underline{appr}}_{Z_{λ}} (h_{1}) (x) \land {\underline{appr}}_{Z_{λ}} (h_{2}) (x) .$ Or ${\underline{appr}}_{Z_{λ}} (h_{1} \cap h_{2}) \subseteq {\underline{appr}}_{Z_{λ}} (h_{1}) \cap {\underline{appr}}_{Z_{λ}} (h_{2}) .$ Also ${\underline{appr}}_{Z_{η}} (k_{1} \cap k_{2}) (e) \leq {\underline{appr}}_{Z_{η}} (k_{1}) (e)$

and ${\underline{appr}}_{Z_{η}} (k_{1} \cap k_{2}) (e) \leq {\underline{appr}}_{Z_{η}} (k_{2}) (e)$ so ${\underline{appr}}_{Z_{η}} (k_{1} \cap k_{2}) (e) \leq {\underline{appr}}_{Z_{η}} (k_{1}) (e) \land {\underline{appr}}_{Z_{η}}$ (k₂) (e) or ${\underline{appr}}_{Z_{η}} (k_{1} \cap k_{2}) \subseteq {\underline{appr}}_{Z_{η}} (k_{1}) \cap {\underline{appr}}_{Z_{η}} (k_{2}) .$ Hence ${\underline{appr}}_{Z_{λ}} (h_{1} \cap h_{2}) \subseteq {\underline{appr}}_{Z_{λ}} (h_{1}) \cap {\underline{appr}}_{Z_{λ}} (h_{2})$ and ${\underline{appr}}_{Z_{η}} (k_{1} \cap k_{2}) \subseteq {\underline{appr}}_{Z_{η}} (k_{1}) \cap {\underline{appr}}_{Z_{η}} (k_{2}) .$ Now we show the reverse inclusion. It can see seen that $({\underline{appr}}_{Z_{λ}} (h_{1}) \cap {\underline{appr}}_{Z_{λ}} (h_{2})) (x)$

$= {\underline{appr}}_{Z_{λ}} (h_{1}) (x) \land {\underline{appr}}_{Z_{λ}} (h_{2}) (x)$ $\leq {\underline{appr}}_{Z_{λ}} (h_{1}) (x) \leq (h_{1}) (y)$ and

$({\underline{appr}}_{Z_{λ}} (h_{1}) \cap {\underline{appr}}_{Z_{λ}} (h_{2})) (x)$ $= {\underline{appr}}_{Z_{λ}} (h_{1}) (x) \land {\underline{appr}}_{Z_{λ}} (h_{2}) (x)$

$\leq {\underline{appr}}_{Z_{λ}} (h_{2}) (x) \leq (h_{2}) (y)$ so $({\underline{appr}}_{Z_{λ}} (h_{1}) \cap {\underline{appr}}_{Z_{λ}} (h_{2})) (x) \leq (h_{1}) (y) \land (h_{2}) (y)$ Which shows $\begin{matrix} ({\underline{appr}}_{Z_{λ}} (h_{1}) \cap {\underline{appr}}_{Z_{λ}} (h_{2})) (x) \\ \leq & ⋀ {(h_{1}) (y) \land (h_{2}) (y) : y \in V, λ (x) = λ (y)}, \\ = & {\underline{appr}}_{Z_{λ}} (h_{1} \cap h_{2}) (x), \end{matrix}$ so $({\underline{appr}}_{Z_{λ}} (h_{1}) \cap {\underline{appr}}_{Z_{λ}} (h_{2})) \subseteq {\underline{appr}}_{Z_{λ}}$ (h₁ ∩ h₂). Similarly it can be seen that $({\underline{appr}}_{Z_{η}} (k_{1}) \cap$ ${\underline{appr}}_{Z_{η}} (k_{2})) \subseteq {\underline{appr}}_{Z_{η}} (k_{1} \cap k_{2}) .$ As $\begin{matrix} ({\underline{appr}}_{Z_{λ}} (h_{1}) \cap {\underline{appr}}_{Z_{λ}} (h_{2})) \\ \subseteq & {\underline{appr}}_{Z_{λ}} (h_{1} \cap h_{2}) \end{matrix}$ and $({\underline{appr}}_{Z_{λ}} (h_{1}) \cap {\underline{appr}}_{Z_{λ}} (h_{2})) \supseteq {\underline{appr}}_{Z_{λ}} (h_{1} \cap h_{2}),$

so $({\underline{appr}}_{Z_{λ}} (h_{1}) \cap {\underline{appr}}_{Z_{λ}} (h_{2})) = {\underline{appr}}_{Z_{λ}} (h_{1}$ ∩h₂). As $({\underline{appr}}_{Z_{η}} (k_{1}) \cap {\underline{appr}}_{Z_{η}} (k_{2})) \subseteq {\underline{appr}}_{Z_{η}} (k_{1} \cap k_{2})$ and $({\underline{appr}}_{Z_{η}} (k_{1}) \cap {\underline{appr}}_{Z_{η}} (k_{2})) \supseteq {\underline{appr}}_{Z_{η}} (k_{1} \cap k_{2}),$

so $({\underline{appr}}_{Z_{η}} (k_{1}) \cap {\underline{appr}}_{Z_{η}} (k_{2})) = {\underline{appr}}_{Z_{η}} (k_{1} \cap k_{2}) .$ Hence $\underline{appr} (G_{1} \cap G_{2}) = \underline{appr} (G_{1}) \cap \underline{appr} (G_{2}) .$ (6) Suppose $\underline{appr} (G_{1} \cup G_{2})$

$= ({\underline{a p p r}}_{Z_{λ}} (h_{1} \cup^{} h_{2}) , {\underline{a p p r}}_{Z_{η}} (k_{1} \cup^{} k_{2}))$ and $\underline{appr} (G_{i}) = ({\underline{appr}}_{Z_{λ}} (h_{i}), {\underline{appr}}_{Z_{η}} (k_{i})),$ where $h_{i} \in F (V)$ and $k_{i} \in F (E)$ for i = 1, 2. Since ${\underline{a p p r}}_{Z λ} (h_{1}) (x)$ so $\begin{matrix} {\underline{appr}}_{Z_{λ}} (h_{1} \cup h_{2}) (x) \\ = & ⋀ {h_{1} (y) \lor h_{2} (y) : y \in V, λ (x) = λ (y)} \\ \geq & {\underline{appr}}_{Z} (h_{1}) (x) . \end{matrix}$ Similarly it can be seen that ${\underline{appr}}_{Z_{λ}} (h_{1} \cup h_{2}) (x) \geq {\underline{appr}}_{Z_{λ}} (h_{2}) (x) .$ Thus $\begin{matrix} {\underline{appr}}_{Z_{λ}} (h_{1} \cup h_{2}) (x) \\ \geq & {\underline{appr}}_{Z_{λ}} (h_{1}) (x) \lor {\underline{appr}}_{Z_{λ}} (h_{1}) (x) \\ = & ({\underline{appr}}_{Z_{λ}} (h_{1}) \lor {\underline{appr}}_{Z_{λ}} (h_{2})) (x) \end{matrix}$ Hence ${\underline{appr}}_{Z_{λ}} (h_{1} \cup h_{2}) \supseteq {\underline{appr}}_{Z_{λ}} (h_{1}) \cup {\underline{appr}}_{Z_{λ}} (h_{2}) .$ Furthermore, ${\underline{appr}}_{Z_{η}} (k_{1}) (e) = ⋀ {k_{1} (f) : f \in E, η (e) = η (f)}$ ≤k₁ (f) ≤ k₁ (f) ∨ k₂ (f) so $\begin{matrix} {\underline{appr}}_{Z_{η}} (k_{1} \cup k_{2}) (e) \\ = & ⋀ {k_{1} (f) \lor k_{2} (f) : f \in E, η (e) = η (f)} \\ \geq & {\underline{appr}}_{Z_{η}} (k_{1}) (e) . \end{matrix}$ Similarly it can be seen ${\underline{appr}}_{Z_{η}} (k_{1} \cup k_{2}) (e) \geq {\underline{appr}}_{Z_{η}} (k_{2}) (f) .$ Thus, $\begin{matrix} {\underline{appr}}_{Z_{η}} (k_{1} \cup k_{2}) (e) \\ \geq & {\underline{appr}}_{Z_{η}} (k_{1}) (e) \lor {\underline{appr}}_{Z_{η}} (k_{2}) (e) \\ = & ({\underline{appr}}_{Z_{η}} (k_{1}) \lor {\underline{appr}}_{Z_{η}} (k_{2})) (e) . \end{matrix}$ Hence ${\underline{appr}}_{Z_{η}} (k_{1} \cup k_{2}) \supseteq {\underline{appr}}_{Z_{η}} (k_{1}) \cup {\underline{appr}}_{Z_{η}} (k_{2}) .$ $\begin{matrix} ({\underline{appr}}_{Z_{λ}} (h_{1} \cup h_{2}), {\underline{appr}}_{Z_{η}} (k_{1} \cup k_{2})) \\ \supseteq & ({\underline{appr}}_{Z_{λ}} (h_{1}) \cup {\underline{appr}}_{Z_{λ}} (h_{1}), {\underline{appr}}_{Z_{η}} (k_{1}) \cup {\underline{appr}}_{Z_{η}} (k_{2})), \end{matrix}$ showing that $\underline{a p p r} (G_{1} \cup^{} G_{2}) \supseteq \underline{a p p r} (G_{1}) \cup^{} \underline{a p p r}$ (G₂). (7) Can be proved with the help of (6) (8) Can be proved with the help of (5).□

4 Application of Z-soft rough fuzzy graphs

In this section, an algorithm is formulated for decision making problems based on Z-soft rough fuzzy graphs. To show the application of Z-soft rough fuzzy graphs in decision making, an example is constructed. Suppose V ={ x₁, x₂, x₃,..., x_r } is the set of r vertices(animals) and A ={ ρ₁, ρ₂, ρ₃,..., ρ_m } is the set of m parameters(infectiousdiseases), then G^∗ = (V, E) , a simple digraph whose vertex set is V and edge set is E. Let (ψ, A) and (ξ, A) be two soft sets over V and E respectively such that for i = 1, 2, 3,..., m and j = 1, 2, 3,..., r. $\begin{matrix} ψ (ρ_{i}) & = & {x_{j} \in V : x_{j} may have infectious disease ρ_{i}}, \\ ξ (ρ_{i}) & = & {x_{i} x_{j} \in E : x_{i} with disease ρ_{i} has affected x_{j}} . \end{matrix}$ For each i, G_i = (ψ (ρ_i) , ξ (ρ_i)) is a subgraph of G^∗ showing that $G = (G^{*}, ψ, ξ, A)$ is a soft graph. For basic evaluation, we compute the lower/upper Z-soft rough fuzzy vertex approximations for a fuzzy subset h of V. Suppose Φ (x_j) be a optimality choice function Φ (x_j) is given as follows; $\begin{matrix} {\underline{{\underline{apr}}_{Z_{λ}} h (x_{j}) + {\bar{apr}}_{Z_{λ}} h (x_{j})}}_{\cdot} \\ {\underline{apr}}_{Z_{λ}} h_{(x_{j})} + {\bar{apr}}_{Z_{λ}} h_{(x_{j})} - {\underline{apr}}_{Z_{λ}} h_{(x_{j})} {\bar{apr}}_{Z_{λ}} h_{(x_{j})} \end{matrix}$ Now we consider the set of edges which indicates the affectedness of each other. The marginal weight function Ω (x_j) for each x_j can be computed by; $Ω (x_{j}) = [Ω_{r} (x_{j}) + Ω_{c} (x_{j})]$ for j = 1, 2, 3,..., r, where $Ω_{r} (x_{j}) = \underset{i = 1}{\overset{m}{\frac{1}{r} \sum}} χ_{E} (x_{j} x_{i}),$ is the measures the affectedness of x_jwith x_m, and $Ω_{c} (x_{j}) = \frac{1}{r} \underset{i = 1}{\sum^{m}} χ_{E} (x_{i} x_{j}),$ is the measures the affectedness of x_m with x_j. Where χ_E is an indicator function on E, defined by $χ_{E} (x_{k} x_{j}) = {\begin{matrix} 1 if x_{m} x_{j} makes an edge \\ 0, otherwise \end{matrix} .$ Henceforth the marginal weight function Ω for each x_j, in both ways, actually measures the degree of affectedness. Finally an evaluation function Θ is defined on V by $Θ (x_{j}) = Φ (x_{j}) . Ω (x_{j})$ for j = 1, 2, 3,..., r. For a threshold β ∈ [0, 1], it can be seen that the all vertices x_j are at optimum for all j, in which Θ (x_j) ≥ β. The vertex x_p is best optimal if $Θ (x_{p}) = \max_{j} {Θ (x_{j})}$ . This algorithm involved both the individuals evaluations as well as the effects of interaction between them. This can be an interaction of two poles of transportation or network problems. This algorithms can be applied by someone to other related problems. A real life application for diagnosing diseases from a group of animals(sheep) has been considered below. A decision making algorithm for Z-soft rough fuzzy graphs can be presented as follows:

Pseudo code

Input is the soft graph $G = (G^{*}, ψ, ξ, A)$ with soft sets (ψ, A) and (ξ, A) over V and E respectively.

Find lower/upper Z-soft rough fuzzy vertex approximations for a fuzzy subset h of V.

Compute the optimality choice function Φ (x_j).

Calculate the weights for each vertex x_j, given by $Ω (x_{j}) = [Ω_{r} (x_{j}) + Ω_{c} (x_{j})]$

Finally calculate the evaluation function Θ, defined on set of vertices V by $Θ (x_{j}) = Φ (x_{j}) . Ω (x_{j})$ The vertex x_p ∈ V is best optimal if $Θ (x_{p}) = \max_{j} {Θ (x_{j})}$ j = 1, 2,... r.

Example 2. Suppose during the medical checkup of animals(sheep) kept in a farm house, five infectious and parasitic diseases found in a group of 20 sheep V ={ x₁, x₂, x₃,..., x₂₀ }, through different sources such as “caused by toxins produced by Clostridium botulinum”, “caused by Corynebacterium pyogenes”, “caused by Clostridium septicum”, “caused by Erysipelothrix insidiosa” and “caused by Bacillus anthracis”. The above process of epidemic results in a diversification of symptoms that vary in severity and character, depending upon the individual factor and the kind of viral infection. Suppose A ={ ρ₁, ρ₂, ρ₃, ρ₄, ρ₅ } be the set of parameters such that ρ₁ represents “animal affected by toxins produced by Clostridium botulinum”, ρ₂ represents “animal affected by Corynebacterium pyogenes”, ρ₃ represents “animal affected by Clostridium septicum, ρ₄ represents ” animal affected by Erysipelothrix insidiosa and ρ₅ represents “animal affected by Bacillus anthracis”. It is also assumed that an animal may have more than one infectious disease. Suppose G^∗ is a simple digraph having vertex set V of 20 animals and edge set E. Let (ψ, A) and (ξ, A) be two soft sets over V and E such that for i = 1, 2, 3, 4, 5 and j = 1, 2, 3,..., 20. $\begin{matrix} ψ (ρ_{i}) & = & {x_{j} \in V : x_{j} may have infectious disease ρ_{i}}, \\ ξ (ρ_{i}) & = & {x_{i} x_{j} \in E : x_{i} with disease ρ_{i} has affected x_{j}} . \end{matrix}$ Let $\begin{matrix} ψ (ρ_{1}) & = & {x_{1}, x_{2}, x_{6}, x_{7}, x_{9}, x_{13}, x_{16}, x_{20}}, \\ ψ (ρ_{2}) & = & {x_{2}, x_{3}, x_{4}, x_{8}, x_{10}, x_{11}, x_{14}}, \end{matrix}$ $\begin{matrix} ψ (ρ_{3}) & = & {x_{5}, x_{6}, x_{12}, x_{14}, x_{15}, x_{17}, x_{18}, x_{19}}, \\ ψ (ρ_{4}) & = & {x_{1}, x_{3}, x_{6}, x_{8}, x_{9}, x_{10}, x_{13}, x_{19}}, \end{matrix}$ $ψ (ρ_{5}) = {x_{5}, x_{7}, x_{15}, x_{19}} .$ Let $ξ (ρ_{1}) = {\begin{matrix} x_{20} x_{7}, x_{1} x_{13}, x_{1} x_{16}, x_{2} x_{3}, x_{1} \\ x_{7}, x_{6} x_{16}, x_{7} x_{16}, x_{20} x_{1}, x_{6} x_{1}, \\ x_{16} x_{9}, x_{6} x_{13}, x_{7} x_{1}, x_{7} x_{2}, \\ x_{7} x_{20}, x_{9} x_{1}, x_{9} x_{2}, x_{9} x_{7}, x_{9} x_{13}, \\ x_{13} x_{6}, x_{16} x_{13}, x_{16} x_{20}, x_{20} x_{6}, \\ x_{20} x_{6}, x_{20} x_{6}, x_{1} x_{2}, x_{1} x_{6}, \\ x_{6} x_{2}, x_{16} x_{1} \end{matrix}}$ $ξ (ρ_{2}) = {\begin{matrix} x_{10} x_{3}, x_{10} x_{11}, x_{2} x_{6}, x_{2} x_{11}, x_{3} x_{8}, \\ x_{3} x_{10}, x_{4} x_{2}, x_{4} x_{3}, x_{4} x_{8}, x_{4} x_{10}, x_{4} x_{14}, \\ x_{14} x_{2}, x_{14} x_{10}, x_{8} x_{3}, x_{8} x_{4}, x_{8} x_{10}, \\ x_{8} x_{14}, x_{10} x_{14}, x_{11} x_{2}, x_{4} x_{11}, x_{11} x_{14} \\ x_{11} x_{3}, x_{11} x_{10}, \end{matrix}},$ $ξ (ρ_{3}) = {\begin{matrix} x_{17} x_{5}, x_{17} x_{6}, x_{5} x_{6}, x_{14} x_{5}, x_{14} x_{6}, \\ x_{5} x_{15}, x_{5} x_{19}, x_{6} x_{5}, x_{6} x_{17}, x_{15} x_{12}, \\ x_{18} x_{17}, x_{5} x_{18}, x_{12} x_{5}, x_{12} x_{6}, x_{12} x_{14}, \\ x_{12} x_{15}, x_{14} x_{15}, x_{14} x_{17}, x_{14} x_{19}, x_{15} x_{5}, \\ x_{15} x_{6}, x_{15} x_{14}, x_{15} x_{17}, x_{6} x_{15}, x_{17} x_{12}, \\ x_{17} x_{14}, x_{17} x_{19}, x_{18} x_{12}, x_{18} x_{14}, x_{18} x_{19}, \\ x_{19} x_{14}, x_{12} x_{17}, x_{12} x_{19}, \end{matrix}},$ $ξ (ρ_{4}) = {\begin{matrix} x_{13} x_{19}, x_{3} x_{1}, x_{3} x_{6}, x_{3} x_{9}, x_{3} x_{13}, \\ x_{3} x_{19}, x_{6} x_{8}, x_{6} x_{19}, x_{8} x_{1}, x_{8} x_{9}, \\ x_{8} x_{19}, x_{9} x_{19}, x_{10} x_{1}, x_{10} x_{9}, x_{10} x_{13}, \\ x_{10} x_{19}, x_{13} x_{10}, x_{9} x_{1}, x_{9} x_{3} \end{matrix}}$ $ξ (ρ_{5}) = {x_{5} x_{7}, x_{7} x_{5}, x_{7} x_{15}, x_{7} x_{19}} .$ Clearly $G = (G^{*}, ψ, ξ, A)$ is a soft graph, i.e., for each i = 1, 2, 3, 4, 5 G_i = (ψ (ρ) , ξ (ρ_i)) is a subgraph of G^∗. Let λ: V → P (A) be a mapping defined as $λ (x) = {ρ \in A : x \in ψ (ρ)}$ such that $λ (x_{1}) = λ (x_{9}) = λ (x_{13}) = {ρ_{1}, ρ 4}$ $λ (x_{2}) = {ρ_{1}, ρ_{2}}$ $λ (x_{3}) = λ (x_{8}) = λ (x_{10}) = {ρ_{2}, ρ_{4}},$ $λ (x_{11}) = λ (x_{4}) = {ρ_{2}}$ λ (x₆) = {ρ₁,ρ₃, ρ₄}, $λ (x_{5}) = λ (x_{15}) = {ρ_{3}, ρ_{5}},$ λ (x₇) = {ρ₁,ρ₅}, $λ (x_{12}) = λ (x_{17}) = λ (x_{18}) = {ρ_{3}},$ λ (x₁₄) = {ρ₂,ρ₃}, $λ (x_{16}) = λ (x_{20}) = {ρ_{1}},$ λ (x₁₉) = {ρ₁,ρ₄, ρ₅}. For a fuzzy subset h∈ $F (x),$ where $h = {\begin{matrix} \frac{x_{1}}{0.3}, \frac{x_{2}}{0.1}, \frac{x_{3}}{0.4}, \frac{x_{4}}{0.8}, \frac{x_{5}}{0.9}, \frac{x_{6}}{0.4}, \frac{x_{7}}{0.9}, \\ \frac{x_{8}}{0.2}, \frac{x_{9}}{0.7}, \frac{x_{10}}{0.6}, \frac{x_{11}}{0.5}, \frac{x_{12}}{0.3}, \frac{x_{13}}{0.1}, \\ \frac{x_{14}}{0.6}, \frac{x_{15}}{0.4}, \frac{x_{16}}{0.7}, \frac{x_{17}}{0.8}, \frac{x_{18}}{0.3}, \frac{x_{19}}{0.4}, \frac{x_{20}}{0.5} \end{matrix}}$ A MATLAB code is developed to perform all the calculations. The calculated values of ${\underline{appr}}_{Z_{λ}} (h)$ and ${\bar{appr}}_{Z_{λ}} (h)$ are as follows: ${\underline{appr}}_{Z_{λ}} (h) = {\begin{matrix} \frac{x_{1}}{0.1}, \frac{x_{2}}{0.1}, \frac{x_{3}}{0.2}, \frac{x_{4}}{0.5}, \frac{x_{5}}{0.4}, \frac{x_{6}}{0.4}, \frac{x_{7}}{0.9}, \frac{x_{8}}{0.2}, \\ \frac{x_{9}}{0.1}, \frac{x_{10}}{0.2}, \frac{x_{11}}{0.5}, \frac{x_{12}}{0.3}, \frac{x_{13}}{0.1}, \\ \frac{x_{14}}{0.6}, \frac{x_{15}}{0.4}, \frac{x_{16}}{0.5}, \frac{x_{17}}{0.3}, \frac{x_{18}}{0.3}, \frac{x_{19}}{0.4}, \frac{x_{20}}{0.5} \end{matrix}}$ ${\bar{appr}}_{Z_{λ}} (h) = {\begin{matrix} \frac{x_{1}}{0.7}, \frac{x_{2}}{0.1}, \frac{x_{3}}{0.6}, \frac{x_{4}}{0.8}, \frac{x_{5}}{0.9}, \frac{x_{6}}{0.4}, \frac{x_{7}}{0.9}, \\ \frac{x_{8}}{0.6}, \frac{x_{9}}{0.7}, \frac{x_{10}}{0.6}, \frac{x_{11}}{0.8}, \frac{x_{12}}{0.8}, \frac{x_{13}}{0.7}, \\ \frac{x_{14}}{0.6}, \frac{x_{15}}{0.9}, \frac{x_{16}}{0.7}, \frac{x_{17}}{0.8}, \frac{x_{18}}{0.8}, \frac{x_{19}}{0.4}, \frac{x_{20}}{0.7} \end{matrix}} .$ Suppose Φ (x_j) be a optimality choice function as defined above. By simple calculations, it is found that $\begin{matrix} Φ (x_{1}) & = & 1.096, Φ (x_{2}) = 1.053, \\ Φ (x_{3}) & = & 1.176 = Φ (x_{8}) = Φ (x_{10}), \end{matrix}$ $\begin{matrix} Φ (x_{4}) & = & 1.444, Φ (x_{5}) = 1.383 = Φ (x_{15}), \\ Φ (x_{6}) & = & 1.250 = Φ (x_{19}), \end{matrix}$ $\begin{matrix} Φ (x_{7}) & = & 1.818, Φ (x_{9}) = 1.096 = Φ (x_{13}), \\ Φ (x_{11}) & = & 1.444, Φ (x_{14}) = 0.429 \end{matrix}$ $\begin{matrix} Φ (x_{12}) & = & 1.279 = Φ (x_{17}) = Φ (x_{18}), \\ Φ (x_{16}) & = & 1.412 = Φ (x_{20}) \end{matrix}$ The marginal weight function Ω (x_j) for each x_j can be computed by; $Ω (x_{j}) = [Ω_{r} (x_{j}) + Ω_{c} (x_{j})]$ for j = 1, 2, 3,..., r, where $Ω_{r} (x_{j}) = \frac{1}{r} \underset{i = 1}{\sum^{m}} χ_{E} (x_{j} x_{i}),$ is the measures the interaction of x_j with x_m, and $Ω_{c} (x_{j}) = \frac{1}{r} \underset{i = 1}{\sum^{m}} χ_{E} (x_{i} x_{j}),$ is the measures the interaction of x_m with x_j. Where χ_E is an indicator function on E, defined by $χ_{E} (x_{m} x_{j}) = {\begin{matrix} 1 if x_{m} x_{j} form an edge \\ 0, otherwise \end{matrix}$ The interaction of all persons with each other and the marginal fuzzy sets for rows and columns are given by Figure 2, Table 3 in Appendix.

Finally an evaluation function Θ defined on V by $Θ (x_{j}) = Φ (x_{j}) . Ω (x_{j})$ gives $\begin{matrix} Θ (x_{1}) & = & 0.822, Θ (x_{2}) = 0.632, \\ Θ (x_{3}) & = & 0.764 = Θ (x_{8}), Θ (x_{4}) = 0.505, \end{matrix}$ $\begin{matrix} Θ (x_{5}) & = & 0.761 = Θ (x_{15}), \\ Θ (x_{6}) & = & 1.313, Θ (x_{7}) = 0.909, Θ (x_{9}) = 0.603 \end{matrix}$ $\begin{matrix} Θ (x_{10}) & = & 0.882, Θ (x_{11}) = 0.578, \\ Θ (x_{12}) & = & 0.576 = Θ (x_{18}), Θ (x_{13}) = 0.548, \end{matrix}$ $\begin{matrix} Θ (x_{14}) & = & 1.286, Θ (x_{16}) = 0.635, \\ Θ (x_{17}) & = & 0.703, Θ (x_{19}) = 0.875, \\ Θ (x_{20}) & = & 0.494 \end{matrix}$ For a threshold β ∈ [0, 1], it can be seen that the all person x_i are at optimum for all i, in which Θ (x_i) ≥ β. The person x_p ∈ V is best optimal if $Θ (x_{p}) = \max_{j} {Θ (x_{j})}$ j = 1, 2,... r. So by calculations, the person x₆ is best optimal.

5 Conclusion

Rough sets, fuzzy sets and soft sets are very important tools for dealing the real life problems involving uncertainties. Graph theory is an other nice tool whose applications are obvious and self evident. This article is devoted to the discussion a novel blend of soft graphs and rough fuzzy sets. The concept of graph approximation is evolved. Due to the inspirational notion of a novel Z-soft rough fuzzy sets [60], the theory of Z-soft rough fuzzy graphs is developed and a real life example is constructed. To validate the applicability of this novel approach this example is a motivation for readers to workout further in this direction. The main difference between this approach involving graphs and the other approaches without involving graphs is; this approach involves the interaction between the objects, while in the previous approaches the decision involves the data from individuals only. All the calculations are made on MATLAB^® program. We hope our results in this paper will constitute a base for real life decision making problems based on Z-soft rough fuzzy graphs. One can obtain interesting results by emerging the concept of Pythagorean fuzzy sets [42] in Z-soft rough fuzzy graphs. In future work, it is under consideration how the lower and upper Z- soft edge approximations can be used for edge set to optimize the algorithm and how can we solve the decision making problems without using marginal fuzzy sets or weights?

Appendix

Fig.1

G^∗ = (V, E).

Fig.2

(Table 3) Marginal fuzzy sets for rows and columns.

References

Akram

and Nawaz

, Operations on soft graphs, Fuzzy Information and Engineering 7 (2015), 423–449.

Akram

and Nawaz

, Fuzzy soft graphs with applications, Journal of Intelligent & Fuzzy Systems 30(6) (2016), 3619–3632.

Aktas

and Çağman

, Soft sets and soft groups, Information Sciences 177(13) (2007), 2726–2735.

Alcantud

J.C.R.

, Biondo

A.E.

and Giarlotta

, Fuzzy politics I: The genesis of parties, Fuzzy Sets and Systems (2018).

Ali

M.I.

, A note on soft sets, rough soft sets and fuzzy soft sets, Applied Soft Computing 11 (2011), 3329–3332.

Ali

M.I.

, Feng

, Liu

, Min

W.K.

and Shabir

, On some new operations in soft set theory, Computers & Mathematics with Applications 57(9) (2009), 1547–1553.

Ali

M.I.

, Shabir

and Feng

, Representation of graphs based on neighborhoods and soft sets, International Journal of Machine Learning and Cybernetics 8(5) (2017), 1525–1535.

Beineke

L.W.

, Wilson

R.J.

, eds, Topics in algebraic graph theory, Cambridge University Press, 102, 2004.

Bonikowski

, Bryniariski

and Skardowska

V.W.

, Extension and intensions in the rough set theory, Information Sciences 107 (1998), 149–167.

10.

Chen

, Wang

C.Z.

and Hu

Q.H.

, A new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets, Information Sciences 177 (2007), 3500–3518.

11.

Chitcharoen

and Pattaraintakorn

, Towards theories of fuzzy set and rough set to flow graphs, IEEE International Conference on Fuzzy Systems 4630596 (2008), 1675–1682.

12.

Dubois

and Prade

, Rough fuzzy sets and fuzzy rough sets, International Journal of General System 17(2–3) (1990), 191–209.

13.

Euler

, Solutio problematis ad geometriam situs pertinentis, Commentarii Academiae Scientiarum Imperialis Petropolitanae 8 (1736), 128–140.

14.

Fatimah, et al., Probabilistic soft sets and dual probabilistic soft sets in decision-making, Neural Computing and Applications (2017), 1–11.

15.

Feng

, Li

, Davvaz

and Ali

M.I.

, Soft sets combined with fuzzy sets and rough sets: A tentative approach, Soft Computing 14 (2010), 899–911.

16.

Feng

, Liu

, Fotea

V.L.

and Jun

Y.B.

, Soft sets and soft rough sets, Information Sciences 181 (2011), 1125–1137.

17.

Feng ,et al. Attribute analysis of information systems based on elementary soft implications, Knowledge-Based Systems 70 (2014), 281–292.

18.

Gong ,et al. Fault-tolerant enhanced bijective soft set with applications, Applied Soft Computing 54 (2017), 431–439.

19.

Greco

, Matarazzo

and Slowinski

, Rough approximation by dominance relations, Journal of Intelligent & Fuzzy Systems 17 (2002), 153–171.

20.

Karaaslan

, Soft classes and soft rough classes with applications in decision making, Mathematical Problems in Engineering (2016), 1–11.

21.

T.J.

, Leung

and Zhang

W.X.

, Generalized fuzzy rough approximation operators based on fuzzy coverings, International Journal of Approximate Reasoning 48 (2008), 836–856.

22.

Liu

and Sai

, A comparison of two types of rough sets induced by coverings, International Journal of Approximate Reasoning 50 (2009), 521–528.

23.

Liu

and Zhu

, The algebraic structures of generalized rough set theory, Information Sciences 178 (2008), 4105–4113.

24.

Maji

P.K.

, Biswas

and Roy

A.R.

, Soft set theory, Computers & Mathematics with Applications 45 (2003), 555–562.

25.

Maji

P.K.

, Roy

A.R.

and Biswas

, An application of soft sets in a decision making problem, Computers & Mathematics with Applications 44 (2002), 1077–1083.

26.

, Zhan

, Ali

M.I.

and Mehmood

, A survey of decision making methods based on two classes of hybrid soft set models, Artificial Intelligence Review 49(4) (2018), 511–529.

27.

, Liu

and Zhan

, A survey of decision making methods based on certain hybrid soft set models, Artificial Intelligence Review 47(4) (2017), 507–530.

28.

Meng

, Zhang

and Qin

, Soft rough fuzzy sets and soft fuzzy rough sets, Computers & Mathematics with Applications 62 (2011), 4635–4645.

29.

Mohinta

and Samanta

T.K.

, An introduction to fuzzy soft graph, Mathematica Moravica 19(2) (2015), 35–48.

30.

Molodtsov

, Soft set theory – First results, Computers & Mathematics with Applications 37 (1999), 19–31.

31.

Molodtsov

, The Theory of Soft Sets (in Russian), URSS Publishers, Moscow, 2004.

32.

Morsi

N.N.

and Yakout

M.M.

, Axiomatics for fuzzy rough sets, Fuzzy sets and Systems 100 (1998), 327–342.

33.

Pawlak

, Rough sets, International Journal of Computer & Information Sciences 11 (1982), 341–356.

34.

Pawlak

, Rough Sets, Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Dordrecht, 1991.

35.

Pawlak

and Skowron

, Rudiments of rough sets, Information Sciences 177 (2007), 3–27.

36.

Pawlak

and Skowron

, Rough sets: Some extensions, Information Sciences (2007), 28–40.

37.

Pedrycz

and Homenda

, From fuzzy cognitive maps to granular cognitive maps, IEEE Transactions on Fuzzy Systems 22(4) (2014), 859–869.

38.

Peng

and Garg

, Algorithms for interval-valued fuzzy soft sets in emergency decision making based on WDBA and CODAS with new information measure, Computers & Industrial Engineering 119 (2018), 439–452.

39.

Peng

and Liu

, Algorithms for neutrosophic soft decision making based on EDAS, new similarity measure and level soft set, Journal of Intelligent & Fuzzy Systems 32(1) (2017), 955–968.

40.

Peng

, Peng

and Dai

, Hesitant fuzzy soft decision making methods based on WASPAS, MABAC and COPRAS with combined weights, Journal of Intelligent & Fuzzy Systems 33(2) (2017), 1313–1325.

41.

Peng

and Yang

, Algorithms for interval-valued fuzzy soft sets in stochastic multi-criteria decision making based on regret theory and prospect theory with combined weight, Applied Soft Computing 54 (2017), 415–430.

42.

Peng

and Selvachandran

, Pythagorean fuzzy set: State of the art and future directions, Artificial Intelligence Review (2017), 1–55.

43.

Quek

S.G.

, et al. Some results on the graph theory for complex neutrosophic sets, Symmetry 10(6) (2018), 190.

44.

Roberts

F.S.

, Graph theory and its applications to problems of society Society for industrial and applied mathematics, 1978.

45.

Rolka

and Rolka

A.M.

, Labeled Fuzzy Rough Sets Versus Fuzzy Flow Graphs, pp, IJCCI- Proceedings of the 8th International Joint Conference on Computational Intelligence (2016), 115–120.

46.

Roy

A.R.

and Maji

P.K.

, A fuzzy soft set theoretic approach to decision making problems, Journal of Computational and Applied Mathematics 203 (2007), 412–418.

47.

Shah

and Hussain

, Neutrosophic soft graphs, Neutrosophic Sets and Systems 11 (2016), 31–44.

48.

Shah

, Rehman

, Shabir

, Ali

M.I.

Another approach to roughness of soft graphs with applications in decision making, (2018) 10.3390/sym10050145.

49.

Słowiński

and Vanderpooten

, A generalized definition of rough approximations based on similarity, IEEE Trans Knowledge Data Eng 12 (2000), 331–336.

50.

Skowron

and Stepaniuk

, Tolerance approximation spaces, Fundamenta Informaticae 27 (1996), 245–253.

51.

West

D.B.

, Introduction to graph theory. Upper Saddle River, Prentice hall, Vol. 2, 2001.

52.

W.X.

and Zhang

W.X.

, Measuring roughness of generalized rough sets induced by a covering, Fuzzy Sets and Systems 158 (2007), 2443–2455.

53.

Yao

Y.Y.

and Lin

T.Y.

, Generalization of rough sets using modal logic, Intelligent Automation & Soft Computing 2 (1996), 103–120.

54.

Yao

Y.Y.

, Constructive and algebraic methods of the theory of rough sets, Information Sciences 109 (1998), 21–47.

55.

Zadeh

L.A.

, Fuzzy sets, Information Sciences 8 (1965), 338–353.

56.

Zhan

and Alcantud

J.C.R.

, A novel type of soft rough covering and its application to multicriteria group decision making, Artificial Intelligence Review (2018), 1–30.

57.

Zhan

and Alcantud

J.C.R.

, A survey of parameter reduction of soft sets and corresponding algorithms, Artificial Intelligence Review (2017), 1–34.

58.

Zhan

, Liu

and Herawan

, A novel soft rough set: Soft rough hemirings and corresponding multicriteria group decision making, Applied Soft Computing 54 (2017), 393–402.

59.

Zhan

, Ali

M.I.

and Mehmood

, On a novel uncertain soft set model: Z-soft fuzzy rough set model and corresponding decision making methods, Applied Soft Computing 56 (2017), 446–457.

60.

Zhan

and Zhu

, A novel soft rough fuzzy set: Z-soft rough fuzzy ideals of hemirings and corresponding decision making, Soft Computing 21(8) (2017), 1923–1936.

61.

Zhan

and Wang

, Certain types of soft coverings based rough sets with applications, International Journal of Machine Learning and Cybernetics (2018), 1–12.

62.

Zhan

and Alcantud

J.C.R.

, A novel type of soft rough covering and its application to multicriteria group decision making, Artificial Intelligence Review (2018), 1–30.

63.

Zhan

and R.

J.C.

, Alcantud, A survey of parameter reduction of soft sets and corresponding algorithms, Artificial Intelligence Review (2018), 1–34.

64.

Zhan

, et al. A study on Z-soft rough fuzzy semigroups and its decision-makings, International Journal for Uncertainty Quantification.

65.

Zhang

and Zhan

, Fuzzy soft β-covering based fuzzy rough sets and corresponding decision-making applications, International Journal of Machine Learning and Cybernetics 1–16.

66.

Zhu

, Generalized rough sets based on relations, Information Sciences 177(22) (2007), 4997–5011.

67.

Zhu

, Relationship between generalized rough sets based on binary relation and covering, Information Sciences 179 (2009), 210–225.

Z -soft rough fuzzy graphs: A new approach to decision making

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Z-soft rough fuzzy graphs

Table 1 Tabular representation of soft set (ψ, A) V x 1 x 2 x 3 x 4 x 5 x 6 x 7 ρ 1 1 0 1 0 1 0 1 ρ 2 1 0 1 1 0 1 1 ρ 3 0 1 0 1 0 1 0 ρ 4 1 0 1 1 0 1 0

5 Conclusion

Appendix

References

Table 1
Tabular representation of soft set (ψ, A)

V x ₁ x ₂ x ₃ x ₄ x ₅ x ₆ x ₇

ρ ₁ 1 0 1 0 1 0 1

ρ ₂ 1 0 1 1 0 1 1

ρ ₃ 0 1 0 1 0 1 0

ρ ₄ 1 0 1 1 0 1 0